inner sep=0pt,minimum size=1mm,scale=1]
(B) at (0,1) [vertex,fill=black,label=below:] ;
(A) at (0,0) [vertex,fill=white,label=below:] ;
[-](B) edge node[right]  (A);
[bend right=10](B) edge node[right]  (A);
[bend left=10](B) edge node[right]  (A);
(C1) at (-0.25,0.5) [vertex,fill=black,label=below:] ;
(C2) at (0.25,0.5) [vertex,fill=black,label=below:] ;
(C3) at (0.5,0.5) [vertex,fill=black,label=below:] ;
[-](A) edge node[right]  (C1);
[-](A) edge node[right]  (C2);
[-](A) edge node[right]  (C3);
[-](B) edge node[right]  (C1);
[-](B) edge node[right]  (C2);
[-](B) edge node[right]  (C3);

[baseline=([yshift=-.5ex]current bounding box.center),auto,vertex/.style=circle,draw=black!100,
inner sep=0pt,minimum size=1mm,scale=1]
(B1) at (-0.5,1) [vertex,fill=black,label=below:] ;
(B2) at (0.5,1) [vertex,fill=black,label=below:] ;
(A) at (0,0) [vertex,fill=white,label=below:] ;
[-](B1) edge node[right]  (A);
[-](B2) edge node[right]  (A);
[bend left=10](B2) edge node[right]  (A);
[bend right=10](B2) edge node[right]  (A);
(C1) at (-0.5,0.5) [vertex,fill=black,label=below:] ;
(C2) at (0.75,0.5) [vertex,fill=black,label=below:] ;
(C3) at (0.0,0.5) [vertex,fill=black,label=below:] ;
(C4) at (0.5,0.5) [vertex,fill=black,label=below:] ;
[-](A) edge node[right]  (C1);
[-](A) edge node[right]  (C2);
[-](A) edge node[right]  (C3);
[-](A) edge node[right]  (C4);
[-](B1) edge node[right]  (C1);
[-](B2) edge node[right]  (C2);
[-](B2) edge node[right]  (C3);
[-](B2) edge node[right]  (C4);

[baseline=([yshift=-.5ex]current bounding box.center),auto,vertex/.style=circle,draw=black!100,
inner sep=0pt,minimum size=1mm,scale=1]
(B1) at (-0.5,1) [vertex,fill=black,label=below:] ;
(B2) at (0.5,1) [vertex,fill=black,label=below:] ;
(A) at (0,0) [vertex,fill=white,label=below:] ;
[bend left=7](B1) edge node[right]  (A);
[bend right=7](B1) edge node[right]  (A);
[bend left=7](B2) edge node[right]  (A);
[bend right=7](B2) edge node[right]  (A);
(C1) at (-0.75,0.5) [vertex,fill=black,label=below:] ;
(C2) at (-0.5,0.5) [vertex,fill=black,label=below:] ;
(C3) at (0.5,0.5) [vertex,fill=black,label=below:] ;
(C4) at (0.75,0.5) [vertex,fill=black,label=below:] ;
[-](A) edge node[right]  (C1);
[-](A) edge node[right]  (C2);
[-](A) edge node[right]  (C3);
[-](A) edge node[right]  (C4);
[-](B1) edge node[right]  (C1);
[-](B1) edge node[right]  (C2);
[-](B2) edge node[right]  (C3);
[-](B2) edge node[right]  (C4);

$\lMset 0,(\frac12)^{(s-1)},1^{(r-s)},(\frac32)^{(s+1)}\rMset$
$3$     $3$     $1$   $3$   $3$   $5$   $5$    $2s+1$

[baseline=([yshift=-.5ex]current bounding box.center),auto,vertex/.style=circle,draw=black!100,
inner sep=0pt,minimum size=1mm,scale=1]
(B) at (-.25,0) [vertex,fill=white,label=below:] ;
(A) at (0,1) [vertex,fill=black,label=below:] ;
[-](B) edge node[right]  (A);
(C) at (-0.5,1) [vertex,fill=black,label=below:] ;
[-](A) edge node[right]  (C);
[-](B) edge node[right]  (C);

[baseline=([yshift=-.5ex]current bounding box.center),auto,vertex/.style=circle,draw=black!100,
inner sep=0pt,minimum size=1mm,scale=1]
(B) at (-.25,0) [vertex,fill=white,label=below:] ;
(A) at (0,1) [vertex,fill=black,label=below:] ;
[bend left=7](B) edge node[right]  (A);
[bend right=7](B) edge node[right]  (A);
(C) at (-0.5,1) [vertex,fill=black,label=below:] ;
[bend left=7](A) edge node[right]  (C);
[bend right=7](A) edge node[right]  (C);
[bend left=7](B) edge node[right]  (C);
[bend right=7](B) edge node[right]  (C);

[baseline=([yshift=-.5ex]current bounding box.center),auto,vertex/.style=circle,draw=black!100,
inner sep=0pt,minimum size=1mm,scale=1]
(B) at (-.25,0) [vertex,fill=white,label=below:] ;
(A) at (0,1) [vertex,fill=black,label=below:] ;
[-](B) edge node[right]  (A);
[bend left=10](B) edge node[right]  (A);
[bend right=10](B) edge node[right]  (A);
(C) at (-0.5,1) [vertex,fill=black,label=below:] ;
[-](A) edge node[right]  (C);
[-](B) edge node[right]  (C);
[bend left=10](A) edge node[right]  (C);
[bend right=10](A) edge node[right]  (C);
[bend left=10](B) edge node[right]  (C);
[bend right=10](B) edge node[right]  (C);

[baseline=([yshift=-.5ex]current bounding box.center),auto,vertex/.style=circle,draw=black!100,
inner sep=0pt,minimum size=1mm,scale=1]
(B) at (0,0) [vertex,fill=white,label=below:] ;
(A1) at (-0.2,1) [vertex,fill=black,label=below:] ;
(C1) at (-0.6,1) [vertex,fill=black,label=below:] ;
(A2) at (0.2,1) [vertex,fill=black,label=below:] ;
(C2) at (0.6,1) [vertex,fill=black,label=below:] ;
[-](B) edge node[right]  (A1);
[-](B) edge node[right]  (C1);
[-](B) edge node[right]  (A2);
[bend left=10](B) edge node[right]  (A2);
[bend right=10](B) edge node[right]  (A2);
[-](B) edge node[right]  (C2);
[bend left=10](B) edge node[right]  (C2);
[bend right=10](B) edge node[right]  (C2);
[-](A1) edge node[right]  (C1);
[-](A2) edge node[right]  (C2);
[bend left=12](A2) edge node[right]  (C2);
[bend right=12](A2) edge node[right]  (C2);

[baseline=([yshift=-.5ex]current bounding box.center),auto,vertex/.style=circle,draw=black!100,
inner sep=0pt,minimum size=1mm,scale=1]
(B) at (0,0) [vertex,fill=white,label=below:] ;
(A1) at (-0.2,1) [vertex,fill=black,label=below:] ;
(C1) at (-0.6,1) [vertex,fill=black,label=below:] ;
(A2) at (0.2,1) [vertex,fill=black,label=below:] ;
(C2) at (0.6,1) [vertex,fill=black,label=below:] ;
[bend left=7](B) edge node[right]  (A1);
[bend right=7](B) edge node[right]  (A1);
[bend left=7](B) edge node[right]  (C1);
[bend right=7](B) edge node[right]  (C1);
[bend left=7](B) edge node[right]  (A2);
[bend right=7](B) edge node[right]  (A2);
[bend left=7](B) edge node[right]  (C2);
[bend right=7](B) edge node[right]  (C2);
[bend right=10](A1) edge node[right]  (C1);
[bend left=10](A1) edge node[right]  (C1);
[bend left=10](A2) edge node[right]  (C2);
[bend right=10](A2) edge node[right]  (C2);

$\lMset 0,(\frac12)^{(s-1)},(\frac32)^{(s+1)}\rMset$
$3$     $3$     $2$   $3$   $3$   $4$   $4$    $s+2$

[baseline=([yshift=-.5ex]current bounding box.center),auto,vertex/.style=circle,draw=black!100,
inner sep=0pt,minimum size=1mm,scale=1]
(B) at (-.25,1) [vertex,fill=black,label=below:] ;
(A) at (0,0) [vertex,fill=white,label=below:] ;
[-](B) edge node[right]  (A);
(C) at (-0.5,0) [vertex,fill=white,label=below:] ;
[-](A) edge node[right]  (C);
[-](B) edge node[right]  (C);

[baseline=([yshift=-.5ex]current bounding box.center),auto,vertex/.style=circle,draw=black!100,
inner sep=0pt,minimum size=1mm,scale=1]
(B) at (-.25,1) [vertex,fill=black,label=below:] ;
(A) at (0,0) [vertex,fill=white,label=below:] ;
[bend left=7](B) edge node[right]  (A);
[bend right=7](B) edge node[right]  (A);
(C) at (-0.5,0) [vertex,fill=white,label=below:] ;
[bend left=7](A) edge node[right]  (C);
[bend right=7](A) edge node[right]  (C);
[bend left=7](B) edge node[right]  (C);
[bend right=7](B) edge node[right]  (C);

[baseline=([yshift=-.5ex]current bounding box.center),auto,vertex/.style=circle,draw=black!100,
inner sep=0pt,minimum size=1mm,scale=1]
(B) at (-.25,1) [vertex,fill=black,label=below:] ;
(A) at (0,0) [vertex,fill=white,label=below:] ;
[-](B) edge node[right]  (A);
[bend left=10](B) edge node[right]  (A);
[bend right=10](B) edge node[right]  (A);
(C) at (-0.5,0) [vertex,fill=white,label=below:] ;
[-](A) edge node[right]  (C);
[-](B) edge node[right]  (C);
[bend left=10](A) edge node[right]  (C);
[bend right=10](A) edge node[right]  (C);
[bend left=10](B) edge node[right]  (C);
[bend right=10](B) edge node[right]  (C);

[baseline=([yshift=-.5ex]current bounding box.center),auto,vertex/.style=circle,draw=black!100,
inner sep=0pt,minimum size=1mm,scale=1]
(B1) at (-0.5,1) [vertex,fill=black,label=below:] ;
(B2) at (0,1) [vertex,fill=black,label=below:] ;
(A) at (0,0) [vertex,fill=white,label=below:] ;
[-](B1) edge node[right]  (A);
[-](B2) edge node[right]  (A);
[bend left=10](B2) edge node[right]  (A);
[bend right=10](B2) edge node[right]  (A);
(C) at (-0.5,0) [vertex,fill=white,label=below:] ;
[-](A) edge node[right]  (C);
[-](B1) edge node[right]  (C);
[-](B2) edge node[right]  (C);
[bend left=10](A) edge node[right]  (C);
[bend right=10](A) edge node[right]  (C);
[bend left=25](A) edge node[right]  (C);
[bend right=25](A) edge node[right]  (C);
[bend left=10](B2) edge node[right]  (C);
[bend right=10](B2) edge node[right]  (C);

[baseline=([yshift=-.5ex]current bounding box.center),auto,vertex/.style=circle,draw=black!100,
inner sep=0pt,minimum size=1mm,scale=1]
(B1) at (-0.5,1) [vertex,fill=black,label=below:] ;
(B2) at (0,1) [vertex,fill=black,label=below:] ;
(A) at (0,0) [vertex,fill=white,label=below:] ;
[bend left=7](B1) edge node[right]  (A);
[bend right=7](B1) edge node[right]  (A);
[bend left=7](B2) edge node[right]  (A);
[bend right=7](B2) edge node[right]  (A);
(C) at (-0.5,0) [vertex,fill=white,label=below:] ;
[bend left=7](B1) edge node[right]  (C);
[bend right=7](B1) edge node[right]  (C);
[bend left=10](A) edge node[right]  (C);
[bend right=10](A) edge node[right]  (C);
[bend left=25](A) edge node[right]  (C);
[bend right=25](A) edge node[right]  (C);
[bend left=10](B2) edge node[right]  (C);
[bend right=10](B2) edge node[right]  (C);

$\lMset 0,1^{(s-1)},(\frac32)^{(2)}\rMset$

All examples of non-tree building blocks with three vertices
and their different contracted frame unions (including those with
multiple edges) for the simplest non-trivial partition
$A=\lMset 1,3\rMset$, $B=\lMset 2,2\rMset$.

[frames identified along all vertices]

If $V_0=V(\G)$, then $\wGr F_a(\wG,V(\G))$ is the $r$-fold edge copy
of $\wG$, i.e., the vertex set is the same, but each edge is copied
$r$ times keeping its original weight and magnetic potential.
Examples are the graphs from the building blocks labelled 2.1, 3.3,
3.3', 3.4, 3.4', 3.6, 3.6' in all.ex.3vxall.ex.3vx'.

Moreover, the order of the $V_1$-contracted frame union
$\wGr F_{A,V_1}$ for some $V_1 \subset V_0$ now is

\begin{equation*}
\card{\wGr F_{A,V_1}}
=s(\card{V_0}-\card{V_1}) +\card {V_1},
\end{equation*}

i.e., it does not depend on $r$.  In particular, we obtain an
infinite family of isospectral graphs, as an $s$-partition for
any $r \in \{s+2,s+3, \dots\}$ leads to another
non-isomorphic graph (isomorphy as multigraph, of course).  We have
another class of examples in all.but.1.contracted.

If $\wG=(\G,0,1)$ has standard weight, then the frames for
$V_0=V(\G)$ as in the previous example are the $r$-fold edge copy of
$\G$ (i.e., each edge with standard weight is repeated $r$ times).
Note that $\G$ and its $r$-fold edge copy are isolaplacian (as the
common factor $r$ cancels out in the matrix representation of the

Moreover, each frame member and also the $V_1$-contracted frame
union $\Gr F_{A,V_1}$ for some proper subset $V_1 \subset V_0$ can
be turned into a simple graph (called underlying simple
weighted graph) with edge weights now given by the number of
parallel edges in the original graph, see mult.ed.weighed.gr.
Note that as weighted graphs, two contracted frame unions for
different $s$-partitions are still non-isomorphic (as weighted
graphs), as the weights are given by the partition $A$.

A special case of the last example is the following:

[the smallest non-trivial example]
Let $\G$ be the graph on two vertices and one edge joining them with
standard weight.  Let $V_0=V=\{v_1,v_2\}$ and $V_1=\{v_1\}$ (see the
first row in all.ex.3vx).  Moreover, let
$A=\lMset a_1,\dots,a_s\rMset$ be an $s$-partition of $r$.  Then
$\wGr F_{A,v_1}$ is a star graph with $s+1$ vertices and multiple
edges according to the partition $A$ ($a_1$ edges joining the first
non-central vertex, $a_2$ the second etc.).  Note that for two
different $s$-partitions, the corresponding contracted frame unions
are not isomorphic (as multigraphs).

The spectrum of $\wGr F_{A,v_1}$ is $\lMset 0, 1^{(s-1)},2\rMset$
for any $s$-partition $A$.  For $s=2$, the underlying simple
weighted graph is a path graph with three vertices.  Actually, it
can be seen that the spectrum of this graph is always
$\lMset 0,1,2\rMset$ for any weights on the two edges.

label   $\G$     $\card V$    $\card {V_0}$     $\card {V_1}$     $\Gr F_2$     $\Gr F_3$     $\Gr F_{A,V_1}$
$\Gr F_{B,V_1}$    $\card{\Gr F_{C,V_1}}$, $\spec{\Gr F_{C,V_1}}$

$4$     $2$     $1$   $6$   $8$   $11$   $11$    $2r+s+1$

$4.1$    [baseline=([yshift=-.5ex]current bounding box.center),auto,vertex/.style=circle,draw=black!100,
inner sep=0pt,minimum size=1mm,scale=1]
(B) at (0,1) [vertex,fill=black,label=below:] ;
(A) at (0,0) [vertex,fill=white,label=below:] ;
[-](B) edge node[right]  (A);
(C) at (0,2) [vertex,fill=black,label=below:] ;
[-](B) edge node[right]  (C);
(D) at (0,3) [vertex,fill=black,label=below:] ;
[-](D) edge node[right]  (C);

[baseline=([yshift=-.5ex]current bounding box.center),auto,vertex/.style=circle,draw=black!100,
inner sep=0pt,minimum size=1mm,scale=1]
(B1) at (0.25,1) [vertex,fill=black,label=below:] ;
(B2) at (-0.25,1) [vertex,fill=black,label=below:] ;
(A) at (0,0) [vertex,fill=white,label=below:] ;

[-](B1) edge node[right]  (A);
[-](B2) edge node[right]  (A);
(C1) at (0.25,2) [vertex,fill=black,label=below:] ;
(C2) at (-0.25,2) [vertex,fill=black,label=below:] ;
(D) at (0,3) [vertex,fill=black,label=below:] ;
[-](B1) edge node[right]  (C1);
[-](C1) edge node[right]  (D);
[-](B2) edge node[right]  (C2);
[-](C2) edge node[right]  (D);

[baseline=([yshift=-.5ex]current bounding box.center),auto,vertex/.style=circle,draw=black!100,
inner sep=0pt,minimum size=1mm,scale=1]
(B1) at (0.5,1) [vertex,fill=black,label=below:] ;
(B2) at (0,1) [vertex,fill=black,label=below:] ;
(B3) at (-0.5,1) [vertex,fill=black,label=below:] ;
(A) at (0,0) [vertex,fill=white,label=below:] ;

[-](B1) edge node[right]  (A);
[-](B2) edge node[right]  (A);
[-](B3) edge node[right]  (A);
(C1) at (0.5,2) [vertex,fill=black,label=below:] ;
(C2) at (0,2) [vertex,fill=black,label=below:] ;
(C3) at (-0.5,2) [vertex,fill=black,label=below:] ;
(D) at (0,3) [vertex,fill=black,label=below:] ;
[-](B1) edge node[right]  (C1);
[-](C1) edge node[right]  (D);
[-](B2) edge node[right]  (C2);
[-](C2) edge node[right]  (D);
[-](B3) edge node[right]  (C3);
[-](C3) edge node[right]  (D);

[baseline=([yshift=-.5ex]current bounding box.center),auto,vertex/.style=circle,draw=black!100,
inner sep=0pt,minimum size=1mm,scale=1]
(B1) at (1,1) [vertex,fill=black,label=below:] ;
(B2) at (0.5,1) [vertex,fill=black,label=below:] ;
(B3) at (0,1) [vertex,fill=black,label=below:] ;
(B4) at (-0.5,1) [vertex,fill=black,label=below:] ;
(A) at (0,0) [vertex,fill=white,label=below:] ;
(C1) at (1,2) [vertex,fill=black,label=below:] ;
(C2) at (0.5,2) [vertex,fill=black,label=below:] ;
(C3) at (0,2) [vertex,fill=black,label=below:] ;
(C4) at (-0.5,2) [vertex,fill=black,label=below:] ;

[-](B1) edge node[right]  (A);
[-](B2) edge node[right]  (A);
[-](B3) edge node[right]  (A);
[-](B4) edge node[right]  (A);
(D1) at (-0.5,3) [vertex,fill=black,label=below:] ;
(D2) at (0.5,3) [vertex,fill=black,label=below:] ;
[-](B1) edge node[right]  (C1);
[-](B2) edge node[right]  (C2);
[-](B3) edge node[right]  (C3);
[-](B4) edge node[right]  (C4);
[-](C1) edge node[right]  (D2);
[-](C2) edge node[right]  (D2);
[-](C3) edge node[right]  (D2);
[-](C4) edge node[right]  (D1);

[baseline=([yshift=-.5ex]current bounding box.center),auto,vertex/.style=circle,draw=black!100,
inner sep=0pt,minimum size=1mm,scale=1]
(B1) at (1,1) [vertex,fill=black,label=below:] ;
(B2) at (0.5,1) [vertex,fill=black,label=below:] ;
(B3) at (0,1) [vertex,fill=black,label=below:] ;
(B4) at (-0.5,1) [vertex,fill=black,label=below:] ;
(C1) at (1,2) [vertex,fill=black,label=below:] ;
(C2) at (0.5,2) [vertex,fill=black,label=below:] ;
(C3) at (0,2) [vertex,fill=black,label=below:] ;
(C4) at (-0.5,2) [vertex,fill=black,label=below:] ;
(A) at (0,0) [vertex,fill=white,label=below:] ;

[-](B1) edge node[right]  (A);
[-](B2) edge node[right]  (A);
[-](B3) edge node[right]  (A);
[-](B4) edge node[right]  (A);
(D1) at (-0.25,3) [vertex,fill=black,label=below:] ;
(D2) at (0.75,3) [vertex,fill=black,label=below:] ;
[-](B1) edge node[right]  (C1);
[-](B2) edge node[right]  (C2);
[-](B3) edge node[right]  (C3);
[-](B4) edge node[right]  (C4);
[-](C1) edge node[right]  (D2);
[-](C2) edge node[right]  (D2);
[-](C3) edge node[right]  (D1);
[-](C4) edge node[right]  (D1);

0.26$\lMset 0,(1\pm\frac{\sqrt3}2)^{(s-1)},\frac12^{(r-s+1)}, $$1^{(s-1)},(\frac32)^{(r-s+1)},2\rMset$

[baseline=([yshift=-.5ex]current bounding box.center),auto,vertex/.style=circle,draw=black!100,
inner sep=0pt,minimum size=1mm,scale=1]
(B) at (0,1) [vertex,fill=black,label=below:] ;
(BB) at (0,2) [vertex,fill=black,label=below:] ;
(A) at (0,0) [vertex,fill=white,label=below:] ;
[-](BB) edge node[right]  (A);
(C) at (-0.5,1) [vertex,fill=black,label=below:] ;
[-](BB) edge node[right]  (C);

[baseline=([yshift=-.5ex]current bounding box.center),auto,vertex/.style=circle,draw=black!100,
inner sep=0pt,minimum size=1mm,scale=1]
(B1) at (0.25,1) [vertex,fill=black,label=below:] ;
(B2) at (-0.25,1) [vertex,fill=black,label=below:] ;
(BB) at (0,2) [vertex,fill=black,label=below:] ;
(A) at (0,0) [vertex,fill=white,label=below:] ;
[-](B1) edge node[right]  (A);
[-](B2) edge node[right]  (A);
[-](B1) edge node[right]  (BB);
[-](B2) edge node[right]  (BB);
(C1) at (-0.5,1) [vertex,fill=black,label=below:] ;
(C2) at (0.5,1) [vertex,fill=black,label=below:] ;
[-](BB) edge node[right]  (C1);
[-](BB) edge node[right]  (C2);

[baseline=([yshift=-.5ex]current bounding box.center),auto,vertex/.style=circle,draw=black!100,
inner sep=0pt,minimum size=1mm,scale=1]
(B1) at (0.5,1) [vertex,fill=black,label=below:] ;
(B2) at (-0.25,1) [vertex,fill=black,label=below:] ;
(B0) at (0,1) [vertex,fill=black,label=below:] ;
(BB) at (0,2) [vertex,fill=black,label=below:] ;
(A) at (0,0) [vertex,fill=white,label=below:] ;
[-](B1) edge node[right]  (A);
[-](B2) edge node[right]  (A);
[-](B0) edge node[right]  (A);
[-](B1) edge node[right]  (BB);
[-](B2) edge node[right]  (BB);
[-](B0) edge node[right]  (BB);
(C1) at (-0.5,1) [vertex,fill=black,label=below:] ;
(C2) at (0.25,1) [vertex,fill=black,label=below:] ;
(C0) at (0.75,1) [vertex,fill=black,label=below:] ;
[-](BB) edge node[right]  (C1);
[-](BB) edge node[right]  (C2);
[-](BB) edge node[right]  (C0);

[baseline=([yshift=-.5ex]current bounding box.center),auto,vertex/.style=circle,draw=black!100,
inner sep=0pt,minimum size=1mm,scale=1]
(B1) at (0.5,1) [vertex,fill=black,label=below:] ;
(B2) at (-0.25,1) [vertex,fill=black,label=below:] ;
(B0) at (0,1) [vertex,fill=black,label=below:] ;
(B-1) at (-0.75,1) [vertex,fill=black,label=below:] ;
(BB1) at (-0.75,2) [vertex,fill=black,label=below:] ;
(BB2) at (0,2) [vertex,fill=black,label=below:] ;
(A) at (0,0) [vertex,fill=white,label=below:] ;
[-](B1) edge node[right]  (A);
[-](B2) edge node[right]  (A);
[-](B0) edge node[right]  (A);
[-](B-1) edge node[right]  (A);
[-](B-1) edge node[right]  (BB1);
[-](B1) edge node[right]  (BB2);
[-](B2) edge node[right]  (BB2);
[-](B0) edge node[right]  (BB2);
(C1) at (-0.5,1) [vertex,fill=black,label=below:] ;
(C2) at (0.25,1) [vertex,fill=black,label=below:] ;
(C0) at (0.75,1) [vertex,fill=black,label=below:] ;
(C-1) at (-1,1) [vertex,fill=black,label=below:] ;
[-](BB1) edge node[right]  (C-1);
[-](BB2) edge node[right]  (C1);
[-](BB2) edge node[right]  (C2);
[-](BB2) edge node[right]  (C0);

[baseline=([yshift=-.5ex]current bounding box.center),auto,vertex/.style=circle,draw=black!100,
inner sep=0pt,minimum size=1mm,scale=1]
(B1) at (0.5,1) [vertex,fill=black,label=below:] ;
(B2) at (-0.25,1) [vertex,fill=black,label=below:] ;
(B0) at (0,1) [vertex,fill=black,label=below:] ;
(B-1) at (-0.75,1) [vertex,fill=black,label=below:] ;
(BB1) at (-0.75,2) [vertex,fill=black,label=below:] ;
(BB2) at (0.5,2) [vertex,fill=black,label=below:] ;
(A) at (0,0) [vertex,fill=white,label=below:] ;
[-](B1) edge node[right]  (A);
[-](B2) edge node[right]  (A);
[-](B0) edge node[right]  (A);
[-](B-1) edge node[right]  (A);
[-](B-1) edge node[right]  (BB1);
[-](B1) edge node[right]  (BB2);
[-](B2) edge node[right]  (BB1);
[-](B0) edge node[right]  (BB2);
(C1) at (-0.5,1) [vertex,fill=black,label=below:] ;
(C2) at (0.25,1) [vertex,fill=black,label=below:] ;
(C0) at (0.75,1) [vertex,fill=black,label=below:] ;
(C-1) at (-1,1) [vertex,fill=black,label=below:] ;
[-](BB1) edge node[right]  (C-1);
[-](BB1) edge node[right]  (C1);
[-](BB2) edge node[right]  (C2);
[-](BB2) edge node[right]  (C0);

0.26$\lMset 0,(1\pm\frac{\sqrt3}2)^{(s-1)}, $$\frac12,1^{(2r-s-1)},\frac32,2\rMset$

[baseline=([yshift=-.5ex]current bounding box.center),auto,vertex/.style=circle,draw=black!100,
inner sep=0pt,minimum size=1mm,scale=1]
(B) at (0,1) [vertex,fill=black,label=below:] ;
(BB) at (0,0) [vertex,fill=white,label=below:] ;
(A) at (0,2) [vertex,fill=black,label=below:] ;
[-](BB) edge node[right]  (A);
(C) at (-0.5,1) [vertex,fill=black,label=below:] ;
[-](BB) edge node[right]  (C);

[baseline=([yshift=-.5ex]current bounding box.center),auto,vertex/.style=circle,draw=black!100,
inner sep=0pt,minimum size=1mm,scale=1]
(B1) at (0.25,1) [vertex,fill=black,label=below:] ;
(B2) at (-0.25,1) [vertex,fill=black,label=below:] ;
(BB) at (0,0) [vertex,fill=white,label=below:] ;
(A) at (0,2) [vertex,fill=black,label=below:] ;
[-](B1) edge node[right]  (A);
[-](B2) edge node[right]  (A);
[-](B1) edge node[right]  (BB);
[-](B2) edge node[right]  (BB);
(C1) at (-0.5,1) [vertex,fill=black,label=below:] ;
(C2) at (0.5,1) [vertex,fill=black,label=below:] ;
[-](BB) edge node[right]  (C1);
[-](BB) edge node[right]  (C2);

[baseline=([yshift=-.5ex]current bounding box.center),auto,vertex/.style=circle,draw=black!100,
inner sep=0pt,minimum size=1mm,scale=1]
(B1) at (0.5,1) [vertex,fill=black,label=below:] ;
(B2) at (-0.25,1) [vertex,fill=black,label=below:] ;
(B0) at (0,1) [vertex,fill=black,label=below:] ;
(BB) at (0,0) [vertex,fill=white,label=below:] ;
(A) at (0,2) [vertex,fill=black,label=below:] ;
[-](B1) edge node[right]  (A);
[-](B2) edge node[right]  (A);
[-](B0) edge node[right]  (A);
[-](B1) edge node[right]  (BB);
[-](B2) edge node[right]  (BB);
[-](B0) edge node[right]  (BB);
(C1) at (-0.5,1) [vertex,fill=black,label=below:] ;
(C2) at (0.25,1) [vertex,fill=black,label=below:] ;
(C0) at (0.75,1) [vertex,fill=black,label=below:] ;
[-](BB) edge node[right]  (C1);
[-](BB) edge node[right]  (C2);
[-](BB) edge node[right]  (C0);

[baseline=([yshift=-.5ex]current bounding box.center),auto,vertex/.style=circle,draw=black!100,
inner sep=0pt,minimum size=1mm,scale=1]
(B1) at (0.5,1) [vertex,fill=black,label=below:] ;
(B2) at (-0.25,1) [vertex,fill=black,label=below:] ;
(B0) at (0,1) [vertex,fill=black,label=below:] ;
(B-1) at (-0.75,1) [vertex,fill=black,label=below:] ;
(BB) at (0,0) [vertex,fill=white,label=below:] ;
(A1) at (-0.75,2) [vertex,fill=black,label=below:] ;
(A2) at (0,2) [vertex,fill=black,label=below:] ;
[-](B1) edge node[right]  (A2);
[-](B2) edge node[right]  (A2);
[-](B0) edge node[right]  (A2);
[-](B-1) edge node[right]  (A1);
[-](B-1) edge node[right]  (BB);
[-](B1) edge node[right]  (BB);
[-](B2) edge node[right]  (BB);
[-](B0) edge node[right]  (BB);
(C1) at (-0.5,1) [vertex,fill=black,label=below:] ;
(C2) at (0.25,1) [vertex,fill=black,label=below:] ;
(C0) at (0.75,1) [vertex,fill=black,label=below:] ;
(C-1) at (-1,1) [vertex,fill=black,label=below:] ;
[-](BB) edge node[right]  (C-1);
[-](BB) edge node[right]  (C1);
[-](BB) edge node[right]  (C2);
[-](BB) edge node[right]  (C0);

[baseline=([yshift=-.5ex]current bounding box.center),auto,vertex/.style=circle,draw=black!100,
inner sep=0pt,minimum size=1mm,scale=1]
(B1) at (0.5,1) [vertex,fill=black,label=below:] ;
(B2) at (-0.25,1) [vertex,fill=black,label=below:] ;
(B0) at (0,1) [vertex,fill=black,label=below:] ;
(B-1) at (-0.75,1) [vertex,fill=black,label=below:] ;
(BB) at (0,0) [vertex,fill=white,label=below:] ;
(A1) at (-0.75,2) [vertex,fill=black,label=below:] ;
(A2) at (0.5,2) [vertex,fill=black,label=below:] ;
[-](B1) edge node[right]  (A2);
[-](B2) edge node[right]  (A1);
[-](B0) edge node[right]  (A2);
[-](B-1) edge node[right]  (A1);
[-](B-1) edge node[right]  (BB);
[-](B1) edge node[right]  (BB);
[-](B2) edge node[right]  (BB);
[-](B0) edge node[right]  (BB);
(C1) at (-0.5,1) [vertex,fill=black,label=below:] ;
(C2) at (0.25,1) [vertex,fill=black,label=below:] ;
(C0) at (0.75,1) [vertex,fill=black,label=below:] ;
(C-1) at (-1,1) [vertex,fill=black,label=below:] ;
[-](BB) edge node[right]  (C-1);
[-](BB) edge node[right]  (C1);
[-](BB) edge node[right]  (C2);
[-](BB) edge node[right]  (C0);

0.26$\lMset 0,(1\pm\frac1{\sqrt2})^{(s-1)}, $$\frac12,1^{(2r-s-1)},\frac32,2\rMset$

$4$     $2$     $1$   $6$   $8$   $11$   $11$    $2r+s+1$

[baseline=([yshift=-.5ex]current bounding box.center),auto,vertex/.style=circle,draw=black!100,
inner sep=0pt,minimum size=1mm,scale=1]
(B) at (0,1) [vertex,fill=black,label=below:] ;
(BB) at (-0.25,1) [vertex,fill=black,label=below:] ;
[-](B) edge node[right]  (BB);
(A) at (0,0) [vertex,fill=white,label=below:] ;
[-](B) edge node[right]  (A);
(C) at (0,2) [vertex,fill=black,label=below:] ;
[-](B) edge node[right]  (C);

[baseline=([yshift=-.5ex]current bounding box.center),auto,vertex/.style=circle,draw=black!100,
inner sep=0pt,minimum size=1mm,scale=1]
(B1) at (0.25,1) [vertex,fill=black,label=below:] ;
(B2) at (-0.25,1) [vertex,fill=black,label=below:] ;
(BB1) at (0.5,1) [vertex,fill=black,label=below:] ;
[-](B1) edge node[right]  (BB1);
(BB2) at (0,1) [vertex,fill=black,label=below:] ;
[-](B2) edge node[right]  (BB2);
(A) at (0,0) [vertex,fill=white,label=below:] ;

[-](B1) edge node[right]  (A);
[-](B2) edge node[right]  (A);
(C) at (0,2) [vertex,fill=black,label=below:] ;
[-](B1) edge node[right]  (C);
[-](B2) edge node[right]  (C);

[baseline=([yshift=-.5ex]current bounding box.center),auto,vertex/.style=circle,draw=black!100,
inner sep=0pt,minimum size=1mm,scale=1]
(B1) at (0.5,1) [vertex,fill=black,label=below:] ;
(B2) at (0,1) [vertex,fill=black,label=below:] ;
(B3) at (-0.5,1) [vertex,fill=black,label=below:] ;
(BB1) at (0.25,1) [vertex,fill=black,label=below:] ;
[-](B1) edge node[right]  (BB1);
(BB2) at (-0.25,1) [vertex,fill=black,label=below:] ;
[-](B2) edge node[right]  (BB2);
(BB3) at (-0.75,1) [vertex,fill=black,label=below:] ;
[-](B3) edge node[right]  (BB3);
(A) at (0,0) [vertex,fill=white,label=below:] ;

[-](B1) edge node[right]  (A);
[-](B2) edge node[right]  (A);
[-](B3) edge node[right]  (A);
(C) at (0,2) [vertex,fill=black,label=below:] ;
[-](B1) edge node[right]  (C);
[-](B2) edge node[right]  (C);
[-](B3) edge node[right]  (C);

[baseline=([yshift=-.5ex]current bounding box.center),auto,vertex/.style=circle,draw=black!100,
inner sep=0pt,minimum size=1mm,scale=1]
(B1) at (1,1) [vertex,fill=black,label=below:] ;
(B2) at (0.5,1) [vertex,fill=black,label=below:] ;
(B3) at (0,1) [vertex,fill=black,label=below:] ;
(B4) at (-0.5,1) [vertex,fill=black,label=below:] ;
(BB1) at (0.75,1) [vertex,fill=black,label=below:] ;
[-](B1) edge node[right]  (BB1);
(BB2) at (0.25,1) [vertex,fill=black,label=below:] ;
[-](B2) edge node[right]  (BB2);
(BB3) at (-0.25,1) [vertex,fill=black,label=below:] ;
[-](B3) edge node[right]  (BB3);
(BB4) at (-0.75,1) [vertex,fill=black,label=below:] ;
[-](B4) edge node[right]  (BB4);
(A) at (0.25,0) [vertex,fill=white,label=below:] ;

[-](B1) edge node[right]  (A);
[-](B2) edge node[right]  (A);
[-](B3) edge node[right]  (A);
[-](B4) edge node[right]  (A);
(C1) at (-0.5,2) [vertex,fill=black,label=below:] ;
(C2) at (0.5,2) [vertex,fill=black,label=below:] ;
[-](B1) edge node[right]  (C2);
[-](B2) edge node[right]  (C2);
[-](B3) edge node[right]  (C2);
[-](B4) edge node[right]  (C1);

[baseline=([yshift=-.5ex]current bounding box.center),auto,vertex/.style=circle,draw=black!100,
inner sep=0pt,minimum size=1mm,scale=1]
(B1) at (1,1) [vertex,fill=black,label=below:] ;
(B2) at (0.5,1) [vertex,fill=black,label=below:] ;
(B3) at (0,1) [vertex,fill=black,label=below:] ;
(B4) at (-0.5,1) [vertex,fill=black,label=below:] ;
(BB1) at (0.75,1) [vertex,fill=black,label=below:] ;
[-](B1) edge node[right]  (BB1);
(BB2) at (0.25,1) [vertex,fill=black,label=below:] ;
[-](B2) edge node[right]  (BB2);
(BB3) at (-0.25,1) [vertex,fill=black,label=below:] ;
[-](B3) edge node[right]  (BB3);
(BB4) at (-0.75,1) [vertex,fill=black,label=below:] ;
[-](B4) edge node[right]  (BB4);
(A) at (0.25,0) [vertex,fill=white,label=below:] ;

[-](B1) edge node[right]  (A);
[-](B2) edge node[right]  (A);
[-](B3) edge node[right]  (A);
[-](B4) edge node[right]  (A);
(C1) at (-0.25,2) [vertex,fill=black,label=below:] ;
(C2) at (0.75,2) [vertex,fill=black,label=below:] ;
[-](B1) edge node[right]  (C2);
[-](B2) edge node[right]  (C2);
[-](B3) edge node[right]  (C1);
[-](B4) edge node[right]  (C1);

0.26$\lMset 0,(1\pm\frac1{\sqrt3})^{(r-s)}, $$(1\pm\frac{\sqrt2}{\sqrt3})^{(s-1)}, 1^{(s+1)},2\rMset$
$4$     $3$     $1$   $5$   $6$   $9$   $9$    $r+2s+1$

[baseline=([yshift=-.5ex]current bounding box.center),auto,vertex/.style=circle,draw=black!100,
inner sep=0pt,minimum size=1mm,scale=1]
(B) at (0,1) [vertex,fill=black,label=below:] ;
(BB) at (-0.25,2) [vertex,fill=black,label=below:] ;
[-](B) edge node[right]  (BB);
(A) at (0,0) [vertex,fill=white,label=below:] ;
[-](B) edge node[right]  (A);
(C) at (0.25,2) [vertex,fill=black,label=below:] ;
[-](B) edge node[right]  (C);

[baseline=([yshift=-.5ex]current bounding box.center),auto,vertex/.style=circle,draw=black!100,
inner sep=0pt,minimum size=1mm,scale=1]
(B1) at (0.25,1) [vertex,fill=black,label=below:] ;
(B2) at (-0.25,1) [vertex,fill=black,label=below:] ;
(BB) at (-0.25,2) [vertex,fill=black,label=below:] ;
[-](B1) edge node[right]  (BB);
[-](B2) edge node[right]  (BB);
(A) at (0,0) [vertex,fill=white,label=below:] ;

[-](B1) edge node[right]  (A);
[-](B2) edge node[right]  (A);
(C) at (0.25,2) [vertex,fill=black,label=below:] ;
[-](B1) edge node[right]  (C);
[-](B2) edge node[right]  (C);

[baseline=([yshift=-.5ex]current bounding box.center),auto,vertex/.style=circle,draw=black!100,
inner sep=0pt,minimum size=1mm,scale=1]
(B1) at (0.5,1) [vertex,fill=black,label=below:] ;
(B2) at (0,1) [vertex,fill=black,label=below:] ;
(B3) at (-0.5,1) [vertex,fill=black,label=below:] ;
(BB) at (-0.25,2) [vertex,fill=black,label=below:] ;
[-](B1) edge node[right]  (BB);
[-](B2) edge node[right]  (BB);
[-](B3) edge node[right]  (BB);
(A) at (0,0) [vertex,fill=white,label=below:] ;

[-](B1) edge node[right]  (A);
[-](B2) edge node[right]  (A);
[-](B3) edge node[right]  (A);
(C) at (0.25,2) [vertex,fill=black,label=below:] ;
[-](B1) edge node[right]  (C);
[-](B2) edge node[right]  (C);
[-](B3) edge node[right]  (C);

[baseline=([yshift=-.5ex]current bounding box.center),auto,vertex/.style=circle,draw=black!100,
inner sep=0pt,minimum size=1mm,scale=1]
(B1) at (-0.5,1) [vertex,fill=black,label=below:] ;
(B2) at (0,1) [vertex,fill=black,label=below:] ;
(B3) at (0.5,1) [vertex,fill=black,label=below:] ;
(B4) at (1,1) [vertex,fill=black,label=below:] ;
(BB1) at (-0.25,2) [vertex,fill=black,label=below:] ;
[-](B1) edge node[right]  (BB1);
(BB2) at (0.25,2) [vertex,fill=black,label=below:] ;
[-](B2) edge node[right]  (BB2);
[-](B3) edge node[right]  (BB2);
[-](B4) edge node[right]  (BB2);
(A) at (0.25,0) [vertex,fill=white,label=below:] ;

[-](B1) edge node[right]  (A);
[-](B2) edge node[right]  (A);
[-](B3) edge node[right]  (A);
[-](B4) edge node[right]  (A);
(C1) at (-0.75,2) [vertex,fill=black,label=below:] ;
(C2) at (0.75,2) [vertex,fill=black,label=below:] ;
[-](B1) edge node[right]  (C1);
[-](B2) edge node[right]  (C2);
[-](B3) edge node[right]  (C2);
[-](B4) edge node[right]  (C2);

[baseline=([yshift=-.5ex]current bounding box.center),auto,vertex/.style=circle,draw=black!100,
inner sep=0pt,minimum size=1mm,scale=1]
(B1) at (-0.5,1) [vertex,fill=black,label=below:] ;
(B2) at (0,1) [vertex,fill=black,label=below:] ;
(B3) at (0.5,1) [vertex,fill=black,label=below:] ;
(B4) at (1,1) [vertex,fill=black,label=below:] ;
(BB1) at (0,2) [vertex,fill=black,label=below:] ;
[-](B1) edge node[right]  (BB1);
[-](B2) edge node[right]  (BB1);
(BB2) at (1,2) [vertex,fill=black,label=below:] ;
[-](B3) edge node[right]  (BB2);
[-](B4) edge node[right]  (BB2);
(A) at (0.25,0) [vertex,fill=white,label=below:] ;

[-](B1) edge node[right]  (A);
[-](B2) edge node[right]  (A);
[-](B3) edge node[right]  (A);
[-](B4) edge node[right]  (A);
(C1) at (-0.5,2) [vertex,fill=black,label=below:] ;
(C2) at (0.5,2) [vertex,fill=black,label=below:] ;
[-](B1) edge node[right]  (C1);
[-](B2) edge node[right]  (C1);
[-](B3) edge node[right]  (C2);
[-](B4) edge node[right]  (C2);

0.26$\lMset 0,(1\pm\frac{\sqrt2}{\sqrt3})^{(s-1)}, 1^{(r+1)},2\rMset$
$4$     $3$     $2$   $5$   $6$   $8$   $8$    $r+s+2$

[baseline=([yshift=-.5ex]current bounding box.center),auto,vertex/.style=circle,draw=black!100,
inner sep=0pt,minimum size=1mm,scale=1]
(B) at (0,1) [vertex,fill=black,label=below:] ;
(BB) at (-0.25,0) [vertex,fill=white,label=below:] ;
[-](B) edge node[right]  (BB);
(A) at (0.25,0) [vertex,fill=white,label=below:] ;
[-](B) edge node[right]  (A);
(C) at (0,2) [vertex,fill=black,label=below:] ;
[-](B) edge node[right]  (C);

[baseline=([yshift=-.5ex]current bounding box.center),auto,vertex/.style=circle,draw=black!100,
inner sep=0pt,minimum size=1mm,scale=1]
(B1) at (0.25,1) [vertex,fill=black,label=below:] ;
(B2) at (-0.25,1) [vertex,fill=black,label=below:] ;
(BB) at (-0.25,0) [vertex,fill=white,label=below:] ;
[-](B1) edge node[right]  (BB);
[-](B2) edge node[right]  (BB);
(A) at (0.25,0) [vertex,fill=white,label=below:] ;
[-](B1) edge node[right]  (A);
[-](B2) edge node[right]  (A);
(C) at (0,2) [vertex,fill=black,label=below:] ;
[-](B1) edge node[right]  (C);
[-](B2) edge node[right]  (C);

[baseline=([yshift=-.5ex]current bounding box.center),auto,vertex/.style=circle,draw=black!100,
inner sep=0pt,minimum size=1mm,scale=1]
(B1) at (0.5,1) [vertex,fill=black,label=below:] ;
(B2) at (0,1) [vertex,fill=black,label=below:] ;
(B3) at (-0.5,1) [vertex,fill=black,label=below:] ;
(BB) at (-0.25,0) [vertex,fill=white,label=below:] ;
[-](B1) edge node[right]  (BB);
[-](B2) edge node[right]  (BB);
[-](B3) edge node[right]  (BB);
(A) at (0.25,0) [vertex,fill=white,label=below:] ;

[-](B1) edge node[right]  (A);
[-](B2) edge node[right]  (A);
[-](B3) edge node[right]  (A);
(C) at (0,2) [vertex,fill=black,label=below:] ;
[-](B1) edge node[right]  (C);
[-](B2) edge node[right]  (C);
[-](B3) edge node[right]  (C);

[baseline=([yshift=-.5ex]current bounding box.center),auto,vertex/.style=circle,draw=black!100,
inner sep=0pt,minimum size=1mm,scale=1]
(B1) at (1,1) [vertex,fill=black,label=below:] ;
(B2) at (0.5,1) [vertex,fill=black,label=below:] ;
(B3) at (0,1) [vertex,fill=black,label=below:] ;
(B4) at (-0.5,1) [vertex,fill=black,label=below:] ;
(BB) at (0,0) [vertex,fill=white,label=below:] ;
[-](B1) edge node[right]  (BB);
[-](B2) edge node[right]  (BB);
[-](B3) edge node[right]  (BB);
[-](B4) edge node[right]  (BB);
(A) at (0.5,0) [vertex,fill=white,label=below:] ;

[-](B1) edge node[right]  (A);
[-](B2) edge node[right]  (A);
[-](B3) edge node[right]  (A);
[-](B4) edge node[right]  (A);
(C1) at (-0.5,2) [vertex,fill=black,label=below:] ;
(C2) at (0.5,2) [vertex,fill=black,label=below:] ;
[-](B1) edge node[right]  (C2);
[-](B2) edge node[right]  (C2);
[-](B3) edge node[right]  (C2);
[-](B4) edge node[right]  (C1);

[baseline=([yshift=-.5ex]current bounding box.center),auto,vertex/.style=circle,draw=black!100,
inner sep=0pt,minimum size=1mm,scale=1]
(B1) at (1,1) [vertex,fill=black,label=below:] ;
(B2) at (0.5,1) [vertex,fill=black,label=below:] ;
(B3) at (0,1) [vertex,fill=black,label=below:] ;
(B4) at (-0.5,1) [vertex,fill=black,label=below:] ;
(BB) at (0,0) [vertex,fill=white,label=below:] ;
[-](B1) edge node[right]  (BB);
[-](B2) edge node[right]  (BB);
[-](B3) edge node[right]  (BB);
[-](B4) edge node[right]  (BB);
(A) at (0.5,0) [vertex,fill=white,label=below:] ;

[-](B1) edge node[right]  (A);
[-](B2) edge node[right]  (A);
[-](B3) edge node[right]  (A);
[-](B4) edge node[right]  (A);
(C1) at (-0.25,2) [vertex,fill=black,label=below:] ;
(C2) at (0.75,2) [vertex,fill=black,label=below:] ;
[-](B1) edge node[right]  (C2);
[-](B2) edge node[right]  (C2);
[-](B3) edge node[right]  (C1);
[-](B4) edge node[right]  (C1);

0.26$\lMset 0,(1\pm\frac1{\sqrt3})^{(s-1)}, $$1^{(r-s+2))},2\rMset$

All examples of tree building blocks with four
vertices and their different contracted frame unions leading to
simple graphs.

§.§ Other small contracted frame unions

We now list all possible cases with standard weights leading to
isospectral and non-isomorphic examples starting from building blocks
$\G$ with two, three or four vertices.  We start with a certain graph
$\G$ as building block, and draw the vertices in $V_0$ on top and
bottom; while the vertices in $V_1$ (the “distinguished” vertices,
along the frames are contracted) are drawn on bottom and outlined).
For one graph $\G$ one might hence have several possibilities of frames
and contracted frame unions.

If $\G$ has two or three vertices, we allow that the isospectral
graphs have parallel edges (see all.ex.3vxall.ex.3vx').  If $\G$ has four
vertices, we restrict ourselves to examples leading only to
simple graphs, see simple.frames.  all.ex.4vx
contains all examples with a building block being a tree with four
vertices while all.ex.4vx' lists all remaining cases.

As simplest non-trivial partitions in all.ex.3vxall.ex.4vx'
we choose $A=\lMset 1,3\rMset$ and $B=\lMset 2,2\rMset$.  Moreover,
$C$ denotes an arbitrary $s$-partition of $r$.

[all examples with building blocks with three

Examples of building blocks with three vertices and standard
weights can be seen in all.ex.3vxall.ex.3vx' (labels starting with 3).
Only one building block leads to simple isospectral examples (label
3.1); this example class was already presented
in <cit.> under the name inflated
stars there.

All other examples leads to graphs with parallel edges.  Note that
one can always replace parallel edges by a single weighted edge
with the number of parallel edges as weight, see
mult.ed.weighed.gr.  The corresponding Laplacians are the
same, hence we also obtain isospectral, but non-isomorphic
examples; these examples are not isomorphic as weighted graphs as
their lists of weights are different.

label   $\G$     $\card V$    $\card {V_0}$     $\card {V_1}$     $\Gr F_2$     $\Gr F_3$     $\Gr F_{A,V_1}$
$\Gr F_{B,V_1}$     $\card{\Gr F_{C,V_1}}$, $\spec{\Gr F_{C,V_1}}$

$4$     $2$     $1$   $6$   $8$   $11$   $11$   $2r+s+1$

[baseline=([yshift=-.5ex]current bounding box.center),auto,vertex/.style=circle,draw=black!100,
inner sep=0pt,minimum size=1mm,scale=1]
(B) at (0,1) [vertex,fill=black,label=below:] ;
(BB) at (-0.25,1) [vertex,fill=black,label=below:] ;
[-](B) edge node[right]  (BB);
(A) at (0,0) [vertex,fill=white,label=below:] ;
[-](B) edge node[right]  (A);
(C) at (0,2) [vertex,fill=black,label=below:] ;
[-](B) edge node[right]  (C);
[-](BB) edge node[right]  (C);

[baseline=([yshift=-.5ex]current bounding box.center),auto,vertex/.style=circle,draw=black!100,
inner sep=0pt,minimum size=1mm,scale=1]
(B1) at (0.25,1) [vertex,fill=black,label=below:] ;
(B2) at (-0.25,1) [vertex,fill=black,label=below:] ;
(BB1) at (0.5,1) [vertex,fill=black,label=below:] ;
[-](B1) edge node[right]  (BB1);
(BB2) at (0,1) [vertex,fill=black,label=below:] ;
[-](B2) edge node[right]  (BB2);
(A) at (0,0) [vertex,fill=white,label=below:] ;

[-](B1) edge node[right]  (A);
[-](B2) edge node[right]  (A);
(C) at (0,2) [vertex,fill=black,label=below:] ;
[-](B1) edge node[right]  (C);
[-](B2) edge node[right]  (C);
[-](BB1) edge node[right]  (C);
[-](BB2) edge node[right]  (C);

[baseline=([yshift=-.5ex]current bounding box.center),auto,vertex/.style=circle,draw=black!100,
inner sep=0pt,minimum size=1mm,scale=1]
(B1) at (0.5,1) [vertex,fill=black,label=below:] ;
(B2) at (0,1) [vertex,fill=black,label=below:] ;
(B3) at (-0.5,1) [vertex,fill=black,label=below:] ;
(BB1) at (0.25,1) [vertex,fill=black,label=below:] ;
[-](B1) edge node[right]  (BB1);
(BB2) at (-0.25,1) [vertex,fill=black,label=below:] ;
[-](B2) edge node[right]  (BB2);
(BB3) at (-0.75,1) [vertex,fill=black,label=below:] ;
[-](B3) edge node[right]  (BB3);
(A) at (0,0) [vertex,fill=white,label=below:] ;

[-](B1) edge node[right]  (A);
[-](B2) edge node[right]  (A);
[-](B3) edge node[right]  (A);
(C) at (0,2) [vertex,fill=black,label=below:] ;
[-](B1) edge node[right]  (C);
[-](B2) edge node[right]  (C);
[-](B3) edge node[right]  (C);
[-](BB1) edge node[right]  (C);
[-](BB2) edge node[right]  (C);
[-](BB3) edge node[right]  (C);

[baseline=([yshift=-.5ex]current bounding box.center),auto,vertex/.style=circle,draw=black!100,
inner sep=0pt,minimum size=1mm,scale=1]
(B1) at (1,1) [vertex,fill=black,label=below:] ;
(B2) at (0.5,1) [vertex,fill=black,label=below:] ;
(B3) at (0,1) [vertex,fill=black,label=below:] ;
(B4) at (-0.5,1) [vertex,fill=black,label=below:] ;
(BB1) at (0.75,1) [vertex,fill=black,label=below:] ;
[-](B1) edge node[right]  (BB1);
(BB2) at (0.25,1) [vertex,fill=black,label=below:] ;
[-](B2) edge node[right]  (BB2);
(BB3) at (-0.25,1) [vertex,fill=black,label=below:] ;
[-](B3) edge node[right]  (BB3);
(BB4) at (-0.75,1) [vertex,fill=black,label=below:] ;
[-](B4) edge node[right]  (BB4);
(A) at (0,0) [vertex,fill=white,label=below:] ;

[-](B1) edge node[right]  (A);
[-](B2) edge node[right]  (A);
[-](B3) edge node[right]  (A);
[-](B4) edge node[right]  (A);
(C1) at (-0.5,2) [vertex,fill=black,label=below:] ;
(C2) at (0.5,2) [vertex,fill=black,label=below:] ;
[-](B1) edge node[right]  (C2);
[-](B2) edge node[right]  (C2);
[-](B3) edge node[right]  (C2);
[-](B4) edge node[right]  (C1);
[-](BB4) edge node[right]  (C1);
[-](BB1) edge node[right]  (C2);
[-](BB2) edge node[right]  (C2);
[-](BB3) edge node[right]  (C2);

[baseline=([yshift=-.5ex]current bounding box.center),auto,vertex/.style=circle,draw=black!100,
inner sep=0pt,minimum size=1mm,scale=1]
(B1) at (1,1) [vertex,fill=black,label=below:] ;
(B2) at (0.5,1) [vertex,fill=black,label=below:] ;
(B3) at (0,1) [vertex,fill=black,label=below:] ;
(B4) at (-0.5,1) [vertex,fill=black,label=below:] ;
(BB1) at (0.75,1) [vertex,fill=black,label=below:] ;
[-](B1) edge node[right]  (BB1);
(BB2) at (0.25,1) [vertex,fill=black,label=below:] ;
[-](B2) edge node[right]  (BB2);
(BB3) at (-0.25,1) [vertex,fill=black,label=below:] ;
[-](B3) edge node[right]  (BB3);
(BB4) at (-0.75,1) [vertex,fill=black,label=below:] ;
[-](B4) edge node[right]  (BB4);
(A) at (0,0) [vertex,fill=white,label=below:] ;

[-](B1) edge node[right]  (A);
[-](B2) edge node[right]  (A);
[-](B3) edge node[right]  (A);
[-](B4) edge node[right]  (A);
(C1) at (-0.25,2) [vertex,fill=black,label=below:] ;
(C2) at (0.75,2) [vertex,fill=black,label=below:] ;
[-](B1) edge node[right]  (C2);
[-](B2) edge node[right]  (C2);
[-](B3) edge node[right]  (C1);
[-](B4) edge node[right]  (C1);
[-](BB4) edge node[right]  (C1);
[-](BB3) edge node[right]  (C1);
[-](BB2) edge node[right]  (C2);
[-](BB1) edge node[right]  (C2);

0.28$\lMset 0,(\frac34\pm\frac{\sqrt{57}}{12})^{(s-1)},$$(1\pm\frac1{\sqrt6})^{(r-s)},(\frac54\pm\frac{\sqrt{33}}{12}),(\frac32)^{(s)}\rMset$

[baseline=([yshift=-.5ex]current bounding box.center),auto,vertex/.style=circle,draw=black!100,
inner sep=0pt,minimum size=1mm,scale=1]
(B) at (0,1) [vertex,fill=black,label=below:] ;
(BB) at (-0.25,1) [vertex,fill=black,label=below:] ;
[-](B) edge node[right]  (BB);
(A) at (0,0) [vertex,fill=white,label=below:] ;
[-](B) edge node[right]  (A);
(C) at (0,2) [vertex,fill=black,label=below:] ;
[-](B) edge node[right]  (C);
[-](BB) edge node[right]  (A);

[baseline=([yshift=-.5ex]current bounding box.center),auto,vertex/.style=circle,draw=black!100,
inner sep=0pt,minimum size=1mm,scale=1]
(B1) at (0.25,1) [vertex,fill=black,label=below:] ;
(B2) at (-0.25,1) [vertex,fill=black,label=below:] ;
(BB1) at (0.5,1) [vertex,fill=black,label=below:] ;
[-](B1) edge node[right]  (BB1);
(BB2) at (0,1) [vertex,fill=black,label=below:] ;
[-](B2) edge node[right]  (BB2);
(A) at (0,0) [vertex,fill=white,label=below:] ;

[-](B1) edge node[right]  (A);
[-](B2) edge node[right]  (A);
(C) at (0,2) [vertex,fill=black,label=below:] ;
[-](B1) edge node[right]  (C);
[-](B2) edge node[right]  (C);
[-](BB1) edge node[right]  (A);
[-](BB2) edge node[right]  (A);

[baseline=([yshift=-.5ex]current bounding box.center),auto,vertex/.style=circle,draw=black!100,
inner sep=0pt,minimum size=1mm,scale=1]
(B1) at (0.5,1) [vertex,fill=black,label=below:] ;
(B2) at (0,1) [vertex,fill=black,label=below:] ;
(B3) at (-0.5,1) [vertex,fill=black,label=below:] ;
(BB1) at (0.25,1) [vertex,fill=black,label=below:] ;
[-](B1) edge node[right]  (BB1);
(BB2) at (-0.25,1) [vertex,fill=black,label=below:] ;
[-](B2) edge node[right]  (BB2);
(BB3) at (-0.75,1) [vertex,fill=black,label=below:] ;
[-](B3) edge node[right]  (BB3);
(A) at (0,0) [vertex,fill=white,label=below:] ;

[-](B1) edge node[right]  (A);
[-](B2) edge node[right]  (A);
[-](B3) edge node[right]  (A);
(C) at (0,2) [vertex,fill=black,label=below:] ;
[-](B1) edge node[right]  (C);
[-](B2) edge node[right]  (C);
[-](B3) edge node[right]  (C);
[-](BB1) edge node[right]  (A);
[-](BB2) edge node[right]  (A);
[-](BB3) edge node[right]  (A);

[baseline=([yshift=-.5ex]current bounding box.center),auto,vertex/.style=circle,draw=black!100,
inner sep=0pt,minimum size=1mm,scale=1]
(B1) at (1,1) [vertex,fill=black,label=below:] ;
(B2) at (0.5,1) [vertex,fill=black,label=below:] ;
(B3) at (0,1) [vertex,fill=black,label=below:] ;
(B4) at (-0.5,1) [vertex,fill=black,label=below:] ;
(BB1) at (0.75,1) [vertex,fill=black,label=below:] ;
[-](B1) edge node[right]  (BB1);
(BB2) at (0.25,1) [vertex,fill=black,label=below:] ;
[-](B2) edge node[right]  (BB2);
(BB3) at (-0.25,1) [vertex,fill=black,label=below:] ;
[-](B3) edge node[right]  (BB3);
(BB4) at (-0.75,1) [vertex,fill=black,label=below:] ;
[-](B4) edge node[right]  (BB4);
(A) at (0,0) [vertex,fill=white,label=below:] ;

[-](B1) edge node[right]  (A);
[-](B2) edge node[right]  (A);
[-](B3) edge node[right]  (A);
[-](B4) edge node[right]  (A);
(C1) at (-0.5,2) [vertex,fill=black,label=below:] ;
(C2) at (0.5,2) [vertex,fill=black,label=below:] ;
[-](B1) edge node[right]  (C2);
[-](B2) edge node[right]  (C2);
[-](B3) edge node[right]  (C2);
[-](B4) edge node[right]  (C1);
[-](BB4) edge node[right]  (A);
[-](BB1) edge node[right]  (A);
[-](BB2) edge node[right]  (A);
[-](BB3) edge node[right]  (A);

[baseline=([yshift=-.5ex]current bounding box.center),auto,vertex/.style=circle,draw=black!100,
inner sep=0pt,minimum size=1mm,scale=1]
(B1) at (1,1) [vertex,fill=black,label=below:] ;
(B2) at (0.5,1) [vertex,fill=black,label=below:] ;
(B3) at (0,1) [vertex,fill=black,label=below:] ;
(B4) at (-0.5,1) [vertex,fill=black,label=below:] ;
(BB1) at (0.75,1) [vertex,fill=black,label=below:] ;
[-](B1) edge node[right]  (BB1);
(BB2) at (0.25,1) [vertex,fill=black,label=below:] ;
[-](B2) edge node[right]  (BB2);
(BB3) at (-0.25,1) [vertex,fill=black,label=below:] ;
[-](B3) edge node[right]  (BB3);
(BB4) at (-0.75,1) [vertex,fill=black,label=below:] ;
[-](B4) edge node[right]  (BB4);
(A) at (0,0) [vertex,fill=white,label=below:] ;

[-](B1) edge node[right]  (A);
[-](B2) edge node[right]  (A);
[-](B3) edge node[right]  (A);
[-](B4) edge node[right]  (A);
(C1) at (-0.25,2) [vertex,fill=black,label=below:] ;
(C2) at (0.75,2) [vertex,fill=black,label=below:] ;
[-](B1) edge node[right]  (C2);
[-](B2) edge node[right]  (C2);
[-](B3) edge node[right]  (C1);
[-](B4) edge node[right]  (C1);
[-](BB4) edge node[right]  (A);
[-](BB3) edge node[right]  (A);
[-](BB2) edge node[right]  (A);
[-](BB1) edge node[right]  (A);

0.28$\lMset 0,(1\pm\frac1{\sqrt6})^{(r-s)},(1\pm\frac1{\sqrt2})^{(s-1)},$$(\frac54\pm\frac{\sqrt{33}}{12}),1^{(s-1)},\frac32\rMset$
$4$     $2$     $1$   $6$   $8$   $11$   $11$    $2r+s+1$

[baseline=([yshift=-.5ex]current bounding box.center),auto,vertex/.style=circle,draw=black!100,
inner sep=0pt,minimum size=1mm,scale=1]
(A) at (0,0) [vertex,fill=white,label=below:] ;
(BB) at (0,2) [vertex,fill=black,label=below:] ;
(B) at (-0.25,1) [vertex,fill=black,label=below:] ;
(C) at (0.25,1) [vertex,fill=black,label=below:] ;
[-](A) edge node[right]  (B);
[-](A) edge node[right]  (C);
[-](BB) edge node[right]  (B);
[-](BB) edge node[right]  (C);

[baseline=([yshift=-.5ex]current bounding box.center),auto,vertex/.style=circle,draw=black!100,
inner sep=0pt,minimum size=1mm,scale=1]
(B1) at (0.25,1) [vertex,fill=black,label=below:] ;
(B2) at (-0.25,1) [vertex,fill=black,label=below:] ;
(BB) at (0,2) [vertex,fill=black,label=below:] ;
(A) at (0,0) [vertex,fill=white,label=below:] ;
[-](B1) edge node[right]  (A);
[-](B2) edge node[right]  (A);
[-](B1) edge node[right]  (BB);
[-](B2) edge node[right]  (BB);
(C1) at (-0.5,1) [vertex,fill=black,label=below:] ;
(C2) at (0.5,1) [vertex,fill=black,label=below:] ;
[-](BB) edge node[right]  (C1);
[-](BB) edge node[right]  (C2);
[-](C1) edge node[right]  (A);
[-](C2) edge node[right]  (A);

[baseline=([yshift=-.5ex]current bounding box.center),auto,vertex/.style=circle,draw=black!100,
inner sep=0pt,minimum size=1mm,scale=1]
(B1) at (0.5,1) [vertex,fill=black,label=below:] ;
(B2) at (-0.25,1) [vertex,fill=black,label=below:] ;
(B0) at (0,1) [vertex,fill=black,label=below:] ;
(BB) at (0,2) [vertex,fill=black,label=below:] ;
(A) at (0,0) [vertex,fill=white,label=below:] ;
[-](B1) edge node[right]  (A);
[-](B2) edge node[right]  (A);
[-](B0) edge node[right]  (A);
[-](B1) edge node[right]  (BB);
[-](B2) edge node[right]  (BB);
[-](B0) edge node[right]  (BB);
(C1) at (-0.5,1) [vertex,fill=black,label=below:] ;
(C2) at (0.25,1) [vertex,fill=black,label=below:] ;
(C0) at (0.75,1) [vertex,fill=black,label=below:] ;
[-](BB) edge node[right]  (C1);
[-](BB) edge node[right]  (C2);
[-](BB) edge node[right]  (C0);
[-](C1) edge node[right]  (A);
[-](C2) edge node[right]  (A);
[-](C0) edge node[right]  (A);

[baseline=([yshift=-.5ex]current bounding box.center),auto,vertex/.style=circle,draw=black!100,
inner sep=0pt,minimum size=1mm,scale=1]
(B1) at (0.5,1) [vertex,fill=black,label=below:] ;
(B2) at (-0.25,1) [vertex,fill=black,label=below:] ;
(B0) at (0,1) [vertex,fill=black,label=below:] ;
(B-1) at (-0.75,1) [vertex,fill=black,label=below:] ;
(BB1) at (-0.75,2) [vertex,fill=black,label=below:] ;
(BB2) at (0,2) [vertex,fill=black,label=below:] ;
(A) at (0,0) [vertex,fill=white,label=below:] ;
[-](B1) edge node[right]  (A);
[-](B2) edge node[right]  (A);
[-](B0) edge node[right]  (A);
[-](B-1) edge node[right]  (A);
[-](B-1) edge node[right]  (BB1);
[-](B1) edge node[right]  (BB2);
[-](B2) edge node[right]  (BB2);
[-](B0) edge node[right]  (BB2);
(C1) at (-0.5,1) [vertex,fill=black,label=below:] ;
(C2) at (0.25,1) [vertex,fill=black,label=below:] ;
(C0) at (0.75,1) [vertex,fill=black,label=below:] ;
(C-1) at (-1,1) [vertex,fill=black,label=below:] ;
[-](BB1) edge node[right]  (C-1);
[-](BB2) edge node[right]  (C1);
[-](BB2) edge node[right]  (C2);
[-](BB2) edge node[right]  (C0);
[-](C1) edge node[right]  (A);
[-](C2) edge node[right]  (A);
[-](C0) edge node[right]  (A);
[-](C-1) edge node[right]  (A);

[baseline=([yshift=-.5ex]current bounding box.center),auto,vertex/.style=circle,draw=black!100,
inner sep=0pt,minimum size=1mm,scale=1]
(B1) at (0.5,1) [vertex,fill=black,label=below:] ;
(B2) at (-0.25,1) [vertex,fill=black,label=below:] ;
(B0) at (0,1) [vertex,fill=black,label=below:] ;
(B-1) at (-0.75,1) [vertex,fill=black,label=below:] ;
(BB1) at (-0.75,2) [vertex,fill=black,label=below:] ;
(BB2) at (0.5,2) [vertex,fill=black,label=below:] ;
(A) at (0,0) [vertex,fill=white,label=below:] ;
[-](B1) edge node[right]  (A);
[-](B2) edge node[right]  (A);
[-](B0) edge node[right]  (A);
[-](B-1) edge node[right]  (A);
[-](B-1) edge node[right]  (BB1);
[-](B1) edge node[right]  (BB2);
[-](B2) edge node[right]  (BB1);
[-](B0) edge node[right]  (BB2);
(C1) at (-0.5,1) [vertex,fill=black,label=below:] ;
(C2) at (0.25,1) [vertex,fill=black,label=below:] ;
(C0) at (0.75,1) [vertex,fill=black,label=below:] ;
(C-1) at (-1,1) [vertex,fill=black,label=below:] ;
[-](BB1) edge node[right]  (C-1);
[-](BB1) edge node[right]  (C1);
[-](BB2) edge node[right]  (C2);
[-](BB2) edge node[right]  (C0);
[-](C1) edge node[right]  (A);
[-](C2) edge node[right]  (A);
[-](C0) edge node[right]  (A);
[-](C-1) edge node[right]  (A);

0.28$\lMset 0,(1\pm\frac1{\sqrt2})^{(s-1)},1^{(2r-s+1)},2\rMset$
$4$     $2$     $1$   $6$   $8$   $11$   $11$   $2r+s+1$

[baseline=([yshift=-.5ex]current bounding box.center),auto,vertex/.style=circle,draw=black!100,
inner sep=0pt,minimum size=1mm,scale=1]
(A) at (0,0) [vertex,fill=white,label=below:] ;
(BB) at (0,2) [vertex,fill=black,label=below:] ;
(B) at (-0.25,1) [vertex,fill=black,label=below:] ;
(C) at (0.25,1) [vertex,fill=black,label=below:] ;
[-](B) edge node[right]  (C);
[-](A) edge node[right]  (B);
[-](A) edge node[right]  (C);
[-](BB) edge node[right]  (B);
[-](BB) edge node[right]  (C);

[baseline=([yshift=-.5ex]current bounding box.center),auto,vertex/.style=circle,draw=black!100,
inner sep=0pt,minimum size=1mm,scale=1]
(B1) at (0.25,1) [vertex,fill=black,label=below:] ;
(B2) at (-0.25,1) [vertex,fill=black,label=below:] ;
(BB) at (0,2) [vertex,fill=black,label=below:] ;
(A) at (0,0) [vertex,fill=white,label=below:] ;
[-](B1) edge node[right]  (A);
[-](B2) edge node[right]  (A);
[-](B1) edge node[right]  (BB);
[-](B2) edge node[right]  (BB);
(C1) at (0.5,1) [vertex,fill=black,label=below:] ;
(C2) at (-0.5,1) [vertex,fill=black,label=below:] ;
[-](B1) edge node[right]  (C1);
[-](B2) edge node[right]  (C2);
[-](BB) edge node[right]  (C1);
[-](BB) edge node[right]  (C2);
[-](C1) edge node[right]  (A);
[-](C2) edge node[right]  (A);

[baseline=([yshift=-.5ex]current bounding box.center),auto,vertex/.style=circle,draw=black!100,
inner sep=0pt,minimum size=1mm,scale=1]
(B1) at (0.5,1) [vertex,fill=black,label=below:] ;
(B2) at (-0.25,1) [vertex,fill=black,label=below:] ;
(B0) at (0,1) [vertex,fill=black,label=below:] ;
(BB) at (0,2) [vertex,fill=black,label=below:] ;
(A) at (0,0) [vertex,fill=white,label=below:] ;
[-](B1) edge node[right]  (A);
[-](B2) edge node[right]  (A);
[-](B0) edge node[right]  (A);
[-](B1) edge node[right]  (BB);
[-](B2) edge node[right]  (BB);
[-](B0) edge node[right]  (BB);
(C1) at (0.75,1) [vertex,fill=black,label=below:] ;
(C2) at (-0.5,1) [vertex,fill=black,label=below:] ;
(C0) at (0.25,1) [vertex,fill=black,label=below:] ;
[-](B0) edge node[right]  (C0);
[-](B1) edge node[right]  (C1);
[-](B2) edge node[right]  (C2);
[-](BB) edge node[right]  (C1);
[-](BB) edge node[right]  (C2);
[-](BB) edge node[right]  (C0);
[-](C1) edge node[right]  (A);
[-](C2) edge node[right]  (A);
[-](C0) edge node[right]  (A);

[baseline=([yshift=-.5ex]current bounding box.center),auto,vertex/.style=circle,draw=black!100,
inner sep=0pt,minimum size=1mm,scale=1]
(B1) at (0.5,1) [vertex,fill=black,label=below:] ;
(B2) at (-0.25,1) [vertex,fill=black,label=below:] ;
(B0) at (0,1) [vertex,fill=black,label=below:] ;
(B-1) at (-0.75,1) [vertex,fill=black,label=below:] ;
(BB1) at (-0.75,2) [vertex,fill=black,label=below:] ;
(BB2) at (0,2) [vertex,fill=black,label=below:] ;
(A) at (0,0) [vertex,fill=white,label=below:] ;
[-](B1) edge node[right]  (A);
[-](B2) edge node[right]  (A);
[-](B0) edge node[right]  (A);
[-](B-1) edge node[right]  (A);
[-](B-1) edge node[right]  (BB1);
[-](B1) edge node[right]  (BB2);
[-](B2) edge node[right]  (BB2);
[-](B0) edge node[right]  (BB2);
(C1) at (-0.5,1) [vertex,fill=black,label=below:] ;
(C2) at (0.25,1) [vertex,fill=black,label=below:] ;
(C0) at (0.75,1) [vertex,fill=black,label=below:] ;
(C-1) at (-1,1) [vertex,fill=black,label=below:] ;
[-](B0) edge node[right]  (C2);
[-](B1) edge node[right]  (C0);
[-](B2) edge node[right]  (C1);
[-](B-1) edge node[right]  (C-1);
[-](BB1) edge node[right]  (C-1);
[-](BB2) edge node[right]  (C1);
[-](BB2) edge node[right]  (C2);
[-](BB2) edge node[right]  (C0);
[-](C1) edge node[right]  (A);
[-](C2) edge node[right]  (A);
[-](C0) edge node[right]  (A);
[-](C-1) edge node[right]  (A);

[baseline=([yshift=-.5ex]current bounding box.center),auto,vertex/.style=circle,draw=black!100,
inner sep=0pt,minimum size=1mm,scale=1]
(B1) at (0.5,1) [vertex,fill=black,label=below:] ;
(B2) at (-0.25,1) [vertex,fill=black,label=below:] ;
(B0) at (0,1) [vertex,fill=black,label=below:] ;
(B-1) at (-0.75,1) [vertex,fill=black,label=below:] ;
(BB1) at (-0.75,2) [vertex,fill=black,label=below:] ;
(BB2) at (0.5,2) [vertex,fill=black,label=below:] ;
(A) at (0,0) [vertex,fill=white,label=below:] ;
[-](B1) edge node[right]  (A);
[-](B2) edge node[right]  (A);
[-](B0) edge node[right]  (A);
[-](B-1) edge node[right]  (A);
[-](B-1) edge node[right]  (BB1);
[-](B1) edge node[right]  (BB2);
[-](B2) edge node[right]  (BB1);
[-](B0) edge node[right]  (BB2);
(C1) at (-0.5,1) [vertex,fill=black,label=below:] ;
(C2) at (0.25,1) [vertex,fill=black,label=below:] ;
(C0) at (0.75,1) [vertex,fill=black,label=below:] ;
(C-1) at (-1,1) [vertex,fill=black,label=below:] ;
[-](B0) edge node[right]  (C2);
[-](B1) edge node[right]  (C0);
[-](B2) edge node[right]  (C1);
[-](B-1) edge node[right]  (C-1);
[-](BB1) edge node[right]  (C-1);
[-](BB1) edge node[right]  (C1);
[-](BB2) edge node[right]  (C2);
[-](BB2) edge node[right]  (C0);
[-](C1) edge node[right]  (A);
[-](C2) edge node[right]  (A);
[-](C0) edge node[right]  (A);
[-](C-1) edge node[right]  (A);

0.28$\lMset 0,(\frac56\pm\frac{\sqrt{13}}6)^{(s-1)},$$(\frac23)^{(r-s)},1,(\frac43)^{(r)},\frac53\rMset$

All examples of non-tree building blocks with four vertices
and their different contracted frame unions leading to
simple graphs.

[all examples with building blocks with four vertices]

For simplicity, we list here only those cases leading to
simple isospectral examples.  all.ex.4vx contains all
cases where the building block $\G$ is a tree while
all.ex.4vx' contains the remaining cases with building
blocks not being a tree.

Note that only the cases with label 4.2', 4,4 and 4.4' can be
treated using Theorem 1 of [11].  In all other
cases, the graphs $G_j[A',B]$ (in the notation
of [11]) are not complete bipartite graphs as
required by <cit.>.

The case with label 4.4 can also be treated with the perturbative
technique described in <cit.>, where we
used the name fuzzy complete bipartite graph.  With the
method in the present paper, it is based on the frames used in

§.§ Some other examples with possible non-trivial magnetic

Here, we give examples where the underlying building block is not a
tree, hence there exist magnetic potentials having an effect on the

[a diamond-like graph with magnetic potential]

Taking as building block the magnetic graph $\wG^\theta$ of
diamond.mag, and choosing as set $V_1=\{v_1\}$ just the bottom
(outlined) vertex, then the spectrum of
$\DirGr {(\wG^\theta)} {V_1}$ is given by

\begin{equation*}
\bigspec{\DirGr {(\wG^\theta)} {V_1}}
1-\frac{\cos\theta}4 - \frac12\sqrt{1+\frac{\cos^2\theta}4},
1-\frac{\cos\theta}4 + \frac12\sqrt{1+\frac{\cos^2\theta}4}
\Bigr\rMset.
\end{equation*}

For $A=\lMset 1,3\rMset$ and $B=\lMset 2,2\rMset$, we have the
isospectral magnetic graphs as in diamond-mag.iso.

[auto, vertex/.style=circle,draw=black!100,fill=black!100,
inner sep=0pt,minimum size=1mm,scale=1]
in 0
(C) at (0,1) [vertex,draw=black,fill=none,label=below:] ;
(B) at (*.5,2) [vertex,label=below:] ;
[-] (B) to[min distance=8mm,in=45,out=325,looseness=1]  node[right]  (B);
[-](C) edge node[right]  (B);
(A) at (0,3) [vertex,label=below:] ;
[-](B) edge node[right]  (A);

in 0,2
(C) at (0,1) [vertex,draw=black,fill=none,label=below:] ;
(B) at (*.5,0) [vertex,label=below:] ;
[-] (B) to[min distance=8mm,in=45,out=325,looseness=1]  node[right]  (B);
[-](C) edge node[right]  (B);
(A) at (0,-1) [vertex,label=below:] ;
[-](B) edge node[right]  (A);

in -2
(C) at (0,1) [vertex,draw=black,fill=none,label=below:] ;
(B) at (*.5,0) [vertex,label=below:] ;
[-] (B) to[min distance=8mm,in=135,out=225,looseness=1]  node[right]  (B);
[-](C) edge node[right]  (B);
(A) at (0,-1) [vertex,label=below:] ;
[-](B) edge node[right]  (A);

[] (.3,.-1.5)node[left] $\wGr F_{A,V_1}(\wG^\theta,V_0)$;
[auto, vertex/.style=circle,draw=black!100,fill=black!100,
inner sep=0pt,minimum size=1mm,scale=1]
in -1
(C) at (0,1) [vertex,draw=black,fill=none,label=below:] ;
(B) at (*.5,2) [vertex,label=below:] ;
[-] (B) to[min distance=8mm,in=135,out=225,looseness=1]  node[right]  (B);
[-](C) edge node[right]  (B);
(A) at (0,3) [vertex,label=below:] ;
[-](B) edge node[right]  (A);

in 1
(C) at (0,1) [vertex,draw=black,fill=none,label=below:] ;
(B) at (*.5,2) [vertex,label=below:] ;
[-] (B) to[min distance=8mm,in=45,out=325,looseness=1]  node[right]  (B);
[-](C) edge node[right]  (B);
(A) at (0,3) [vertex,label=below:] ;
[-](B) edge node[right]  (A);

in 1
(C) at (0,1) [vertex,draw=black,fill=none,label=below:] ;
(B) at (*.5,0) [vertex,label=below:] ;
[-] (B) to[min distance=8mm,in=45,out=325,looseness=1]  node[right]  (B);
[-](C) edge node[right]  (B);
(A) at (0,-1) [vertex,label=below:] ;
[-](B) edge node[right]  (A);

in -1
(C) at (0,1) [vertex,draw=black,fill=none,label=below:] ;
(B) at (*.5,0) [vertex,label=below:] ;
[-] (B) to[min distance=8mm,in=135,out=225,looseness=1]  node[right]  (B);
[-](C) edge node[right]  (B);
(A) at (0,-1) [vertex,label=below:] ;
[-](B) edge node[right]  (A);

[] (.3,.-1.5)node[left] $\wGr F_{B,V_1}(\wG^\theta,V_0)$;

Two magnetic isospectral graphs. Note that the magnetic
potential is the same on each loop.

[kite graphs]

Let $\G$ be the complete graph on four vertices with one pendant
vertex added (so $\G$ has five vertices) as in the frame
construction of kite.frame.  The vertex set $V_0$ consists now
of the pendant vertex and one of the remaining four.  The set of
distinguished vertices $V_1$ consists just of the pendant vertex.
Let $\wG$ be the corresponding weighted graph with standard weights
and no magnetic potential, then

\begin{align*}
\spec{\DirGr \wG {V_1}}
= \Bigl\lMset \frac{4-\sqrt {13}}6,\frac 43,\frac 43,\frac{4+\sqrt {13}}6
\Bigr\rMset.
\end{align*}

An example of isospectral graphs is given in kite.isospec
(right hand side).  Note that this construction also works for any
magnetic potential, see kite.frame.

[all non-trivial magnetic graphs with building block
with four vertices]
Note that all examples of all.ex.4vx' can carry also a
magnetic potential on the building block leading to a non-trivial
family with one parameter (labels 4.5, 4.5' and 4.6) respectively
two parameters (label 4.7) in

§.§ A class of weighted isospectral graphs

[contracted along all but one vertex]

Let $\G$ be a graph with standard weights and $V_0=V(\G)$.  Then the
frame members $\Gr F_a(\wG,V_0)$ are the $r$-fold edge copy of $\G$
as in all.vx.id and in particular isolaplacian with $\G$ (see
second and third graph in all.but.1.identified).  Note that
Butler [10] also uses the idea of changing weights.
Special cases are actually the graphs with label 2.1, 3.3', 3.4' and
3.6' in all.ex.3vxall.ex.3vx' where $\G$ is a path graph
with two or three vertices or a cycle graph with three vertices.

Let now $\Gr F_{A,V_1}=\Gr F_{A,V_1}(\G,V(\G))$ be the contracted
frame union for $V_1=V(\G) \setminus \{v^*\}$ and some $s$-partition
of $r$.  Here, $v^* \in V(\G)$ is one chosen vertex.  Its order is
$\card{\Gr F_{A,V_1}}=s+\card{V_1}=\card \wG +s-1$ independently of
$r$ as in all.vx.id.  The spectrum of $\DirGr \G {V_1}$ is just
$\lMset 1 \rMset$, as $V(\G) \setminus V_1=\{v^*\}$ contains just
one point.  From spec.contr.frame.union we conclude that

\begin{equation*}
\spec{\Gr F_{A,V_1}}
= \spec \G \uplus \lMset 1\rMset^{(s-1)}.
\end{equation*}

Note that the matrix representation of Laplacian on $\Gr F_{A,V_1}$
(with standard weights) has the matrix representation of $\G^*$ as
principal submatrix, where $\G^*$ denotes the graph $\G$ with $v^*$
and all adjacent edges removed.  It could also be observed that
passing to the corresponding simple weighted graphs (with weights
$r$ on all edges in $\G^*$) and weights $a \in A$ on the edges
joining the $s$ copies of $v^*$.  Dividing the weights by the common
factor $1/r$ (this does not change the Laplacian), we end up with an
isolaplacian graph $\G_{A,v^*}$ with standard weights ($w_e=1$) for
all edges inside $\G^*$, and weights $a/r$ ($a \in A$) for the
remaining edges joining the extra $s$ copies of $v^*$ (see
all.but.1.identified).  In particular, we have a way of
keeping the spectrum of $\G$ and adding eigenvalues $1$.

inner sep=0pt,minimum size=1mm,scale=1]
in 0
(C) at (*0.5,1) [vertex,fill=black,label=left:] ;
(B) at (-0.5,0) [vertex,fill=white,label=left:] ;
(BB) at (0.5,0) [vertex,fill=white,label=left:] ;
[-](C) edge node[right]  (B);
[-](C) edge node[right]  (BB);
[-](B) edge node[right]  (BB);
(A) at (-0.5,-1) [vertex,fill=white,label=left:] ;
(AA) at (0.5,-1) [vertex,fill=white,label=left:] ;
[-](B) edge node[right]  (A);
[-](BB) edge node[right]  (AA);
[-](A) edge node[right]  (AA);

[] (0.2,.-1.5)node[left] $\G$;
[] (0.5,1.3)node[left] $v^*$;
inner sep=0pt,minimum size=1mm,scale=1]
in 0
(C) at (*0.5,1) [vertex,fill=black,label=left:] ;
(B) at (-0.5,0) [vertex,fill=white,label=left:] ;
(BB) at (0.5,0) [vertex,fill=white,label=left:] ;
[bend left=7](C) edge node[right]  (B);
[bend right=7](C) edge node[right]  (B);
[bend left=7](C) edge node[right]  (BB);
[bend right=7](C) edge node[right]  (BB);
(A) at (-0.5,-1) [vertex,fill=white,label=left:] ;
(AA) at (0.5,-1) [vertex,fill=white,label=left:] ;
[bend left=7](B) edge node[right]  (BB);
[bend right=7](B) edge node[right]  (BB);
[bend left=7](B) edge node[right]  (A);
[bend right=7](B) edge node[right]  (A);
[bend left=7](BB) edge node[right]  (AA);
[bend right=7](BB) edge node[right]  (AA);
[bend left=7](A) edge node[right]  (AA);
[bend right=7](A) edge node[right]  (AA);

[] (1,.-1.5)node[left] $\Gr F_2(\G,V_0)$;
inner sep=0pt,minimum size=1mm,scale=1]
in 0
(C) at (*0.5,1) [vertex,fill=black,label=left:] ;
(B) at (-0.5,0) [vertex,fill=white,label=left:] ;
(BB) at (0.5,0) [vertex,fill=white,label=left:] ;
[-](C) edge node[right]  (B);
[bend left=10](C) edge node[right]  (B);
[bend right=10](C) edge node[right]  (B);
[-](C) edge node[right]  (BB);
[bend left=10](C) edge node[right]  (BB);
[bend right=10](C) edge node[right]  (BB);
(A) at (-0.5,-1) [vertex,fill=white,label=left:] ;
(AA) at (0.5,-1) [vertex,fill=white,label=left:] ;
[-](B) edge node[right]  (BB);
[bend left=10](B) edge node[right]  (BB);
[bend right=10](B) edge node[right]  (BB);
[-](B) edge node[right]  (A);
[bend left=10](B) edge node[right]  (A);
[bend right=10](B) edge node[right]  (A);
[-](BB) edge node[right]  (AA);
[bend left=10](BB) edge node[right]  (AA);
[bend right=10](BB) edge node[right]  (AA);
[-](A) edge node[right]  (AA);
[bend left=10](A) edge node[right]  (AA);
[bend right=10](A) edge node[right]  (AA);

[] (1,.-1.5)node[left]  $\Gr F_3(\G,V_0)$;
inner sep=0pt,minimum size=1mm,scale=1]
(B) at (-0.5,0) [vertex,fill=white,label=left:] ;
(BB) at (0.5,0) [vertex,fill=white,label=left:] ;
(A) at (-0.5,-1) [vertex,fill=white,label=left:] ;
(AA) at (0.5,-1) [vertex,fill=,label=left:] ;
[bend left=5](B) edge node[right]  (BB);
[bend right=5](B) edge node[right]  (BB);
[bend left=5](B) edge node[right]  (A);
[bend right=5](B) edge node[right]  (A);
[bend left=5](BB) edge node[right]  (AA);
[bend right=5](BB) edge node[right]  (AA);
[bend left=5](A) edge node[right]  (AA);
[bend right=5](A) edge node[right]  (AA);
[bend left=14](B) edge node[right]  (BB);
[bend right=14](B) edge node[right]  (BB);
[bend left=14](B) edge node[right]  (A);
[bend right=14](B) edge node[right]  (A);
[bend left=14](BB) edge node[right]  (AA);
[bend right=14](BB) edge node[right]  (AA);
[bend left=14](A) edge node[right]  (AA);
[bend right=14](A) edge node[right]  (AA);
(C1) at (-0.5,1) [vertex,fill=black,label=left:] ;
(C2) at (0.5,1) [vertex,fill=black,label=left:] ;
[-](C1) edge node[right]  (B);
[-](C1) edge node[right]  (BB);
[bend right=10](C2) edge node[right]  (B);
[bend left=10](C2) edge node[right]  (B);
[bend right=10](C2) edge node[right]  (BB);
[bend left=10](C2) edge node[right]  (BB);
[-](C2) edge node[right]  (B);
[-](C2) edge node[right]  (BB);
[] (1,.-1.5)node[left]  $\Gr F_{A,V_1}(\G,V_0)$;
inner sep=0pt,minimum size=1mm,scale=1]
(B) at (-0.5,0) [vertex,fill=white,label=left:] ;
(BB) at (0.5,0) [vertex,fill=white,label=left:] ;
(A) at (-0.5,-1) [vertex,fill=white,label=left:] ;
(AA) at (0.5,-1) [vertex,fill=,label=left:] ;
[bend left=5](B) edge node[right]  (BB);
[bend right=5](B) edge node[right]  (BB);
[bend left=5](B) edge node[right]  (A);
[bend right=5](B) edge node[right]  (A);
[bend left=5](BB) edge node[right]  (AA);
[bend right=5](BB) edge node[right]  (AA);
[bend left=5](A) edge node[right]  (AA);
[bend right=5](A) edge node[right]  (AA);
[bend left=14](B) edge node[right]  (BB);
[bend right=14](B) edge node[right]  (BB);
[bend left=14](B) edge node[right]  (A);
[bend right=14](B) edge node[right]  (A);
[bend left=14](BB) edge node[right]  (AA);
[bend right=14](BB) edge node[right]  (AA);
[bend left=14](A) edge node[right]  (AA);
[bend right=14](A) edge node[right]  (AA);
(C1) at (-0.5,1) [vertex,fill=black,label=left:] ;
(C2) at (0.5,1) [vertex,fill=black,label=left:] ;
[bend right=7](C1) edge node[right]  (B);
[bend right=7](C1) edge node[right]  (BB);
[bend right=7](C2) edge node[right]  (B);
[bend right=7](C2) edge node[right]  (BB);
[bend left=7](C1) edge node[right]  (B);
[bend left=7](C1) edge node[right]  (BB);
[bend left=7](C2) edge node[right]  (B);
[bend left=7](C2) edge node[right]  (BB);
[] (1,.-1.5)node[left]  $\Gr F_{B,V_1}(\G,V_0)$;
inner sep=0pt,minimum size=1mm,scale=1]
(B) at (-0.5,0) [vertex,fill=white,label=left:] ;
(BB) at (0.5,0) [vertex,fill=white,label=left:] ;
(A) at (-0.5,-1) [vertex,fill=white,label=left:] ;
(AA) at (0.5,-1) [vertex,fill=white,label=left:] ;
(C1) at (-0.5,1) [vertex,fill=black,label=left:] ;
(C2) at (0.5,1) [vertex,fill=black,label=left:] ;
[-](B) edge node[right]  (BB);
[-](B) edge node[right]  (A);
[-](BB) edge node[right]  (AA);
[-](A) edge node[right]  (AA);
[-](C1) edge node[right]  (B);
[-](C1) edge node[right]  (BB);
[-](C2) edge node[right]  (B);
[-](C2) edge node[right]  (BB);
[] (-0.40,0.6)node[left]  $\frac14$;
[] (0.40,0.6)node[right]  $\frac34$;
[] (0.1,1)node[left]  $\frac14$;
[] (-0.1,1)node[right]  $\frac34$;
[] (0.8,.-1.5)node[left]  $\wG_{A,v^*}$;
inner sep=0pt,minimum size=1mm,scale=1]
(B) at (-0.5,0) [vertex,fill=white,label=left:] ;
(BB) at (0.5,0) [vertex,fill=white,label=left:] ;
(A) at (-0.5,-1) [vertex,fill=white,label=left:] ;
(AA) at (0.5,-1) [vertex,fill=white,label=left:] ;
(C1) at (-0.5,1) [vertex,fill=black,label=left:] ;
(C2) at (0.5,1) [vertex,fill=black,label=left:] ;
[-](B) edge node[right]  (BB);
[-](B) edge node[right]  (A);
[-](BB) edge node[right]  (AA);
[-](A) edge node[right]  (AA);
[-](C1) edge node[right]  (B);
[-](C1) edge node[right]  (BB);
[-](C2) edge node[right]  (B);
[-](C2) edge node[right]  (BB);
[] (-0.40,0.6)node[left]  $\frac12$;
[] (0.40,0.6)node[right]  $\frac12$;
[] (0.1,1)node[left]  $\frac12$;
[] (-0.1,1)node[right]  $\frac12$;
[] (0.8,.-1.5)node[left]  $\wG_{B,v^*}$;

All vertices in a frame are identified ($V_0=V(\G)$), and for
$V_1=V_0 \setminus \{v^*\}$.  On the right, the pair of isospectral
weighted graphs with edge weights indicated (if different from $1$).

§ CONCLUSIONS

Sunada's method [50] as well as some of its
generalisations [4, 46, 3] use
representation theory as a key ingredient to construct the isospectral
manifolds or graphs which appear as quotient graphs. It is worth to point
out differences and similarities between the method presented here and
the one given, for example, in [4] even if we think that
both methods are different in nature. We do not exclude though that
both approaches may be combined.

In [4] the metric graph considered first carries the action
of a finite (in general non-commutative) group.  Two different
subgroups are chosen which will determine the isospectral quotient
graphs provided the corresponding induced representations are
equivalent. Neumann and Dirichlet conditions on the metric graph
appear naturally.

The method presented in this article has a more combinatorial nature
even if the underlying symmetry is, of course, expressed in terms of
cyclic (commutative) groups acting on the copies of the building block
as mentioned in dualgroup.  But the construction of the quotient
here goes through for any choice of the pair of subsets $V_0$,
$V_1$, and we see no way to implement this freedom through
subgroups of $\Z_a$.  Another instance is that the action of the cyclic
groups is not free on the quotient, as the vertices in $V_0$ are fixed
points.  An additional difference is the importance of the eigenfunctions
with Dirichlet conditions on the vertices $V_0$ and $V_1$ and the
specific form of the partition that appears explicitly in the degree
lists of the different quotients but not in the corresponding spectra.
Nevertheless, Brooks result on isospectral (regular) graphs not
arising from Sunada's method (cf. [7]) as well as a
theory of groups acting not freely on graphs might give a hint how to
understand our examples in the spirit of representation theory.

Last but not least the approach using Dirichlet-to-Neumann maps of
subgraphs of quantum graphs in [34] produces
some of our examples of isospectral graphs using different methods; and
it may also apply to discrete graphs.  Moreover, as the authors also
write in [34] it is not obvious how their
approach can be linked with representation theory.

§ MULTISETS AND PARTITIONS

§.§ Multisets

Multisets will be a convenient tool to deal with spectra and degree
lists of graphs.

* A multiset is an ordered pair $(A,m)$ where $A$ is a set
and $\map m A {\N_0=\{0,1,2,\dots\}}$ is a map on $A$.  We say that
$x \in A$ appears $m(x)$-times in $A$.  In order to simplify
notation, we set $m_A(x)=0$ whenever $x \notin A$.

* We often simply refer to $A$ as a multiset without mentioning
the map $m$, and hence refer to the multiplicity map of $A$ as
$m_A$.  Sets $A$ can be considered as multisets by setting
$m_A(x)=1$ whenever $x \in A$ and $0$ otherwise.

* The cardinality of the multiset $A$ is the sum of the
multiplicities of all its elements, i.e.,
$\card A =\sum_{x\in A} m_A(x)$.

* When $A$ is finite (i.e., $\card A < \infty$) we write
$A=\{a_1,a_2,\dots,a_s\}$, and a multiset $(A,m)$ will then also be
written as

\begin{align*}
A&=\lMset a_1^{(m(a_1))},a_2^{(m(a_2))},\dots,a_s^{(m(a_s))} \rMset
\;=\;\lMset \underbrace{a_1,\dots,a_1}_{m(a_1)},
\underbrace{a_2,\dots,a_2}_{m(a_2)},\dots,
\underbrace{a_s,\dots,a_s}_{m(a_s)}\rMset.
\end{align*}

Multisets with elements in $\N$ or $\R$ (or any other linearly
ordered set) can be also seen as ordered lists and sometimes it will
be convenient for the exposition to take this point of view.

* The sum of two multisets $A$ and $B$ denoted as
$ A \uplus B$ is the set $A \cup B$ with multiplicity map
$m_{A_\uplus B}(x):=m_A(x)+m_B(x)$ for all $x \in A \cup B$.

* The difference of two multisets, denoted by $A \ominus B$
is the set $A \setminus B$ with multiplicity map
$m_{A \ominus B}(x) := \max\{m_A(x)-m_B(x),0\}$.

* The $k$-th multiple of a multiset $A$ (with multiplicity
map $m_A$), denoted by $A^{(k)}$, is the multiset $A$ with
multiplicity map $m_{A^{(k)}}(x)=km_A(x)$ for all $x\in A$, i.e.,
the multiplicity of each element in $A^{(k)}$ is multiplied by $k$.

§.§ Partitions of a natural number

The notion of a partition of a natural number will be important for
our construction of isospectral graphs.

A partition of a natural number $r$ of length $s$ (or an $s$-partition $r$ for short) is a multiset $A$ of natural numbers with sum $r$, i.e.,

\begin{equation*}
A= \lMset a_1,a_2,\dots,a_s \rMset
\quadtext{such that}
\sum_{i= 1}^s a_i =r.
\end{equation*}

The number $s$ is called the length of the partition.

We frequently use the following equations

\begin{equation}
\label{eq:part.number}
\sum_{a \in A} 1=s
\qquadtext{and}
\sum_{a \in A} a=r,
\end{equation}

where $A$ is understood as a multiset.

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