# The $M$-matrix group inverse problem
for distance-biregular graphs

Aida Abiad Ángeles Carmona Andrés M. Encinas María José Jiménez
<EMAIL_ADDRESS>Department of Mathematics and Computer Science,
Eindhoven University of Technology, The Netherlands
Department of Mathematics: Analysis, Logic and Discrete Mathematics, Ghent
University, Belgium
Department of Mathematics and Data Science of Vrije Universiteit Brussel,
<EMAIL_ADDRESS>Department of Mathematics, Polytechnic
University of Catalonia<EMAIL_ADDRESS>Department of
Mathematics, Polytechnic University of Catalonia,
<EMAIL_ADDRESS>Department of Mathematics, Polytechnic
University of Catalonia, Spain

###### Abstract

In this paper we provide the group inverse of the combinatorial Laplacian
matrix of distance–biregular graphs using the so–called equilibrium measures
for sets obtained by deleting a vertex. We also show that the two equilibrium
arrays characterizing a distance–biregular graph can be expressed in terms of
the mentioned equilibrium measures. As a consequence of the minimum principle,
we show a characterization of when the group inverse of the combinatorial
Laplacian matrix of a distance–biregular graph is an $M$–matrix.

AMS Classification: 15A09, 05C50, 51E05, 51E12

##  1 Introduction

One problem with the theory of distance-regular graphs is that it does not
apply directly to the graphs of generalised polygons. Godsil and Shawe–Taylor
[24] overcame this difficulty by introducing the class of distance-regularised
graphs, a natural common generalisation. These graphs are shown to either be
distance-regular or distance-biregular. This family includes the generalised
polygons and other interesting graphs. Distance-biregular graphs, which were
introduced by Delorme [20] in 1983, can be viewed as a bipartite variant of
distance-regular graph: the graphs are bipartite and for each vertex there
exists an intersection array depending on the stable component of the vertex.
Thus such graphs are to distance-regular graphs as bipartite regular graphs
are to regular graphs. They also are to non-symmetric association schemes as
distance-regular graphs are to symmetric association schemes. Since their
introduction, distance-biregular graphs have received quite some attention,
see [1, 18, 19, 21, 23, 25, 31] or [14, Chapter 4] for an overview.

In the first part of this paper we obtain the group inverse of the
combinatorial Laplacian matrix of distance-biregular graphs. The group inverse
matrix can be seen in the framework of discrete potential theory as the
Green’s functions associated with the Laplacian operator and it can be used to
deal with diffusion-type problems on graphs, such as chip-firing, load
balancing, and discrete Markov chains. For some graph classes, the group
inverse is known. Instances of it are the work of Urakawa [34], Bendito et al.
[8, 10] or more recently the study of the Green function for forests by Chung
and Zeng [16]. Other generalized inverses, such as the Moore-Penrose inverse,
have been studied. For instance, the Moore-Penrose inverse of the incidence
matrix of several graphs has been investigated by Azami and Bapat [2, 3, 4].
Nevertheless, the problem of computing group inverses still remains wide open
for most graph classes. We show an explicit expression for the group inverse
of the combinatorial Laplacian matrix of a distance-biregular graph in terms
of its intersection numbers. This result, together with the group inverse of a
distance-regular graph found by Bendito, Carmona and Encinas [10], and
independently, by Chung and Yau [17], completes the corresponding study for
distance-regularised graphs.

In matrix theory, the Laplacian matrix is known to be a symmetric $M$-matrix
(a symmetric positive semi-definite matrix with non-positive off-diagonal
elements). Nonnegative matrices and $M$-matrices have become a staple in
contemporary linear algebra, and they arise frequently in its applications.
Such matrices are encountered not only in matrix analysis, but also in
stochastic processes, graph theory, electrical networks, and demographic
models [28]. A fundamental problem related with $M$-matrices is the so-called
inverse $M$-matrix problem, that consists in characterizing all non-negative
matrices whose inverses are $M$-matrices. For singular matrices, the inverse
problem was originally posted by Neumann, Poole and Werner as follows:

######  Question 1.

[22, 32][27, Question 3.3.8] Characterize all singular and irreducible
$M$-matrices for which its group inverse is also an $M$-matrix.

This question has only been answered for specific matrix classes. In the graph
setting, this question has been solved for weighted trees by Kirkland and
Neumann [29], and for distance-regular graphs by Bendito, Carmona and Encinas
[12]. In a more general setting, Question 1 has been investigated for
nonnegative matrices having few eigenvalues by Kirkland and Neumann [28], for
periodic and nonperiodic Jacobi matrices by Chen, Kirkland and Neumann [15]
and for general symmetric $M$-matrices whose underlying graphs are paths by
Bendito, Carmona and Encinas [6] and Carmona, Encinas and Mitjana [7].
Recently, matrices whose group inverses are $M$-matrices were investigated by
Kalauch, Lavanya and Sivakumar [26].

We answer Question 1 for distance-biregular graphs, completing, together with
the known results for distance-regular graphs [12], the characterization of
when the group inverse of the combinatorial Laplacian matrix of a
distance–regularised graph is an $M$–matrix.

##  2 Preliminaries

The triple $\Gamma=(V,E,c)$ denotes a finite network; that is, a finite
connected graph without loops or multiple edges, with vertex set $V$, whose
cardinality equals $n\geq 2$, and edge set $E$, in which each edge $\\{x,y\\}$
has been assigned a conductance $c(x,y)>0$. The conductance can be considered
as a symmetric function $c\colon V\times V\longrightarrow[0,+\infty)$ such
that $c(x,x)=0$ for any $x\in V$ and moreover, $x\sim y$, that is vertex $x$
is adjacent to vertex $y$, iff $c(x,y)>0$. We define the degree function $k$
as

$k(x)=\sum\limits_{y\in V}c(x,y)$

for each $x\in V$. The usual distance from vertex $x$ to vertex $y$ is denoted
by $d(x,y)$ and $D=\max\\{d(x,y):x,y\in V\\}$ stands for the diameter of
$\Gamma$. We denote as $\Gamma_{i}(x)$ the set of vertices at distance $i$
from vertex $x$, $\Gamma_{i}(x)=\\{y:d(x,y)=i\\},\,0\leq i\leq D$ and define
$k_{i}(x)=\big{|}\Gamma_{i}(x)\big{|}$. Then,

$B_{i}(x)=\sum\limits_{j=0}^{i}k_{j}(x)$

is the cardinal of the $i$-ball centered at $x$. The complement of $\Gamma$ is
defined as the graph $\overline{\Gamma}$ on the same vertices such that two
vertices are adjacent iff they are not adjacent in $\Gamma$; that is $x\sim y$
in $\overline{\Gamma}$ iff $c(x,y)=0$. More generally, for any $i=1,\ldots,D$,
we denote by $\Gamma_{i}$ the graph whose vertices are those of $\Gamma$ and
in which two vertices are adjacent iff they are at distance $i$ in $\Gamma$.
Therefore for any $x\in V$, $\Gamma_{i}(x)$ is the set of adjacent vertices to
$x$ in $\Gamma_{i}$. Clearly $\Gamma_{1}$ is the graph subjacent to the
network $\Gamma$ and $\Gamma_{2}=\overline{\Gamma}$ when $D=2$.

The set of real-valued functions on $V$ is denoted by ${\cal C}(V)$. When
necessary, we identify the functions in ${\cal C}(V)$ with vectors in
$\mathbb{R}^{|V|}$ and the endomorphisms of ${\cal C}(V)$ with $|V|$-order
square matrices.

The combinatorial Laplacian or simply the Laplacian of the graph $\Gamma$ is
the endomorphism of ${\cal C}(V)$ that assigns to each $u\in{\cal C}(V)$ the
function

${\cal L}(u)(x)\displaystyle=\sum\limits_{y\in
V}c(x,y)\,\Big{(}u(x)-u(y)\Big{)}=k(x)u(x)-\sum\limits_{y\in
V}c(x,y)\,u(y),\hskip 7.11317ptx\in V.$ (1)

It is well-known that ${\cal L}$ is a positive semi-definite self-adjoint
operator and has $0$ as its lowest eigenvalue whose associated eigenfunctions
are constant. So, ${\cal L}$ can be interpreted as an irreducible, symmetric,
diagonally dominant and singular $M$-matrix, that in the sequel will be
denoted as ${\sf L}$. Therefore, the Poisson equation ${\cal L}(u)=f$ on $V$
has solution iff

$\sum\limits_{x\in V}f(x)=0$

and, when this happens, there exists a unique solution $u\in{\cal C}(V)$ such
that $\sum\limits_{x\in V}u(x)=0,$ see [8].

The Green operator is the linear operator ${\cal G}:{\cal
C}(V)\longrightarrow{\cal C}(V)$ that assigns to any $f\in{\cal C}(V)$ the
unique solution of the Poisson equation ${\cal
L}(u)=f-\frac{1}{n}\sum\limits_{x\in V}f(x)$ such that $\sum\limits_{x\in
V}u(x)=0$. It is easy to prove that ${\cal G}$ is a positive semi-definite
self-adjoint operator and has $0$ as its lowest eigenvalue whose associated
eigenfunctions are constant. Moreover, if ${\cal P}$ denotes the projection on
the subspace of constant functions then,

${\cal L}\circ{\cal G}={\cal G}\circ{\cal L}={\cal I}-{\cal P}.$

In addition, we define the Green function as $G:V\times
V\longrightarrow\mathbb{R}$ given by $G(x,y)={\cal G}(\varepsilon_{y})(x)$,
where $\varepsilon_{y}$ stands for the Dirac function at $y$. Therefore,
interpreting ${\cal G}$, or $G$, as a matrix it is nothing else but ${\sf
L}^{\\#}$ the group inverse inverse of ${\sf L}$, that coincides with its
Moore-Penrose inverse. In consequence, ${\sf L^{\\#}}$ is a _$M$ -matrix_ iff
${\sf L^{\\#}}(x,y)\leq 0$ for any $x,y\in V$ with $x\neq y$ and then ${\sf
L^{\\#}}$ can be identified with the combinatorial Laplacian matrix of a new
connected network with the same vertex set, that we denote by $\Gamma^{\\#}$.

From now on we will say that a network $\Gamma$ has the $M$-property iff ${\sf
L}^{\\#}$ is an $M$-matrix; that is, if ${\sf L}$ provides an answer to
Question 1.

In [8] it was proved that for any $y\in V$, there exists a unique
$\nu^{y}\in{\cal C}(V)$ such that $\nu^{y}(y)=0$, $\nu^{y}(x)>0$ for any
$x\not=y$ and satisfying

${\cal L}(\nu^{y})={\sf 1}-n\varepsilon_{y}\hskip 7.11317pt\hbox{on $V$}.$ (2)

We call $\nu^{y}$ the equilibrium measure of $V\setminus\\{y\\}$ and then we
define capacity as the function ${\sf cap}\in{\cal C}(V)$ given by ${\sf
cap}(y)=\sum\limits_{x\in V}\nu^{y}(x)$.

Following the ideas in [8, 12, 34], we define, for any $y\in V$, the
equilibrium array for $y$ as the set $\\{\nu^{y}(x):x\in V\\}$ of different
values taken by the equilibrium measure of $y$, and we consider the length of
the equilibrium array to be $\ell(y)=\big{|}\big{\\{}\nu^{y}(x):x\in
V\setminus\\{y\\}\big{\\}}\big{|}$. Since $\Gamma$ is connected and $n\geq 2$,
we obtain that $\ell(y)\geq 1$ for any $y\in V$. On the other hand, since
$0=\nu^{y}(y)$ we obtain that $\\{\nu^{y}(x):x\in
V\\}=\\{q_{i}(y):i=0,\ldots,\ell(y)\\}$, where
$0=q_{0}(y)<q_{1}(y)<\cdots<q_{\ell(y)}(y)$. In addition, given $y\in V$ for
any $i=0,\ldots,\ell(y)$, we define $m_{i}(y)=\big{|}\\{x\in
V:\nu^{y}(x)=q_{i}(y)\\}\big{|}$. Clearly, for any $y\in V$ we have that

$m_{0}(y)=1,\,\,\,n=\sum\limits_{i=0}^{\ell(y)}m_{i}(y),\,\,\,\mbox{and }{\sf
cap}(y)=\sum\limits_{i=1}^{\ell(y)}m_{i}(y)q_{i}(y).$

In [11, Proposition 3.12] it was shown that, for any $y\in V$, the equilibrium
measure (and hence the equilibrium array) reflects the graph depth from $y$,
since

$\nu^{y}(x)=q_{i}(y)\Longrightarrow d(x,y)\leq i$ (3)

and hence

$\sum\limits_{j=0}^{i}m_{j}(y)\leq B_{i}(y)$

for any $0\leq i\leq D$. In particular, (3) implies that if
$\nu^{y}(x)=q_{1}(y)$ then $x\sim y$; that is, that the minimum values of the
equilibrium measure for $y$ are attained at vertices adjacent to $y$ (in fact
this a formulation of the so-called minimum principle).

In general, when $d(x,y)=i$, Property (3) only assures that $\nu^{y}(x)\geq
q_{i}(y)$, but the inequality can be strict. In particular the length of some
equilibrium arrays could be greater than $D$.

######  Example 2.

To illustrate the above statements, consider the complete graph $K_{3}$ with
vertex set $V=\\{x_{1},x_{2},x_{3}\\}$ and conductances
$c_{1}=c(x_{1},x_{2})$, $c_{2}=c(x_{2},x_{3})$ and $c_{3}=c(x_{3},x_{1})$.
Then,

$\begin{array}[]{rlrlrl}\nu^{x_{1}}(x_{2})=&\dfrac{2c_{2}+c_{3}}{c_{1}c_{2}+c_{2}c_{3}+c_{3}c_{1}},&\nu^{x_{1}}(x_{3})=&\dfrac{2c_{2}+c_{1}}{c_{1}c_{2}+c_{2}c_{3}+c_{3}c_{1}},&{\sf
cap}(x_{1})=&\dfrac{4c_{2}+c_{1}+c_{3}}{c_{1}c_{2}+c_{2}c_{3}+c_{3}c_{1}}\\\\[11.62495pt]
\nu^{x_{2}}(x_{1})=&\dfrac{2c_{3}+c_{2}}{c_{1}c_{2}+c_{2}c_{3}+c_{3}c_{1}},&\nu^{x_{2}}(x_{3})=&\dfrac{2c_{3}+c_{1}}{c_{1}c_{2}+c_{2}c_{3}+c_{3}c_{1}},&{\sf
cap}(x_{1})=&\dfrac{4c_{3}+c_{1}+c_{2}}{c_{1}c_{2}+c_{2}c_{3}+c_{3}c_{1}}\\\\[11.62495pt]
\nu^{x_{3}}(x_{1})=&\dfrac{2c_{1}+c_{2}}{c_{1}c_{2}+c_{2}c_{3}+c_{3}c_{1}},&\nu^{x_{3}}(x_{2})=&\dfrac{2c_{1}+c_{3}}{c_{1}c_{2}+c_{2}c_{3}+c_{3}c_{1}},&{\sf
cap}(x_{1})=&\dfrac{4c_{1}+c_{2}+c_{3}}{c_{1}c_{2}+c_{2}c_{3}+c_{3}c_{1}}.\end{array}$

So, $D=1$, but $\ell(x_{1})=1$ iff $c_{1}=c_{3}$, $\ell(x_{2})=1$ iff
$c_{1}=c_{2}$ and $\ell(x_{3})=1$ iff $c_{2}=c_{3}$.

The group inverse of the Laplacian matrix and the equilibrium measures provide
an equivalent information about the network structure, since the expression of
${\sf L}^{\\#}$ can be obtained from equilibrium measures and conversely.
Specifically, see [8, Proposition 3.9], the group inverse ${\sf L}^{\\#}$ is
given by

${\sf L}^{\\#}(x,y)=\dfrac{1}{n^{2}}\big{(}{\sf cap}(y)-n\,\nu^{y}(x)\big{)}$
(4)

and this equality also implies that ${\sf cap}(y)=n^{2}{\sf L}^{\\#}(y,y)$ and
that

$\nu^{y}(x)=n\,\big{(}{\sf L}^{\\#}(y,y)-{\sf L}^{\\#}(x,y)\big{)},\hskip
7.11317ptx,y\in V.$ (5)

In addition, the symmetry of the group inverse leads to the following relation
for the equilibrium measures

$\nu^{y}(x)-\nu^{x}(y)=\frac{1}{n}\big{(}{\sf cap}(y)-{\sf
cap}(x)\big{)}=n\,\big{(}{\sf L}^{\\#}(y,y)-{\sf L}^{\\#}(x,x)\big{)},\hskip
7.11317ptx,y\in V.$ (6)

From (4) the minimum principle states that a network $\Gamma$ has the
$M$-property iff for any $y\in V$

${\sf cap}(y)\leq n\nu^{y}(x)\quad\hbox{ for any }x\sim y,$ (7)

see [12, Theorem 1]. In this case, $\overline{\Gamma}$ is a subgraph of the
subjacent graph of $\Gamma^{\\#}$. In fact, to achieve the $M$-property it is
sufficient to satisfy that

$\sum\limits_{i=1}^{\ell(y)}m_{i}(y)q_{i}(y)\leq nq_{1}(y)$

for any $y\in V$. Since this inequality trivially holds when $\ell(y)=1$, and
assuming the common agreement that empty sum equals $0$, we have that $\Gamma$
has the $M$-property iff

$\sum\limits_{i=2}^{\ell(y)}m_{i}(y)\big{(}q_{i}(y)-q_{1}(y)\big{)}\leq
q_{1}(y)$ (8)

for any $y\in V$. Therefore, when $\ell(y)=1$ for any $y\in V$, then $\Gamma$
is a complete network and moreover satisfies the $M$-property. As Example 2
shows, a complete network does not necessarily satisfy the $M$-property:
$K_{3}$ has the $M$-property if and only if $3\max\\{c_{1},c_{2},c_{3}\\}\leq
2(c_{1}+c_{2}+c_{3})$. In particular, if $c_{1}=c_{2}=c_{3}$, then $K_{3}$ has
the $M$-property, but if for instance $c_{3}>2(c_{1}+c_{2})$, then $K_{3}$
does not satisfy the $M$-property.

##  3 Group inverse for distance-biregular graphs

We say that the graph $\Gamma=(V,E)$ is _semiregular_ if $\Gamma$ is bipartite
with $V=V_{0}\cup V_{1}$ and there are numbers $k_{0}$ and $k_{1}$ such that
each vertex in $V_{0}$ has $k_{0}$ neighbors and each vertex in $V_{1}$ has
$k_{1}$ neighbors. In this case, we define $D_{\ell}=\max\\{d(x,y):\,\,y\in
V,x\in V_{\ell}\\}$, $\ell=0,1$. Moreover, for $\ell=0,1$, we denote by
$\bar{\ell}=1-\ell.$ In the sequel without loss of generality we always
suppose that $1\leq D_{0}\leq D_{1}$.

A connected graph $\Gamma$ is a _distance-biregular graph_ if $\Gamma$ is
semiregular and for any two vertices $x$ and $y$ at distance $i$, the numbers
$|\Gamma_{i-1}(x)\cap\Gamma_{1}(y)|$ and $|\Gamma_{i+1}(x)\cap\Gamma_{1}(y)|$
only depend on $i$ and on the stable set where $x$ is.

Examples of distance-biregular graphs are the subdivision graph of minimal
$(k,g)$-cages. In particular, the subdivision graph of the Petersen graph is a
distance-biregular graph. Also, any bipartite distance-regular graph is a
distance-biregular graph with $k_{0}=k_{1}$.

For $x\in V_{\ell},$ $\ell=0,1$, we define the intersection numbers by
$c_{\ell,i}=|\Gamma_{i-1}(x)\cap\Gamma_{1}(y)|$ and
$b_{\ell,i}=|\Gamma_{i+1}(x)\cap\Gamma_{1}(y)|$, $i=0,\ldots,D_{\ell}$, with
the usual agreement $c_{\ell,0}=b_{\ell,D_{\ell}}=0$. Clearly, for $\ell=0,1$
it is satisfied that $b_{\ell,0}=k_{\ell}$, $c_{\ell,1}=1$,
$b_{\ell,1}=k_{\bar{\ell}}-1$, $\ell=0,1$ and more generally for any
$i\in\\{0,\ldots,D_{\ell}\\}$ the following holds

$\displaystyle
c_{\ell,i}+b_{\ell,i}=\left\\{\begin{array}[]{cc}k_{\ell}&\text{if }i\text{ is
even,}\\\ k_{\bar{\ell}}&\text{if }i\text{ is odd.}\\\ \end{array}\right.$

Therefore, a distance-biregular graph has a double intersection array which
will be denoted by

$\displaystyle\big{\\{}k_{\ell};c_{\ell,1},\ldots
c_{\ell,D_{\ell}}\big{\\}},\,\,\,\ell=0,1.$

If, for $i\in\\{0,\ldots,D_{\ell}\\}$ and $x\in V_{\ell}$ we define
$k_{\ell,i}=|\Gamma_{i}(x)|$ and $B_{\ell,i}=\sum\limits_{j=0}^{i}k_{\ell,j}$.
Then, $k_{\ell,0}=1$, $k_{\ell,1}=k_{\ell}$ and $n=B_{\ell,D_{\ell}}$,
$\ell=0,1$ and moreover the following relationships hold, see the Lemmas
2.1-2.8 in [1, Section 2.1] and the references therein.

######  Lemma 3.

If $\Gamma$ is a distance-biregular graphs with intersection arrays
$\big{\\{}k_{\ell};c_{\ell,1},\ldots c_{\ell,D_{\ell}}\big{\\}}$, $\ell=0,1$.
Then,

  * $(i)$

$0\leq D_{1}-D_{0}\leq 1$ and when $D_{1}=D_{0}+1$, then $D_{0}$ is odd.

  * $(ii)$

For $\ell=0,1$,

$\displaystyle
k_{\ell,i}=\prod_{j=0}^{i-1}\frac{b_{\ell,j}}{c_{\ell,j+1}},\quad
i=0,\ldots,D_{\ell}$

and hence,

$k_{\ell,i}b_{\ell,i}=k_{\ell,i+1}c_{\ell,i+1},\quad i=0,\ldots,D_{\ell}.$

  * $(iii)$

$k_{0}k_{1,2i+1}=k_{1}k_{0,2i+1}$, for any
$i=0,\ldots,\lfloor\frac{D_{0}-1}{2}\rfloor$.

  * $(iv)$

$c_{0,2i}c_{0,2i+1}=c_{1,2i}c_{1,2i+1}$ and
$b_{0,2i-1}b_{0,2i}=b_{1,2i-1}b_{1,2i}$ for any
$i=1,\ldots,\lfloor\frac{D_{0}-1}{2}\rfloor$.

  * $(v)$

For $\ell=0,1$, $1\leq c_{\ell,i}\leq c_{\bar{\ell},i+1}$ and $b_{\ell,i}\geq
b_{\bar{\ell},i+1}$ for any $i=0,\ldots,D_{\bar{\ell}}-1$. Moreover,
$b_{\ell,i}\geq c_{\bar{\ell},i+1},\hskip
7.11317pti=1,\ldots,D_{\bar{\ell}}-2$.

  * $(vi)$

For $\ell=0,1$, $c_{\ell,2}\leq{c_{\ell,3}-1\choose c_{\bar{\ell},2}-1}$.

  * $(vii)$

For $\ell=0,1$, if $i+j$ is even and $i+j\leq D_{\ell}$, then $c_{\ell,i}\leq
b_{\ell,j}.$

  * $(viii)$

For $\ell=0,1$, if $i+j$ is odd and $i+j\leq D_{0}$, then $c_{\ell,i}\leq
b_{\bar{\ell},j}$ and $c_{\bar{\ell},i}\leq b_{\ell,j}$.

The properties $(ii)$, $(iii)$ and $(iv)$ imply that for $\ell=0,1$, the
intersection numbers $\\{c_{\ell,i},b_{\ell,i}\\}$ are determined by the
intersection numbers $\\{c_{\bar{\ell},i},b_{\bar{\ell},i}\\}$. In particular
both sequences are the same iff $k_{0}=k_{1}$ and in this case $\Gamma$ is a
(bipartite) distance-regular graph.

######  Lemma 4.

[1, Corollary 2.11] Let $\Gamma$ be a distance-bipartite regular graph. We can
assume w.l.o.g. that one of the following holds

  1. 1.

$D_{0}=D_{1}$ and $k_{0}=k_{1}$; so $\Gamma$ is a bipartite distance-regular
graph.

  2. 2.

$D_{0}=D_{1}-1$ is odd and $k_{0}>k_{1}$.

  3. 3.

$D_{0}=D_{1}$ is even and $k_{0}>k_{1}$.

We display some preliminary results about the intersection parameters of a
distance-biregular graphs, whose proofs are omitted since they follow
trivially from [1].

######  Lemma 5.

If $k_{0}>k_{1}$, then

$\begin{array}[]{rl}\dfrac{b_{1,i}}{b_{0,i}}<\dfrac{k_{1}}{k_{0}}<\dfrac{c_{1,i}}{c_{0,i}},&\mbox{
if }i\mbox{ is even},\\\\[8.61108pt]
\dfrac{b_{0,i}}{b_{1,i}}<\dfrac{k_{1}}{k_{0}}<\dfrac{c_{0,i}}{c_{1,i}},&\mbox{
if }i\mbox{ is odd}.\end{array}$

The result provides an explicit expression of the equilibrium measure for sets
$V\setminus\\{y\\},\forall y\in V$ of distance-biregular graphs.

######  Proposition 6.

Let $\Gamma$ be a distance-biregular graph with $V=V_{0}\cup V_{1}$. Then, for
any $\ell=0,1$, there exists an array $q_{\ell}$ of length $D_{\ell}$ such
that if $x\in V_{\ell}$, for any $y\in V$ it holds

$\nu^{x}(y)=q_{\ell,m}\,\,\Longleftrightarrow\,\,d(x,y)=m,\,\,m=0,\ldots,D_{\ell}.$

Moreover,

$q_{\ell,m}=\displaystyle\sum\limits_{j=0}^{m-1}\dfrac{n-B_{\ell,j}}{k_{\ell,j}b_{\ell,j}}=\sum\limits_{j=1}^{m}\dfrac{n-B_{\ell,j-1}}{k_{\ell,j}c_{\ell,j}}.$

In particular, $q_{\ell,1}=\dfrac{n-1}{k_{\ell}}$ and
$q_{\ell,2}=\dfrac{n-1}{k_{\ell}}+\dfrac{n-1-k_{\ell}}{k_{\ell}(k_{\bar{\ell}}-1)}.$

###### Proof.

Take $x\in V_{\ell}$ with $\ell=0,1$. Assume that the value $\nu^{x}(y)$
depends only on the distance from $x$ to $y$, that is, there exists
$q_{\ell,i}$, $i=1,\ldots,D_{\ell}$ such that
$\nu^{x}(y)=q_{\ell,i}\,\,\Longleftrightarrow\,\,d(x,y)=i.$ Moreover, we
define $q_{{}_{\ell,D_{\ell}+1}}=0$. Note that, since the equilibrium system
${\cal L}\nu^{x}(y)=1$ for all $y\in V\setminus\\{x\\}$ has a unique solution,
then if with our hypothesis we can solve the system, such solution must
correspond to the equilibrium measure $\nu^{x}(y)=q_{\ell,i}$.

In our case, ${\cal L}\nu^{x}(y)=1$ for all $y\in V\setminus\\{x\\}$ is
equivalent to the system

$(b_{\ell,i}+c_{\ell,i})q_{\ell,i}-c_{\ell,i}q_{\ell,i-1}-b_{\ell,i}q_{\ell,i+1}=1,\,\,\,\,i=1,\ldots,D_{\ell}$

where $\ell=0,1$. Multiplying by $k_{\ell,i}$, we obtain

$k_{\ell,i}c_{\ell,i}(q_{\ell,i}-q_{\ell,i-1})-k_{\ell,i}b_{\ell,i}(q_{\ell,i+1}-q_{\ell,i})=k_{\ell,i},\,\,\,\,i=1,\ldots,D_{\ell}.$

Since $k_{\ell,i}c_{\ell,i}=k_{\ell,i-1}b_{\ell,i-1}$ and denoting
$\gamma_{\ell,i}=k_{\ell,i}b_{\ell,i}(q_{\ell,i+1}-q_{\ell,i})$, then

$\gamma_{\ell,{i-1}}-\gamma_{\ell,i}=k_{\ell,i},\qquad\text{for
}i=1,\ldots,D_{\ell}.$

Observing that $\gamma_{{}_{\ell,D_{\ell}}}=0$, then summing up

$n-B_{\ell,j}=\sum\limits_{i=j+1}^{D_{\ell}}k_{\ell,i}=\gamma_{\ell,j}-\gamma_{{}_{\ell,D_{\ell}}}=\gamma_{\ell,j}$

for $j=0,\ldots,D_{\ell}-1$, it follows

$q_{\ell,i+1}-q_{\ell,i}=\frac{n-B_{\ell,i}}{k_{\ell,i}b_{\ell,i}},\qquad\text{for
}i=0,\ldots,D_{\ell}-1.$

Finally, since $q_{\ell,0}=0$, it follows

$q_{\ell,m}=\sum\limits_{j=0}^{m-1}\frac{n-B_{\ell,j}}{k_{\ell,j}b_{\ell,j}},\qquad\text{for
}m=0,\ldots,D_{\ell}.$

The expression for $q_{\ell,m}$ in terms of $c_{\ell,i}$ follows from Lemma 3
$(ii)$. ∎

The above proposition motives the definition of equilibrium arrays for a
distance-biregular graph. If $\Gamma$ is a distance-biregular graph, we call
_equilbrium arrays_ to the values $q_{\ell,i},\,\,\ell=0,1$ and
$i=0,\ldots,D_{\ell}$. Denote $m_{\ell,i}=\big{|}\\{y\in
V:\nu^{x}(y)=q_{\ell,i}\\}\big{|}.$

######  Corollary 7.

Let $\Gamma$ be a distance-biregular graph with $y\in V_{\ell}$ and $x\in
V_{\hat{\ell}}$, $\ell,\hat{\ell}=0,1$. Then,

$q_{\ell,d(x,y)}=q_{\hat{\ell},d(x,y)}+(n-1)\left(\dfrac{1}{k_{\ell}}-\dfrac{1}{k_{\hat{\ell}}}\right).$

###### Proof.

From (6) we know that $\nu^{y}(x)-\nu^{x}(y)=\frac{1}{n}\big{(}{\sf
cap}(y)-{\sf cap}(x)\big{)}$. On the other hand, ${\sf cap}(y)={\sf cap}(z)$
for any $z\in V_{\ell}$ and ${\sf cap}(x)={\sf cap}(w)$ for any $w\in
V_{\hat{\ell}}.$ So, for $\ell\not=\hat{\ell}$, we can choose $z\in V_{\ell}$
and $w\in V_{\hat{\ell}}$ such that $d(z,w)=1$, then

$\frac{1}{n}\big{(}{\sf cap}(y)-{\sf cap}(x)\big{)}=\frac{1}{n}\big{(}{\sf
cap}(z)-{\sf
cap}(w)\big{)}=q_{\ell,1}-q_{\hat{\ell},1}=(n-1)\left(\dfrac{1}{k_{\ell}}-\dfrac{1}{k_{\hat{\ell}}}\right),$

which implies that

$q_{\ell,d(x,y)}=q_{\hat{\ell},d(x,y)}+(n-1)\left(\dfrac{1}{k_{\ell}}-\dfrac{1}{k_{\hat{\ell}}}\right).$

If $\ell=\hat{\ell}$, the result trivially holds. ∎

As an straightforward application of Proposition 6 we can find the
intersection array of a distance-biregular graph in terms of the equilibrium
arrays, analogously as it was done in [11, Proposition 4.5] for distance-
regular graphs.

######  Proposition 8.

Let $\Gamma$ be a distance-biregular graph with equilibrium arrays
$q_{\ell,i}$ for $\ell=0,1$ and $i=0,\ldots,D_{\ell}$. Then, for any
$i=0,\ldots,D_{\ell}-1$, it holds

$\displaystyle k_{\ell,i}=m_{\ell,i},$ $\displaystyle
b_{\ell,i}=\frac{1}{m_{\ell,i}(q_{\ell,i+1}-q_{\ell,i})}\sum_{j=i+1}^{D_{\ell}}m_{\ell,j},$
$\displaystyle
c_{\ell,i+1}=\frac{1}{m_{\ell,i+1}(q_{\ell,i+1}-q_{\ell,i})}\sum_{j=i+1}^{D_{\ell}}m_{\ell,j}.$

The computation of the equilibrium measure is usually done using linear
programming [11]. In this regard, Proposition 8 provides a tool to calculate
the intersection arrays of a distance-biregular graph solving one linear
system.

Another application of the equilibrium measure concerns the estimation of the
effective resistance of a resistive electrical network. As a consequence of
Proposition 6, we show the the effective resistance of distance-biregular
graphs.

######  Corollary 9.

Let $\Gamma$ be a distance-biregular graph with $y\in V_{\ell}$ and $x\in
V_{\hat{\ell}}$, $\ell,\hat{\ell}=0,1$. Then, the effective resistance bewteen
$x,y$ is

$R(x,y)=\frac{2}{n}q_{\ell,d(x,y)}+\frac{(n-1)}{n}\left(\dfrac{1}{k_{\hat{\ell}}}-\dfrac{1}{k_{\ell}}\right).$

###### Proof.

The result follows from Corollary 7 taking into account that
$R(x,y)=\dfrac{\nu^{x}(y)+\nu^{y}(x)}{n}$, see [9]. ∎

The next results shows the group inverse of the Laplacian of a distance-
biregular graphs in terms of the intersection arrays.

######  Theorem 10.

Let $\Gamma$ be a distance-biregular graph. Then, for each $y\in V_{\ell}$
with $\ell=0,1$, the group inverse of ${\sf L}$ is given by

${\sf
L}^{\\#}(x,y)=\displaystyle\displaystyle\dfrac{1}{n}\sum\limits_{j=d(x,y)+1}^{D_{\ell}}\frac{n-B_{\ell,j-1}}{k_{\ell,j}c_{\ell,j}}-\displaystyle\dfrac{1}{n^{2}}\sum\limits_{j=1}^{D_{\ell}}\frac{B_{\ell,j-1}(n-B_{\ell,j-1})}{k_{\ell,j}c_{\ell,j}}.$

###### Proof.

From (4), we know that ${\sf L}^{\\#}(x,y)=\frac{1}{n^{2}}({\sf
cap}(y)-n\nu^{y}(x)).$ Take $y\in V_{\ell}$ with $\ell=0,1$. Now, using
Proposition 6,

$\displaystyle{\sf cap}(y)$ $\displaystyle=\displaystyle\sum\limits_{x\in
V}\nu^{y}(x)=\displaystyle\sum\limits_{m=0}^{D_{\ell}}k_{\ell,m}q_{\ell,m}=\sum\limits_{m=0}^{D_{\ell}}k_{\ell,m}\sum\limits_{j=1}^{m}\dfrac{n-B_{\ell,j-1}}{k_{\ell,j}c_{\ell,j}}$
$\displaystyle=\displaystyle\sum\limits_{j=1}^{D_{\ell}}\sum\limits_{m=j}^{D_{\ell}}\frac{k_{\ell,m}(n-B_{\ell,j-1})}{k_{\ell,j}c_{\ell,j}}=\displaystyle\sum\limits_{j=1}^{D_{\ell}}\frac{(n-B_{\ell,j-1})^{2}}{k_{\ell,j}c_{\ell,j}}.$

Therefore,

$\displaystyle{\sf L}^{\\#}(x,y)=$
$\displaystyle\displaystyle\sum\limits_{j=1}^{D_{\ell}}\frac{(n-B_{\ell,j-1})^{2}}{n^{2}k_{\ell,j}c_{\ell,j}}-\sum\limits_{j=1}^{d(x,y)}\frac{n-B_{\ell,j-1}}{nk_{\ell,j}c_{\ell,j}}$
$\displaystyle=$
$\displaystyle\displaystyle\sum\limits_{j=d(x,y)+1}^{D_{\ell}}\frac{n-B_{\ell,j-1}}{nk_{\ell,j}c_{\ell,j}}-\sum\limits_{j=1}^{D_{\ell}}\frac{B_{\ell,j-1}(n-B_{\ell,j-1})}{n^{2}k_{\ell,j}c_{\ell,j}}.$

∎

######  Remark 11.

Observe that, since the intersection numbers of a distance-biregular graph are
related, from Theorem 10, the expression of the group inverse is equivalent to

${\sf
L}^{\\#}(x,y)=\displaystyle\dfrac{1}{n}\sum\limits_{j=d(x,y)}^{D_{\ell}-1}\frac{n-B_{\ell,j}}{k_{\ell,j}b_{\ell,j}}-\displaystyle\dfrac{1}{n^{2}}\sum\limits_{j=0}^{D_{\ell}-1}\frac{B_{\ell,j}(n-B_{\ell,j})}{k_{\ell,j}b_{\ell,j}}.$

######  Example 12.

As an application of Theorem 10 we obtain the group inverse of the Laplacian
of a complete bipartite graph using its parameters:

$\left.\begin{array}[]{ll}D_{0}=2,k_{0},c_{0,1}=1,c_{0,2}=k_{0},b_{0,0}=k_{0},b_{0,1}=k_{1}-1,\\\\[4.30554pt]
D_{1}=2,k_{1},c_{1,1}=1,c_{1,2}=k_{1},b_{1,0}=k_{1},b_{1,1}=k_{0}-1,\end{array}\right\\}\Longrightarrow
n=k_{0}+k_{1}.$

Take $x,\hat{x}\in V_{0}$, $\hat{x}\not=x$ and $y,\hat{y}\in V_{1}$,
$\hat{y}\not=y$. Then,

$\begin{array}[]{rl}{\sf
L}^{\\#}(x,x)=&\dfrac{1}{n^{2}}\bigg{[}\dfrac{(k_{0}+k_{1}-1)^{2}}{k_{0}}+\dfrac{k_{1}-1}{k_{0}}\bigg{]}=\dfrac{(n-1)^{2}+n-k_{0}-1}{n^{2}k_{0}}=\dfrac{n^{2}-n-k_{0}}{k_{0}n^{2}},\\\\[12.91663pt]
{\sf
L}^{\\#}(y,y)=&\dfrac{1}{n^{2}}\bigg{[}\dfrac{(k_{0}+k_{1}-1)^{2}}{k_{1}}+\dfrac{k_{0}-1}{k_{1}}\bigg{]}=\dfrac{n^{2}-n-k_{1}}{k_{1}n^{2}},\\\\[12.91663pt]
{\sf
L}^{\\#}(y,x)=&\dfrac{n^{2}-n-k_{0}}{k_{0}n^{2}}-\dfrac{n(n-1)}{k_{0}n^{2}}=-\dfrac{1}{n^{2}}={\sf
L}^{\\#}(x,y),\\\\[12.91663pt] {\sf
L}^{\\#}(\hat{x},x)=&\dfrac{n^{2}-n-k_{0}}{k_{0}n^{2}}-\dfrac{n(n-1)}{k_{0}n^{2}}-\dfrac{n}{k_{0}n^{2}}=-\dfrac{(n+k_{0})}{k_{0}n^{2}},\\\\[12.91663pt]
{\sf L}^{\\#}(\hat{y},y)=&-\dfrac{(n+k_{1})}{k_{1}n^{2}}.\end{array}$

Observe that for a complete bipartite graph, it holds that ${\sf L}^{\\#}$ is
always an $M$-matrix. The above expression is valid when $D_{0}=D_{1}=1$, and
$D_{0}=1$ and $D_{1}=2$; that is, for the star graph.

##  4 Distance-biregular graphs with the $M$-property

In this section, we answer Question 1 for distance-biregular graphs,
completing, together with the known results for distance-regular graphs [12],
the characterization of when the group inverse of the combinatorial Laplacian
matrix of a distance-regularised graph is an $M$-matrix.

######  Proposition 13.

Let $\Gamma$ be a distance-biregular graph. Then, $\Gamma$ has the
$M$-property if and only if, it holds

$\displaystyle\displaystyle\sum\limits_{j=1}^{D_{0}-1}\frac{1}{k_{0,j}b_{0,j}}\Big{(}\sum\limits_{i=j+1}^{D_{0}}k_{0,i}\Big{)}^{2}\leq\frac{(n-1)}{k_{0}}.$

###### Proof.

We know that a graph $\Gamma$ satisfies the $M$-property if and only if the
entries of ${\sf L}^{\\#}(x,y)\leq 0$ for all $x\sim y$. The result follows
from using that

$k_{0,0}=1,\,\,b_{0,0}=k_{0},\,\,\mbox{ and
}\,\,\sum\limits_{i=1}^{D_{0}}k_{0,i}=n-1.$

∎

######  Remark 14.

The condition from Proposition 13 is equivalent to

$\displaystyle\displaystyle\sum\limits_{j=1}^{D_{1}-1}\frac{1}{k_{1,j}b_{1,j}}\Big{(}\sum\limits_{i=j+1}^{D_{1}}k_{1,i}\Big{)}^{2}\leq\frac{(n-1)}{k_{1}}.$
(9)

Using Proposition 13 we can also obtain the following necessary condition for
a distance-biregular graph having the $M$-property.

######  Corollary 15.

If $\Gamma$ is a distance-biregular graph with the $M$-property and $D_{0}\geq
2$, then

$n<2k_{1}+k_{0}.$

###### Proof.

Since $D_{1}\geq D_{0}\geq 2$, from Proposition 13 we obtain that

$\displaystyle\frac{1}{k_{1,1}b_{1,1}}\Big{(}\sum\limits_{i=2}^{D_{1}}k_{1,i}\Big{)}^{2}\leq\displaystyle\sum\limits_{j=1}^{D_{1}-1}\frac{1}{k_{1,j}b_{1,j}}\Big{(}\sum\limits_{i=j+1}^{D_{1}}k_{1,i}\Big{)}^{2}\leq\frac{(n-1)}{k_{1}}.$

Now, observing that

$\dfrac{1}{k_{1,1}b_{1,1}}\Big{(}\sum\limits_{i=2}^{D_{1}}k_{1,i}\Big{)}^{2}=\dfrac{(n-k_{1}-1)^{2}}{k_{1}b_{1,1}}$

we get

$\displaystyle(n-k_{1}-1)^{2}\leq(n-1)b_{1,1}=(n-1)(k_{0}-1)\Longleftrightarrow$
$\displaystyle(n-1)^{2}-2k_{1}(n-1)+k_{1}^{2}\leq(n-1)(k_{0}-1)\Longleftrightarrow$
$\displaystyle n-2k_{1}+\dfrac{k_{1}^{2}}{n-1}-k_{0}\leq 0,$

and since $\frac{k_{1}^{2}}{n-1}>0$, the result follows. ∎

Note that the inequality $n<2k_{1}+k_{0}$ turns out to be a strong restriction
for a distance-biregular graph to have the $M$-property. Observe that such
condition implies that the distance-biregular graph needs to be quite dense.

The following result generalizes the above observation by showing that only
distance-biregular graphs with small $D_{\ell}$ can satisfy the $M$-property.
A related result appeared in [12, Proposition 5], where it was shown that the
diameter of a distance-regular graphs with the $M$-property must be at most
$3$.

######  Proposition 16.

If $\Gamma$ is a distance-biregular graph with the $M$-property, then
$D_{1}\leq 4$ and $D_{0}\leq 3$.

###### Proof.

By means of a contradiction, assume $D_{0},D_{1}\geq 4$. Then,

$1+k_{\ell}+k_{\ell,2}+k_{\ell,3}<1+k_{\ell}+k_{\ell,2}+k_{\ell,3}+k_{\ell,4}\leq
n.$

We can assume that $k_{0}>k_{1},$ since otherwise $\Gamma$ is a bipartite
distance-regular graphs and hence $D_{0}=D_{1}\leq 3$, see [12, Proposition
5]. Then,

$k_{0,2}=k_{0}\dfrac{b_{0,1}}{c_{0,2}}\geq
k_{0}\dfrac{c_{1,2}}{c_{0,2}}>k_{0}\dfrac{k_{1}}{k_{0}}=k_{1}.$

On the other hand, since $b_{0,1}\geq c_{0,3}$ and $b_{0,2}\geq c_{0,2}$, we
obtain that

$k_{0,3}=k_{0}\dfrac{b_{0,1}b_{0,2}}{c_{0,2}c_{0,3}}\geq k_{0}.$

Finally, from Corollary 15 it follows that $n+1<1+2k_{0}+k_{1}\leq n$, a
contradiction. ∎

As an application of Proposition 16, we classify distance-biregular graphs
having the $M$-property. We follow the notation from [1].

Case 1:

$D_{0}=D_{1}=1$. This case corresponds to a digon that is a distance-regular
and has the $M$-property.

Case 2:

$D_{0}=1$, $D_{1}=2$. This case corresponds to star graphs, which are known to
have the $M$-property, see [13, 29] or Example 12.

Case 3:

$D_{0}=D_{1}=2$. This case corresponds to a bipartite complete graph (see
Example 12).

Case 4:

$D_{0}=D_{1}=3$. This case corresponds to a bipartite distance-regular graph
[1, Section 5.1], and thus it was already studied in [12]. In this case, the
intersection array is $\\{k,k-1,k-\mu;1,\mu,k\\}$, where $1\leq\mu\leq k-1$
and $\mu$ divides $k(k-1)$. They are antipodal iff $\mu=k-1$. Otherwise, they
are the incidence graphs of nontrivial square
$2-\Big{(}\frac{n}{2},k,\mu\Big{)}$ designs. Therefore, $k-\mu$ must be a
square, see [14, Th. 1.10.4].

######  Proposition 17.

[12, Proposition 13] A bipartite distance-regular graph with $D=3$ satisfies
the $M$-property if and only if

$\dfrac{4k}{5}\leq\mu\leq k-1$

and these inequalities imply that $k\geq 5$. In particular, if $1\leq\mu<k-1$,
then either $\Gamma$ or $\Gamma_{3}$ has the $M$-property, except when
$k-1<5\mu<4k,$ in which case none of them has the $M$-property.

Case 5:

$D_{0}=3,D_{1}=4$ with $k_{0}>k_{1}$. In [1, Proposition 5.3] it is shown that
$\Gamma$ is the point-line incidence graph of a quasi-symmetric design with
$x=0$ if and only if $\Gamma$ is a distance-biregular graph with
$D_{0}=3,D_{1}=4$ and intersection array

$\displaystyle\left\\{\begin{array}[]{lllll}r;&1,&\lambda,&k&\\\
k;&1,&y,&\frac{k\lambda}{y},&k\\\ \end{array}\right\\}.$

Recall that a $2$-$(v,k,\lambda)$ _quasi-symmetric design_ is a design with
two intersection numbers, and we are interested in those having $x=0<y<k.$
Moreover, $k\lambda$ needs to be a multiple of $y$, that is, $k\lambda=\alpha
y$, $\alpha\in\mathbb{N}$. Also, recall that $r>\lambda$. Moreover, since
$B_{0,D_{0}}=B_{1,D_{1}}$, it holds that $(y-1)(r-1)=(k-1)(\lambda-1)$, see
also [5] for a proof based on design techniques.

Next, using the condition in Proposition 13 with $r=k_{0},k=k_{1}$, we obtain
a necessary and sufficient condition for a distance-biregular graph with
$D_{0}=3,\,\,D_{1}=4$ to have the $M$-property.

######  Proposition 18.

A distance-biregular graph with diameters $D_{0}=3,D_{1}=4$ has the
$M$-property if and only if

$\displaystyle(k-1)(r-\lambda)\Big{(}(k+r)^{2}-\lambda k\Big{)}\leq
k^{2}\lambda^{2}.$

###### Proof.

We use Proposition 13 to obtain:

$\displaystyle\displaystyle\frac{1}{k_{0,1}b_{0,1}}\Big{(}k_{0,2}+k_{0,3}\Big{)}^{2}+\frac{1}{k_{0,2}b_{0,2}}\Big{(}k_{0,3}\Big{)}^{2}\leq\frac{(n-1)}{k_{0}}.$

Keeping in mind that

$\begin{array}[]{llll}k_{0}=r,&b_{0,1}=k-1,&b_{0,2}=r-\lambda,&\\\\[12.91663pt]
k_{0,0}=1,&k_{0,1}=r,&k_{0,2}=r\dfrac{(k-1)}{\lambda},&\,\,k_{0,3}=r\dfrac{(k-1)}{\lambda}\dfrac{(r-\lambda)}{k},\end{array}$

we get that

$\dfrac{n-1}{r}=1+\dfrac{(k-1)}{\lambda k}\big{(}k+r-\lambda\big{)}.$

After performing some simplifications on the first inequality, the desired
result follows.

∎

Finally, we study the $M$-property for some classes of distance-biregular
graphs with $D_{0}=3$ and $D_{1}=4$.

######  Example 19.

Consider the point-line incident graph of the affine plane ${\cal A}(2,n)$ of
order $n$, whose intersection array is

$\displaystyle\left\\{\begin{array}[]{lllll}n+1;&1,&1,&n&\\\ n;&1,&1,&n,&n\\\
\end{array}\right\\}.$

It is easy to check that it does not verify the inequality in Proposition 13
and hence it does not verify the $M$-property.

######  Example 20.

Let $\Gamma=S(K_{r+1})$ be the subdivision graph of the the complete graph
$K_{r+1}$. This is a class of distance-biregular graphs with diameters
$D_{0}=4$, $D_{1}=3$ and parameters

$\begin{array}[]{rl}D_{0}=3,\,\,k_{0}=r,&c_{0,1}=c_{0,2}=1,c_{0,3}=2,b_{0,0}=r,b_{0,1}=1,b_{0,2}=r-1;\\\\[6.88889pt]
D_{1}=4,\,\,k_{1}=2,&c_{1,1}=c_{1,2}=1,c_{1,3}=c_{1,4}=2,b_{1,0}=2,b_{1,1}=r-1,b_{1,2}=1,b_{1,3}=r-2.\end{array}$

which does not hold the condition from Proposition 13, and thus does not have
the $M$-property. In fact, we can find the group inverse. We denote ${\sf
L}^{\\#}_{\ell,j}={\sf L}^{\\#}(x,y)$ when $y\in V_{\ell}$ and $d(x,y)=j$.
Then,

$\begin{array}[]{rlrlrl}{\sf
L}^{\\#}_{0,0}=&\dfrac{r(r+3)(2r+3)}{(r+1)^{2}(r+2)^{2}},&{\sf
L}^{\\#}_{0,1}=&\dfrac{(r+3)(r^{2}-2)}{(r+1)^{2}(r+2)^{2}},&{\sf
L}^{\\#}_{0,2}=&-\dfrac{r^{2}+7r+8}{(r+1)^{2}(r+2)^{2}},\\\\[10.33333pt] {\sf
L}^{\\#}_{0,3}=&-\dfrac{2(r^{2}+5r+5)}{(r+1)^{2}(r+2)^{2}},&&&&\end{array}$

and

$\begin{array}[]{rlrlrl}{\sf
L}^{\\#}_{1,0}=&\dfrac{(r+3)(r^{3}+5r^{2}+2r-4)}{2(r+1)^{2}(r+2)^{2}},&{\sf
L}^{\\#}_{1,1}=&\dfrac{(r+3)(r^{2}-2)}{(r+1)^{2}(r+2)^{2}},&{\sf
L}^{\\#}_{1,2}=&\dfrac{r^{3}-r^{2}-18r-20}{2(r+1)^{2}(r+2)^{2}},\\\\[10.33333pt]
{\sf L}^{\\#}_{1,3}=&-\dfrac{2(r^{2}+5r+5)}{(r+1)^{2}(r+2)^{2}},&{\sf
L}^{\\#}_{1,4}=&-\dfrac{(r+3)(3r+4)}{(r+1)^{2}(r+2)^{2}}.&&\end{array}$

For the classification of existing quasi-symmetric 2-designs, see [33, Table
48.25]. We should note that for the existing quasi-symmetric 2-designs with
$x=0$, none passes the condition from Proposition 18.

The above discussion extends the results in [12] and completes the
classification of distance-regularised graphs that have the $M$-property.
There, it was shown that if there are distance-regular graphs with valency
$k\geq 3$ and diameter $D\geq 2$ having the $M$-property, then they have at
most $3k$ vertices and $D\leq 3$. Also in [12], it was conjectured that there
is no primitive distance-regular graph with diameter $3$ having the
$M$-property. This conjecture was shown to be true except possibly for
finitely many primitive distance-regular graphs [30, Theorem 1].

In view of the above results, we conclude this paper with the following
conjecture.

######  Conjecture 21.

There are not point-line incidence graphs of a $2$-quasi-symmetric design with
$x=0$ that have the $M$-property.

## Acknowledgments

The research of A. Abiad is partially supported by the FWO grant 1285921N.

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