# A polynomial time algorithm to find star chromatic index on bounded
treewidth graphs with given maximum degree

Yichen Wang Mei Lu
Department of Mathematical Sciences, Tsinghua University, Beijing 100084,
China. email<EMAIL_ADDRESS>email<EMAIL_ADDRESS>
(2023 3)

###### Abstract

A star edge coloring of a graph $G$ is a proper edge coloring with no
2-colored path or cycle of length four. The star edge coloring problem is to
find an edge coloring of a given graph $G$ with minimum number $k$ of colors
such that $G$ admits a star edge coloring with $k$ colors. This problem is
known to be NP-complete. In this paper, for a bounded treewidth graph with
given maximum degree, we show that it can be solved in polynomial time.

Keywords: edge coloring; star edge coloring; bounded treewidth.

##  1 Introduction

A proper edge coloring of a graph $G$ with vertex set $V$ and edge set $E$ is
an assignment of colors to the edges of $G$ such that no two adjacent edges
have the same color. Under additional constraints on the proper edge coloring,
we have a variety of colorings such as strong edge coloring, vertex
distinguishing coloring and so on. A star edge coloring of $G$ is a proper
edge coloring where at least three distinct colors are used on the edges of
every path and cycle of length four, i.e., there is neither bichromatic path
nor cycle of length four. We say $G$ is $k$-star-edge-colorable if $G$ admits
a star edge coloring using at most $k$ colors and the star chromatic index
$\chi_{st}^{\prime}(G)$ of $G$ is the minimum number $k$ for which $G$ has a
star edge coloring with $k$ colors.

Star edge coloring is invited by Liu and Deng [14] motivated by the vertex
version [1, 7, 9]. In the same paper, Liu and Deng presented an upper bound on
the star chromatic index of graphs with maximum degree $\Delta\geq 7$. In [8],
Dvořák, Mohar and Šámal presented some upper bounds and lower bounds on the
star chromatic index of complete graphs and subcubic graphs (i.e. with maximum
degree at most three). Some bounds on the star chromatic index of subcubic
outerplanar graphs, trees and outerplanar graphs can be find in [11, 13, 15,
17, 20]. The relevant research results of star chromatic index can be referred
to the survey [12].

In [13], Lei, Shi and Song showed that it is $NP$-complete to determine
whether $\chi_{st}^{\prime}(G)\leq 3$ for an arbitrary graph $G$. In [16],
Omoomi, Roshanbin and Dastjerdi presented a polynomial time algorithm that
finds an optimum star edge-coloring for every tree.

The treewidth of a graph is an important invariant in graph theory. The
concept of treewidth was originally introduced by Bertelé and Brioschi [3]
under the name of dimension. It was later rediscovered by Halin [10] in 1976
and by Robertson and Seymour [18] in 1984. The treewidth of a graph gives an
indication of how far away the graph is from being a tree or forest. The
treewidth also is a parameter that plays a fundamental role in various graph
algorithms. It is well-known that many NP-complete problems can be solved in
polynomial time on bounded treewidth graphs [21, 19, 6]. In this paper, for
bounded treewidth graphs with given maximum degree, we propose a polynomial
time algorithm which can determine the star chromatic index of $G$. Our main
idea comes from [19].

The rest of this paper is organized as follows. In section 2, we will give
some terminology and notations that we use in this paper. In section 3, we
give a polynomial time algorithm to determine the star chromatic index of
bounded treewidth graphs with given maximum degree. Some more discussion will
be given in section 4.

##  2 Preliminaries and definitions

In this section, we give definitions involving in treewidth. The treewidth of
a graph is defined through the concept of tree-decomposition.

###### Definition 2.1

A tree-decomposition of a graph $G=(V,E)$ is a pair $(X,T)$, where $T(I,F)$ is
a tree with vertex set $I$ and edge set $F$, and $X=\\{X_{i}\mid i\in I\\}$ is
a family of subsets of $V$, one for each node of $T$, such that:

  * •

$\bigcup\limits_{i\in I}X_{i}=V$.

  * •

for each edge $uv\in E$, there exists an $i\in I$ such that $u,v\in X_{i}$.

  * •

for all $i,j,s\in I$, if $j$ is on the path from $i$ to $s$ in $T$, then
$X_{i}\cap X_{s}\subseteq X_{j}$.

Suppose $i$ is a node of $T$, there exists a vertex set $X_{i}$ corresponds to
$i$. In the following, we also call $X_{i}$ the node of $T$ for convenience.
If $X_{i}$ is a non-leaf node of $T$, we call it an internal node. The width
of a tree-decomposition $(X,T)$ is $\max\limits_{i\in I}|X_{i}|-1$. The
treewidth of a graph $G$ is the minimum treewidth over all possible tree-
decompositions of $G$. The problem of deciding whether a graph has tree-
decomposition of treewidth at most $k$ is NP-complete [2] and Bodlaender [5]
proved that the problem is fixed-parameter tractable, that is, there is a
polynomial $p$ and an algorithm that, for a given graph $G=(V,E)$, computes a
tree-decomposition of $G$ of width $k$ in time at most $2^{p(k)}|V|$.

We say a tree-decomposition $(X,T)$ of treewidth $k$ is smooth, if $T$ is a
binary tree and:

  * •

for all $i\in I$, $|X_{i}|=k+1$.

  * •

every internal node $X_{i}$ has two children, and if $X_{L},X_{R}$ are its
children, then either $X_{L}=X_{i}$ or $X_{R}=X_{i}$.

  * •

for all $(i,j)\in F$: $k\leq|X_{i}\cap X_{j}|\leq k+1$.

  * •

for each edge $uv\in E$, there is at least one leaf $i\in I$, with $u,v\in
X_{i}$.

It can be shown that any tree-decomposition of a graph $G$ can be transformed
into a smooth tree-decomposition of $G$ with the same treewidth and size
$O(n)$ in linear time [4, 5].

For a given graph $G$ with treewidth at most $k$, let $(X,T)$ be its tree-
decomposition. For each node $X_{i}$ of $T$, we define $T_{i}$ to be the
subtree of $T$ rooted at $i$. Let $X_{i}$ be a node of $T$. If $X_{i}$ is a
leaf, then we define $V_{i}=X_{i}$, $E_{i}=\\{uv\in E~{}|~{}u,v\in X_{i}\\}$;
if $X_{i}$ is an internal node with children $X_{L},X_{R}$, then we define
$V_{i}=V_{L}\cup V_{R},E_{i}=E_{L}\cup E_{R}$. The graph $G(V_{i},E_{i})$ is
denoted by $G_{i}$.

For every edge $e\in E$, there is at least one leaf $i$ of $T$ such that
$u,v\in X_{i}$. We choose one such $i$ as the representative of $e$ and denote
it by $rep(e)$. If $X_{i}$ is a leaf, we define $E_{i}^{\prime}=\\{e\in
E~{}|~{}rep(e)\in T_{i}\\}$; if $X_{i}$ is an internal node with children
$X_{L},X_{R}$, we define $E_{i}^{\prime}=E_{L}^{\prime}\cup E_{R}^{\prime}$.
Note that by this definition, for every node $X_{i}$, $E_{i}^{\prime}\subseteq
E_{i}$, and if $X_{i}$ is an internal node with children $X_{L},X_{R}$, then
$E_{L}^{\prime}\cap E_{R}^{\prime}=\emptyset$. We denote the graph on vertex
set $V_{i}$ and edge set $E_{i}^{\prime}$ by $G_{i}^{\prime}$.

In the rest of this paper, when $X_{i}$ is an internal node in a smooth tree-
decomposition, we always use $X_{L}$ and $X_{R}$ to represent its two
children. Without loss of generality, we assume $X_{L}=X_{i}$,
$X_{i}-X_{R}=\\{v^{\prime}\\}$ and $X_{R}-X_{i}=\\{v^{\prime\prime}\\}$. If
$e\in E(G)$, we also use $e$ to denote the subset consisting of two vertices
of $e$ for short. Let $\Delta$ be the maximum degree of $G$. The following
facts are obvious from definition.

###### Fact 2.1

Let $X_{i}$ be an internal node in a smooth tree-decomposition and $X_{L}$ and
$X_{R}$ its two children. If $e\in E_{R}^{\prime}$ (resp. $e\in
E_{L}^{\prime}$), then $v^{\prime}\notin e$ (resp. $v^{\prime\prime}\notin
e$).

###### Fact 2.2

Let $X_{i}$ be an internal node in a smooth tree-decomposition and $X_{L}$ and
$X_{R}$ its two children. If $e_{1}\in E_{L}^{\prime},e_{2}\in E_{R}^{\prime}$
and $\\{v\\}=e_{1}\cap e_{2}$, then $v\in X_{i}-\\{v^{\prime}\\}$.

##  3 The star edge coloring algorithm

In this section, we give a polynomial time algorithm to find the star
chromatic index of bounded treewidth graphs with given maximum degree. Here we
present the main theorem of our paper.

###### Theorem 3.1

For everg graph $G$ of order $n$ with treewidth $k$ and maximum degree
$\Delta$, and integer $c$, there is a deterministic algorithm that determines
in time $O(nc^{2{(k+1)}^{2}\Delta^{6}})$ whether $G$ has a star edge coloring
using at most $c$ colors or not and finds such star edge coloring if it
exists.

Let $(X,T)$ be a smooth tree-decomposition of $G$ and $C=\\{1,2,\ldots,c\\}$
the set of colors. For a node $X_{i}$ of $T$, a mapping
$f:E_{i}^{\prime}\rightarrow C$ is called a partial coloring of
$G_{i}^{\prime}$. We say a partial coloring is a proper edge coloring if no
two adjacent edges have the same color. We say a partial coloring $f$ of
$G_{i}^{\prime}$ is valid if it is a star edge coloring of $G_{i}^{\prime}$,
that is, it is proper and no path or cycle of length $4$ in $G_{i}^{\prime}$
is bicolored.

Consider a partial coloring $f$ of $G_{i}^{\prime}$. For a fixed color pair
$(c_{1},c_{2})\in C^{2}$, we define the neighbour sets as the following
equations.

$\displaystyle N^{(1)}_{f}(c_{1},c_{2})$ $\displaystyle=\\{v_{0}\in
X_{i}~{}|~{}\exists\\{v_{0}v_{1},v_{1}v_{2}\\}\subseteq
E_{i}^{\prime},f(v_{0}v_{1})=c_{1},f(v_{1}v_{2})=c_{2}\\},$ (1) $\displaystyle
N^{(2)}_{f}(c_{1},c_{2})$ $\displaystyle=\\{v_{0}\in
X_{i}~{}|~{}\exists\\{v_{0}v_{1},v_{1}v_{2}\\}\subseteq
E_{i}^{\prime},f(v_{0}v_{1})=c_{2},f(v_{1}v_{2})=c_{1}\\},$ $\displaystyle
N^{(3)}_{f}(c_{1},c_{2})$ $\displaystyle=\\{(v_{0},v_{2})\in
X_{i}^{2}~{}|~{}\exists\\{v_{0}v_{1},v_{1}v_{2}\\}\subseteq
E_{i}^{\prime},f(v_{0}v_{1})=c_{1},f(v_{1}v_{2})=c_{2}\\},$ $\displaystyle
N^{(4)}_{f}(c_{1},c_{2})$ $\displaystyle=\\{v_{0}\in
X_{i}~{}|~{}\exists\\{v_{0}v_{1},v_{1}v_{2},v_{2}v_{3}\\}\subseteq
E_{i}^{\prime},f(v_{0}v_{1})=c_{1},f(v_{1}v_{2})=c_{2},f(v_{2}v_{3})=c_{1}\\},$
$\displaystyle N^{(5)}_{f}(c_{1},c_{2})$ $\displaystyle=\\{v_{0}\in
X_{i}~{}|~{}\exists\\{v_{0}v_{1},v_{1}v_{2},v_{2}v_{3}\\}\subseteq
E_{i}^{\prime},f(v_{0}v_{1})=c_{2},f(v_{1}v_{2})=c_{1},f(v_{2}v_{3})=c_{2}\\},$
$\displaystyle N^{(6)}_{f}(c_{1},c_{2})$ $\displaystyle=\\{v_{0}\in
X_{i}~{}|~{}\exists v_{0}v\in E_{i}^{\prime},f(v_{0}v)=c_{1}\\},$
$\displaystyle N^{(7)}_{f}(c_{1},c_{2})$ $\displaystyle=\\{v_{0}\in
X_{i}~{}|~{}\exists v_{0}v\in E_{i}^{\prime},f(v_{0}v)=c_{2}\\},$
$\displaystyle N^{(8)}_{f}(c_{1},c_{2})$ $\displaystyle=\\{(v_{0},v_{1})\in
X_{i}^{2}~{}|~{}\exists v_{0}v_{1}\in E_{i}^{\prime},f(v_{0}v_{1})=c_{1}\\},$
$\displaystyle N^{(9)}_{f}(c_{1},c_{2})$ $\displaystyle=\\{(v_{0},v_{1})\in
X_{i}^{2}~{}|~{}\exists v_{0}v_{1}\in E_{i}^{\prime},f(v_{0}v_{1})=c_{2}\\}.$

Let $S$ be a set. We use $2^{S}$ to represent the power set of $S$. Denote

$\mathcal{S}(X_{i})=\\{\big{(}A^{(1)},\ldots,A^{(9)}\big{)}~{}|~{}A^{(j)}\subseteq
X_{i},j=1,2,4,5,6,7\mbox{~{}and~{}}A^{(j)}\subseteq X_{i}^{2},j=3,8,9\\}.$

In the following, we abbreviate $\big{(}A^{(1)},\ldots,A^{(9)}\big{)}$ to
${\big{(}A^{(j)}\big{)}}_{1\leq j\leq 9}$. For a partial coloring $f$ of
$G_{i}^{\prime}$, define the color class function
$CL_{f}:\mathcal{S}(X_{i})\rightarrow 2^{C^{2}}$ as Eq. (2). In this case, we
say $CL_{f}$ is the color class function on $X_{i}$.

$\displaystyle CL_{f}\left({\big{(}A^{(j)}\big{)}}_{1\leq j\leq
9}\right)=\\{(c_{1},c_{2})\in
C^{2}~{}|~{}A^{(j)}=N^{(j)}_{f}(c_{1},c_{2}),1\leq j\leq 9\\}.$ (2)

A color class function $CL_{f}$ is active if and only if there exists a valid
partial coloring $g$ such that $CL_{f}=CL_{g}$.

###### Fact 3.1

Let $f$ be a partial coloring of $G_{i}^{\prime}$. Then
$\\{CL_{f}({\big{(}A^{(j)}\big{)}}_{1\leq j\leq
9})~{}|~{}{\big{(}A^{(j)}\big{)}}_{1\leq j\leq 9}\in\mathcal{S}(X_{i})\\}$
forms a partition of $C^{2}$.

Our idea of developing an algorithm is to calculate all active color class
functions on a node $X_{i}$. For any partial coloring $f$ of $G_{i}^{\prime}$,
the color class function is only concerned with the coloring of edges adjacent
to a vertex whose distance is at most $3$ to some vertex in $X_{i}$ (the
distance between two vertices is the minimum length of a path connecting
them). Since at most $(k+1)\Delta^{3}$ vertices have a distance at most $3$ to
some vertex in $X_{i}$, at most ${(k+1)}^{2}\Delta^{6}$ edges have an effect
on the color class function. Therefore, at most $c^{{(k+1)}^{2}\Delta^{6}}$
color class functions use at most $c$ colors.

What we are most concerned about is how to determine whether a color class
function is active. To do this, we first need to analyse the relationship of
color class functions on $X_{i},X_{L}$ and $X_{R}$.

Let $f$ be a partial coloring of $G_{i}^{\prime}$. Define
$f_{L}=f~{}|~{}G_{L}^{\prime}$ and $f_{R}=f~{}|~{}G_{R}^{\prime}$. For any
$(c_{1},c_{2})\in C^{2}$ and $1\leq j\leq 9$, let
$A^{(j)}=N^{(j)}_{f}(c_{1},c_{2}),A_{L}^{(j)}=N^{(j)}_{f_{L}}(c_{1},c_{2})$
and $A_{R}^{(j)}=N^{(j)}_{f_{R}}(c_{1},c_{2})$. Denote

$\displaystyle\bar{A}^{(1)}$ $\displaystyle=\\{v_{0}~{}|~{}(v_{0},v_{1})\in
A_{L}^{(8)},v_{1}\in A_{R}^{(7)}\\}\cup\\{v_{0}~{}|~{}(v_{0},v_{1})\in
A_{R}^{(8)},v_{1}\in A_{L}^{(7)}\\},$ (3) $\displaystyle\bar{A}^{(2)}$
$\displaystyle=\\{v_{0}~{}|~{}(v_{0},v_{1})\in A_{L}^{(9)},v_{1}\in
A_{R}^{(6)}\\}\cup\\{v_{0}~{}|~{}(v_{0},v_{1})\in A_{R}^{(9)},v_{1}\in
A_{L}^{(6)}\\},$ $\displaystyle\bar{A}^{(3)}$
$\displaystyle=\\{(v_{1},v_{2})~{}|~{}\exists(v_{1},v_{3})\in
A_{L}^{(8)},(v_{3},v_{2})\in
A_{R}^{(9)}\\}\cup\\{(v_{1},v_{2})~{}|~{}\exists(v_{1},v_{3})\in
A_{R}^{(8)},(v_{3},v_{2})\in A_{L}^{(9)}\\},$ $\displaystyle\bar{A}^{(4)}$
$\displaystyle=\\{v_{0}~{}|~{}\exists(v_{0},v_{1})\in A_{L}^{(3)},v_{1}\in
A_{R}^{(6)}\\}\cup\\{v_{0}~{}|~{}\exists(v_{0},v_{1})\in
A_{L}^{(8)},(v_{1},v_{2})\in A_{R}^{(9)},v_{2}\in A_{L}^{(6)}\\}$
$\displaystyle\cup\\{v_{0}~{}|~{}\exists(v_{0},v_{1})\in A_{L}^{(8)},v_{1}\in
A_{R}^{(2)}\\}\cup\\{v_{0}~{}|~{}\exists(v_{0},v_{1})\in A_{R}^{(3)},v_{1}\in
A_{L}^{(6)}\\}$ $\displaystyle\cup\\{v_{0}~{}|~{}\exists(v_{0},v_{1})\in
A_{R}^{(8)},(v_{1},v_{2})\in A_{L}^{(9)},v_{2}\in
A_{R}^{(6)}\\}\cup\\{v_{0}~{}|~{}\exists(v_{0},v_{1})\in A_{R}^{(8)},v_{1}\in
A_{L}^{(2)}\\},$ $\displaystyle\bar{A}^{(5)}$
$\displaystyle=\\{v_{0}~{}|~{}\exists(v_{1},v_{0})\in A_{L}^{(3)},v_{1}\in
A_{R}^{(7)}\\}\cup\\{v_{0}~{}|~{}\exists(v_{0},v_{1})\in
A_{L}^{(9)},(v_{1},v_{2})\in A_{R}^{(8)},v_{2}\in A_{L}^{(7)}\\}$
$\displaystyle\cup\\{v_{0}~{}|~{}\exists(v_{0},v_{1})\in A_{L}^{(9)},v_{1}\in
A_{R}^{(1)}\\}\cup\\{v_{0}~{}|~{}\exists(v_{1},v_{0})\in A_{R}^{(3)},v_{1}\in
A_{L}^{(7)}\\}$ $\displaystyle\cup\\{v_{0}~{}|~{}\exists(v_{0},v_{1})\in
A_{R}^{(9)},(v_{1},v_{2})\in A_{L}^{(8)},v_{2}\in
A_{R}^{(7)}\\}\cup\\{v_{0}~{}|~{}\exists(v_{0},v_{1})\in A_{R}^{(9)},v_{1}\in
A_{L}^{(1)}\\},$ $\displaystyle\bar{A}^{(j)}$
$\displaystyle=\emptyset,~{}~{}6\leq j\leq 9.$

Then, we have the following lemmas.

###### Lemma 3.1

For $1\leq j\leq 9$, we have

$\displaystyle A^{(j)}$ $\displaystyle=A_{L}^{(j)}\cup
A_{R}^{(j)}\cup\bar{A}^{(j)}-\\{v^{\prime\prime}\\},$ (4)

where $\\{v^{\prime\prime}\\}=X_{R}-X_{i}$.

Proof: We just show that the result holds for $A^{(1)}$ and the other results
can be showed by the same way. Fixed color pair $(c_{1},c_{2})$, we verify
$A^{(1)}$ one by one.

If $v_{0}\in A^{(1)}$, then $v_{0}\in X_{i}$ and so
$v_{0}\not=v^{\prime\prime}$. By the definition, there exists
$\\{v_{0}v_{1},v_{1}v_{2}\\}\subseteq E_{i}^{\prime}$ such that
$f(v_{0}v_{1})=c_{1}$ and $f(v_{1}v_{2})=c_{2}$.

  1. 1.

If $v_{0}v_{1}\in E_{L}^{\prime}$ and $v_{1}v_{2}\in E_{L}^{\prime}$, then
$v_{0}\in A_{L}^{(1)}$.

  2. 2.

If $v_{0}v_{1}\in E_{R}^{\prime}$ and $v_{1}v_{2}\in E_{R}^{\prime}$, then
$v_{0}\in A_{R}^{(1)}$.

  3. 3.

If $v_{0}v_{1}\in E_{L}^{\prime}$ and $v_{1}v_{2}\in E_{R}^{\prime}$, from
Fact 2.2, we have $v_{1}\in X_{i}-\\{v^{\prime}\\}=X_{L}-\\{v^{\prime}\\}$. By
the definition, $(v_{0},v_{1})\in A_{L}^{(8)},v_{1}\in A_{R}^{(7)}$ and then
we have $v_{0}\in\bar{A}^{(1)}$.

  4. 4.

If $v_{0}v_{1}\in E_{R}^{\prime}$ and $v_{1}v_{2}\in E_{L}^{\prime}$, from
Fact 2.2, we have $v_{1}\in X_{i}-\\{v^{\prime}\\}=X_{L}-\\{v^{\prime}\\}$.
From Fact 2.1, we have $v_{0}\neq v^{\prime}$. By the definition,
$(v_{0},v_{1})\in A_{R}^{(8)},v_{1}\in A_{L}^{(7)}$ and then we have
$v_{0}\in\bar{A}^{(1)}$.

Therefore, we have $v_{0}\in A_{L}^{(1)}\cup
A_{R}^{(1)}\cup\bar{A}^{(1)}-\\{v^{\prime\prime}\\}$ which implies
$A^{(1)}\subseteq A_{L}^{(1)}\cup
A_{R}^{(1)}\cup\bar{A}^{(1)}-\\{v^{\prime\prime}\\}$. Then, we are going to
show $A_{L}^{(1)}\cup
A_{R}^{(1)}\cup\bar{A}^{(1)}-\\{v^{\prime\prime}\\}\subseteq A^{(1)}$. It is
not hard to prove by considering the four situations conversely. The proof is
omitted.

###### Lemma 3.2

Assume $f$ is a partial coloring of $G_{i}^{\prime}$. Let $CL_{f},CL_{f_{L}}$
and $CL_{f_{R}}$ be the color class functions on $X_{i}$, $X_{L}$ and $X_{R}$
respectively. Then, for all ${\big{(}A^{(j)}\big{)}}_{1\leq j\leq
9}\in\mathcal{S}(X_{i})$, we have

$CL_{f}\left({\big{(}A^{(j)}\big{)}}_{1\leq j\leq
9}\right)=\bigcup\left(CL_{f_{L}}\left({\big{(}A_{L}^{(j)}\big{)}}_{1\leq
j\leq 9}\right)\cap CL_{f_{R}}\left({\big{(}A_{R}^{(j)}\big{)}}_{1\leq j\leq
9}\right)\right),$ (5)

where the union is taken over all ${\big{(}A_{L}^{(j)}\big{)}}_{1\leq j\leq
9}$ and ${\big{(}A_{R}^{(j)}\big{)}}_{1\leq j\leq 9}$ satisfying Eq. (4).

Proof: Given ${\big{(}A^{(j)}\big{)}}_{1\leq j\leq 9}\in\mathcal{S}(X_{i})$,
assume $(c_{1},c_{2})\in CL_{f}\left({\big{(}A^{(j)}\big{)}}_{1\leq j\leq
9}\right)$. Then, we have $N_{f}^{(j)}(c_{1},c_{2})=A^{(j)}$ for all $1\leq
j\leq 9$. Let ${\big{(}A_{L}^{(j)}\big{)}}_{1\leq j\leq
9}={\big{(}N_{f_{L}}^{(j)}(c_{1},c_{2})\big{)}}_{1\leq j\leq 9}$ and
${\big{(}A_{R}^{(j)}\big{)}}_{1\leq j\leq
9}={\big{(}N_{f_{R}}^{(j)}(c_{1},c_{2})\big{)}}_{1\leq j\leq 9}$. By Fact 3.1,
$(c_{1},c_{2})$ belongs to exactly one
$CL_{f_{L}}\left({\big{(}A_{L}^{(j)}\big{)}}_{1\leq j\leq 9}\right)$ and
exactly one $CL_{f_{R}}\left({\big{(}A_{R}^{(j)}\big{)}}_{1\leq j\leq
9}\right)$. Hence, we have that $(c_{1},c_{2})\in
CL_{f_{L}}({\big{(}A_{L}^{(j)}\big{)}}_{1\leq j\leq 9})\cap
CL_{f_{R}}({\big{(}A_{R}^{(j)}\big{)}}_{1\leq j\leq 9})$ and by Lemma 3.1, Eq.
(4) is satisfied. Therefore, we have

$CL_{f}\left({\big{(}A^{(j)}\big{)}}_{1\leq j\leq
9}\right)\subseteq\bigcup\left(CL_{f_{L}}\left({\big{(}A_{L}^{(j)}\big{)}}_{1\leq
j\leq 9}\right)\cap CL_{f_{R}}\left({\big{(}A_{R}^{(j)}\big{)}}_{1\leq j\leq
9}\right)\right).$

Assume $(c_{1},c_{2})\in CL_{f_{L}}({\big{(}A_{L}^{(j)}\big{)}}_{1\leq j\leq
9})\cap CL_{f_{R}}({\big{(}A_{R}^{(j)}\big{)}}_{1\leq j\leq 9})$ for some
${\big{(}A_{L}^{(j)}\big{)}}_{1\leq j\leq
9},{\big{(}A_{R}^{(j)}\big{)}}_{1\leq j\leq 9}$ satisfying Eq. (4). Then,
$(c_{1},c_{2})\in CL_{f_{L}}({\big{(}A_{L}^{(j)}\big{)}}_{1\leq j\leq 9})$ and
$(c_{1},c_{2})\in CL_{f_{R}}({\big{(}A_{R}^{(j)}\big{)}}_{1\leq j\leq 9})$. So
${\big{(}N_{f_{L}}^{(j)}(c_{1},c_{2})\big{)}}_{1\leq j\leq
9}={\big{(}A_{L}^{(j)}\big{)}}_{1\leq j\leq 9}$ and
${\big{(}N_{f_{R}}^{(j)}(c_{1},c_{2})\big{)}}_{1\leq j\leq
9}={\big{(}A_{R}^{(j)}\big{)}}_{1\leq j\leq 9}$. From Lemma 3.1, we have
$(c_{1},c_{2})\in CL_{f}({\big{(}A^{(j)}\big{)}}_{1\leq j\leq 9})$ which
implies $\bigcup(CL_{f_{L}}({\big{(}A_{L}^{(j)}\big{)}}_{1\leq j\leq 9})\cap
CL_{f_{R}}({\big{(}A_{R}^{(j)}\big{)}}_{1\leq j\leq 9}))\subseteq
CL_{f}({\big{(}A^{(j)}\big{)}}_{1\leq j\leq 9})$.

Figure 1: Five cases of an invalid but proper star coloring in which the bold
vertices belong to $X_{i}-\\{v^{\prime}\\}$. The two endpoints may be the same
and then it results in a bicolored $C_{4}$ rather than $P_{4}$. Blue and red
represent two different colors and $L$ (resp. $R$) represents that the edge is
from $G_{L}^{\prime}$ (resp. $G_{R}^{\prime}$). The bold vertices must be in
$X_{i}$ from previous facts.

Lemma 3.2 describes the relationship of color class functions of $X_{i},X_{L}$
and $X_{R}$. The remaining problem is how to justify whether a color class
function is active. Lemma 3.3 provides a method.

###### Lemma 3.3

A color class function $CL_{f}$ on $X_{i}$ is active if and only if there
exists active color class functions $CL_{g_{L}}$ and $CL_{g_{R}}$ on $X_{L}$
and $X_{R}$ respectively such that for all ${\big{(}A^{(j)}\big{)}}_{1\leq
j\leq 9}\in\mathcal{S}(X_{i})$,

$CL_{f}\left({\big{(}A^{(j)}\big{)}}_{1\leq j\leq
9}\right)=\bigcup\left(CL_{g_{L}}({\big{(}A_{L}^{(j)}\big{)}}_{1\leq j\leq
9})\cap CL_{g_{R}}({\big{(}A_{R}^{(j)}\big{)}}_{1\leq j\leq 9})\right),$ (6)

where the union is taken over all ${\big{(}A_{L}^{(j)}\big{)}}_{1\leq j\leq
9}$ and ${\big{(}A_{R}^{(j)}\big{)}}_{1\leq j\leq 9}$ satisfying Eq. (4).

Moreover, if $CL_{g_{L}}\left({\big{(}A_{L}^{(j)}\big{)}}_{1\leq j\leq
9}\right)\cap CL_{g_{R}}\left({\big{(}A_{R}^{(j)}\big{)}}_{1\leq j\leq
9}\right)\neq\emptyset$ for some ${\big{(}A_{L}^{(j)}\big{)}}_{1\leq j\leq
9}\in\mathcal{S}(X_{L})$ and ${\big{(}A_{R}^{(j)}\big{)}}_{1\leq j\leq
9}\in\mathcal{S}(X_{R})$, then the following results hold.

  1. (i)

$A_{L}^{(1)}\cap A_{R}^{(2)}=A_{L}^{(2)}\cap A_{R}^{(1)}=\emptyset$ (prevent
invalid case 1 in Fig. 1).

  2. (ii)

If $(v_{0},v_{1})\in A_{L}^{(8)},(v_{1},v_{2})\in A_{R}^{(9)}$, then
$v_{0}\notin A_{R}^{(7)}$ or $v_{2}\notin A_{L}^{(6)}$; if $(v_{0},v_{1})\in
A_{R}^{(8)},(v_{1},v_{2})\in A_{L}^{(9)}$, then $v_{0}\notin A_{L}^{(7)}$ or
$v_{2}\notin A_{R}^{(6)}$ (prevent invalid case 2 in Fig. 1).

  3. (iii)

If $(v_{1},v_{2})\in A_{L}^{(3)}$, then $v_{1}\notin A_{R}^{(7)}$ or
$v_{2}\notin A_{R}^{(6)}$; if $(v_{1},v_{2})\in A_{R}^{(3)}$, then
$v_{1}\notin A_{L}^{(7)}$ or $v_{2}\notin A_{L}^{(6)}$ (prevent invalid case 3
in Fig. 1).

  4. (iv)

$A_{L}^{(4)}\cap A_{R}^{(7)}=A_{R}^{(4)}\cap A_{L}^{(7)}=\emptyset$ (prevent
invalid case 4 in Fig. 1).

  5. (v)

If $(v_{1},v_{2})\in A_{L}^{(8)}$, then $v_{1}\notin A_{R}^{(2)}$ or
$v_{2}\notin A_{R}^{(7)}$; if $(v_{1},v_{2})\in A_{R}^{(8)}$, then
$v_{1}\notin A_{L}^{(2)}$ or $v_{2}\notin A_{L}^{(7)}$ (prevent invalid case 5
in Fig. 1).

  6. (vi)

$A_{L}^{(6)}\cap A_{R}^{(6)}=A_{L}^{(7)}\cap A_{R}^{(7)}=\emptyset$ (assure a
proper edge coloring).

Proof Necessity: Assume that the color class function $CL_{f}$ of $X_{i}$ is
active and $f$ is a valid partial coloring of $G_{i}^{\prime}$. It is obvious
$CL_{f_{L}}$ defined by $f_{L}$ on $X_{L}$ and $CL_{f_{R}}$ defined by $f_{R}$
on $X_{R}$ are active color class functions. Let $g_{L}=f_{L}$ and
$g_{R}=f_{R}$. By Lemma 3.2, Eq. (6) holds. Suppose there is $(c_{1},c_{2})\in
CL_{g_{L}}({\big{(}A_{L}^{(j)}\big{)}}_{1\leq j\leq 9})\cap
CL_{g_{R}}({\big{(}A_{R}^{(j)}\big{)}}_{1\leq j\leq 9})$ for some
${\big{(}A_{L}^{(j)}\big{)}}_{1\leq j\leq 9}\in\mathcal{S}(X_{L})$ and
${\big{(}A_{R}^{(j)}\big{)}}_{1\leq j\leq 9}\in\mathcal{S}(X_{R})$. Then,
$(c_{1},c_{2})\in CL_{f_{L}}({\big{(}A_{L}^{(j)}\big{)}}_{1\leq j\leq 9})$ and
$(c_{1},c_{2})\in CL_{f_{R}}({\big{(}A_{R}^{(j)}\big{)}}_{1\leq j\leq 9})$. We
are going to show that ($i$) to ($vi$) hold. Since $f$ is a valid partial
coloring of $G_{i}^{\prime}$, ($vi$) holds obviously.

($i$) Assume without loss of generality $v_{0}\in A_{L}^{(1)}\cap
A_{R}^{(2)}$. Then there are $v_{1},v_{2}\in X_{L}$ (resp. $v_{3},v_{4}\in
X_{R}$ such that $f(v_{1}v_{0})=c_{1},f(v_{1}v_{2})=c_{2}$ (resp.
$f(v_{3}v_{0})=c_{2},f(v_{3}v_{4})=c_{1}$). Thus we have a bicolored $P_{4}$
or $C_{4}$, a contradiction with $f$ being a valid partial coloring of
$G_{i}^{\prime}$. Hence, ($i$) holds.

The proofs of ($ii$) to ($v$) are the same. We omit here.

Sufficiency: Suppose there exist active color class functions $CL_{g_{L}}$ and
$CL_{g_{R}}$ on $E_{L}^{\prime}$ and $E_{R}^{\prime}$ respectively. Define
$f^{\prime}:E_{i}^{\prime}\rightarrow C$ as following:
$f^{\prime}(e)=g_{L}(e)$ if $e\in E_{L}^{\prime}$ and $f^{\prime}(e)=g_{R}(e)$
if $e\in E_{R}^{\prime}$.

From Lemma 3.2, we have $CL_{f^{\prime}}=CL_{f}$. In order to show that
$CL_{f}$ on $X_{i}$ is active, we just need to prove $f^{\prime}$ is a valid
partial coloring by the definition. By ($vi$), and $CL_{g_{L}}$ and
$CL_{g_{R}}$ are active on $X_{L}$ and $X_{R}$ respectively, we have
$f^{\prime}$ is a proper edge coloring. Hence $f^{\prime}_{L}$ and
$f^{\prime}_{R}$ are valid colorings on $X_{L}$ and $X_{R}$.

Suppose there is a bicolored $P_{4}$ or $C_{4}$ on $G_{i}$ after the coloring
$f^{\prime}$. Then, the $P_{4}$ or $C_{4}$ can not belong to only
$G_{L}^{\prime}$ or $G_{R}^{\prime}$. So it must corresponds to one of the
cases in Fig. 1.

Suppose the bicolored $P_{4}$ or $C_{4}$ is case 1 in Fig. 1. Let $E(P_{4})$
or $E(C_{4})$ = $\\{v_{1}v_{2},v_{2}v_{3},v_{3}v_{4},v_{4}v_{5}\\}$ (when
$v_{1}=v_{5}$, it is the $C_{4}$) where $v_{1}v_{2},v_{2}v_{3}\in
E_{L}^{\prime},v_{3}v_{4},v_{4}v_{5}\in E_{R}^{\prime}$ and
$f^{\prime}(v_{1}v_{2})=f^{\prime}(v_{3}v_{4})=c_{1},f^{\prime}(v_{2}v_{3})=f^{\prime}(v_{4}v_{5})=c_{2}$.
From Fact 2.2, we have $v_{3}\in X_{i}$. Then there exist
${\big{(}A_{L}^{(j)}\big{)}}_{1\leq j\leq 9}\in\mathcal{S}(X_{L})$ and
${\big{(}A_{R}^{(j)}\big{)}}_{1\leq j\leq 9}\in\mathcal{S}(X_{R})$ satisfying
$(c_{1},c_{2})\in CL_{g_{L}}({\big{(}A_{L}^{(j)}\big{)}}_{1\leq j\leq 9})\cap
CL_{g_{R}}({\big{(}A_{R}^{(j)}\big{)}}_{1\leq j\leq 9})$. By the definition of
neighbour sets, we have $v_{3}\in A_{L}^{(1)}\cap B_{R}^{(2)}$, a
contradiction with ($i$). Similar result can be gained if
$v_{1}v_{2},v_{2}v_{3}\in E_{R}^{\prime},v_{3}v_{4},v_{4}v_{5}\in
E_{L}^{\prime}$.

By the same argument, cases 2 to 4 in Fig. 1 can not occur by the conditions
($ii$) to ($v$). Thus $f^{\prime}$ is a valid partial coloring which implies
$CL_{f}$ is active.

Proof of Theorem 3.1 In order to prove Theorem 3.1, we describe our algorithm.
For a given graph $G$ and a number of colors $c$, we first calculate the
smooth tree-decomposition of $G$. Then, we calculate every active color class
functions on each node $X_{i}$ in the following way.

  1. 1.

$X_{i}$ is a leaf node. Enumerate all active $CL_{f}$ of some $f$ on $X_{i}$.
There are at most $c^{{(k+1)}^{2}\Delta^{6}}$ color class functions.

  2. 2.

$X_{i}$ is an internal node. For each active color class functions
$CL_{f_{L}}$ and $CL_{f_{R}}$ on $X_{L}$ and $X_{R}$ respectively, we
calculate all active color class function $CL_{f}$ of $f$ on $X_{i}$ by Lemma
3.3. There are at most $c^{2{(k+1)}^{2}\Delta^{6}}$ pair of active
$CL_{f_{L}},CL_{f_{R}}$ and verify Lemma 3.3 for each color class pair takes
$O(1)$ time.

Suppose $X_{root}$ is the root node in the smooth tree-decomposition. Finally,
there is an active color class function $CL_{f_{r}}$ of $X_{root}$ if and only
if $G$ have a valid star edge coloring within $c$ colors. Since there are at
most $O(n)$ nodes in the smooth tree-decomposition, the whole algorithm takes
$O(nc^{2{(k+1)}^{2}\Delta^{6}})$ time.

From the above Lemmas, we can easily prove that there is a valid partial of
$X_{i}$ if and only if the corresponding color class function $CL$ can be
computed by the above algorithm. We did not describe how to find a star edge
coloring using the specified number of colors. But this can be done easily, by
doing slight modifications in the algorithm (using extra bookkeeping to store
coloring information). We skip the details.

This completes the proof of Theorem 3.1.

It is worth to mention that the algorithm can run parallelly in
$O(c^{2{(k+1)}^{2}\Delta^{6}}\log n)$ time on $O(n)$ terminals.

##  4 Discussion

In our algorithm, $\Delta$ must be considered a constant because we need to
count the number of color class functions. To remove the restriction of
$\Delta$, we define color class count function
$CF\left({\big{(}A^{(j)}\big{)}}_{1\leq j\leq
9}\right)=\left|CL\left({\big{(}A^{(j)}\big{)}}_{1\leq j\leq
9}\right)\right|$. Since $CL$ is a partition of $C^{2}$, we have:

$\sum_{{\big{(}A^{(j)}\big{)}}_{1\leq j\leq
9}\in\mathcal{S}(X_{i})}CF\left({\big{(}A^{(j)}\big{)}}_{1\leq j\leq
9}\right)=c^{2}.$

The number of color class count functions is at most $c^{2^{3k^{2}+6k}}$. If
the following Conjecture 4.1 is proved to be true, we can store $CF$ for each
$X_{i}$ rather than $CL$, and design a polynomial time algorithm.

###### Conjecture 4.1

If $f,g$ are two partial coloring of $G_{i}^{\prime}$ with the same color
class count function, i.e. $CF_{f}\left({\big{(}A^{(j)}\big{)}}_{1\leq j\leq
9}\right)=CF_{g}\left({\big{(}A^{(j)}\big{)}}_{1\leq j\leq 9}\right)$ for all
${\big{(}A^{(j)}\big{)}}_{1\leq j\leq 9}\in\mathcal{S}(X_{i})$, then there
exists a permutation $\pi$ of colors such that $CL_{\pi\circ
f}\left({\big{(}A^{(j)}\big{)}}_{1\leq j\leq
9}\right)=CL_{g}\left({\big{(}A^{(j)}\big{)}}_{1\leq j\leq 9}\right)$, for all
${\big{(}A^{(j)}\big{)}}_{1\leq j\leq 9}\in\mathcal{S}(X_{i})$.

## Acknowledgement

This research was supported by the National Natural Science Foundation of
China (Grant 12171272 &12161141003).

## References

  * [1] M.O. Albertson, G.G. Chappell, H.A. Kierstead, A. Kündgen and R. Ramamurthi. Coloring with no $2$-colored $P_{4}$’s, _the Electronic Journal of Combinatorics_ , 11(2004), #R26.
  * [2] S. Arnborg, D.G. Corneil and A. Proskurowski. Complexity of finding embeddings in a $k$-tree, _SIAM Journal on Algebraic Discrete Methods_ , 8(2)(1987), 277-284.
  * [3] U. Bertele and F. Brioschi. _Nonserial dynamic programming_ , Academic Press, Inc., 1972.
  * [4] H.L. Bodlaender. Polynomial algorithms for graph isomorphism and chromatic index on partial $k$-trees, _Journal of Algorithms_ , 11(4)(1990), 631-643.
  * [5] H.L. Bodlaender. A linear-time algorithm for finding tree-decompositions of small treewidth, _SIAM Journal on Computing_ , 25(6)(1996), 1305-1317.
  * [6] H. Bruhn, R. Lang and M. Stein. List edge-coloring and total coloring in graphs of low treewidth, _Journal of Graph Theory_ , 81(3)(2016), 272-282.
  * [7] T.F. Coleman and J.J. Moré. Estimation of sparse jacobian matrices and graph coloring blems, _SIAM Journal on Numerical Analysis_ , 20(1)(1983), 187-209.
  * [8] Z. Dvořák, B. Mohar and R. Šámal. Star chromatic index, _Journal of Graph Theory_ , 72(3)(2013), 313-326.
  * [9] G. Fertin, A. Raspaud and B. Reed. Star coloring of graphs, _Journal of Graph Theory_ , 47(3)(2004), 163-182.
  * [10] R. Halin. $S$-functions for graphs, _Journal of geometry_ , 8(1976), 171-186.
  * [11] S. Kerdjoudj, K. Pradeep and A. Raspaud. List star chromatic index of sparse graphs, _Discrete Mathematics_ , 341(7)(2018), 1835-1849.
  * [12] H. Lei and Y. Shi. A survey on star edge-coloring of graphs, _arXiv e-prints_ , arXiv:2009.08017, 2020.
  * [13] H. Lei, Y. Shi and Z.X. Song. Star chromatic index of subcubic multigraphs, _Journal of Graph Theory_ , 88(4)(2018), 566-576.
  * [14] X.S. Liu and K. Deng. An upper bound on the star chromatic index of graphs with $\Delta\geq 7$, _Journal-Lanzhou University Natural Science_ , 44(2008), 94-95.
  * [15] B. Lužar, M. Mockovčiaková and R. Soták. On a star chromatic index of subcubic graphs, _Electronic Notes in Discrete Mathematics_ , 61(2017), 835-839.
  * [16] B. Omoomi, E. Roshanbin and M.V. Dastjerdi. A polynomial time algorithm to find the star chromatic index of trees, _Electron. J. Combin._ , 28(1)(2021), #1.6.
  * [17] K. Pradeep and V. Vijayalakshmi. Star chromatic index of subcubic graphs, _Electronic Notes in Discrete Mathematics_ , 53(2016), 155-164.
  * [18] N. Robertson and P.D. Seymour. Graph minors. iii. planar tree-width, _Journal of Combinatorial Theory, Series B_ , 36(1)(1984), 49-64.
  * [19] M.R. Salavatipour. A polynomial time algorithm for strong edge coloring of partial $k$-trees, _Discrete Applied Mathematics_ , 143(1)(2004), 285-291.
  * [20] Y. Wang, W. Wang and Y. Wang. Edge-partition and star chromatic index, _Applied Mathematics and Computation_ , 333(2018), 480-489.
  * [21] X. Zhou, S. Nakano and T. Nishizeki. Edge-coloring partialk-trees, _Journal of Algorithms_ , 21(3)(1996), 598-617.


# Bounds on the Quality-factor of Two-phase Quasi-static Metamaterial
Resonators and Optimal Microstructure Designs

Kshiteej J. Deshmukh*  | * Graeme W. Milton*  |<EMAIL_ADDRESS>
###### Abstract

Material resonances are fundamentally important in the field of nano-photonics
and optics. So it is of great interest to know what are the limits to which
they can be tuned. The bandwidth of the resonances in materials is an
important feature which is commonly characterized by using the quality (Q)
factor. We present bounds on the quality factor of two-phase quasi-static
metamaterial resonators evaluated at a given resonant frequency by introducing
an alternative definition for the Q-factor in terms of the complex effective
permittivity of the composite material. Optimal metamaterial microstrcuture
designs achieving points on these bounds are presented. The most interesting
optimal microstructure, is a limiting case of doubly coated ellipsoids that
attains points on the lower bound. We also obtain bounds on Q for three
dimensional, isotropic, and fixed volume fraction two-phase quasi-static
metamaterials. Some almost optimal isotropic microstructure geometries are
identified.

††preprint: AIP/123-QED

The explosion of interest in metamaterials has been driven by the realization
that they can break expected limits. These limits, or bounds, are based on
assumptions that do not hold for the metamaterials in question. Naturally, one
wants to know what these bounds are, and here we focus on a fundamental
problem: bounding the Q-factor of resonances in metamaterials, under the
assumption that one is in the quasi-static limit.

Resonances in materials have led to many exciting properties and applications
in nano-photonics and optics. A famous example of material resonance is the
"Lycurgus cup", which is a $\displaystyle 4^{th}$ century Roman drinking cup
made of glass with fine particles of gold suspended in it. The resonances of
the gold particles at optical wavelengths cause it to appear either red or
green depending on where the light shines from. By making the gold particles
hollow one can shift the resonant frequencyAden and Kerker (1951); Averitt,
Sarkar, and Halas (1997), even into the infrared where nanoshells have proved
significant in destroying cancer cellsRastinehad  _et al._ (2019). Resonances
are also studied extensively in antenna theory. A large body of work is
available on the design of antennas Gustafsson and Nordebo (2012); Gustafsson,
Capek, and Schab (2019) with applications ranging from electronic sensors to
radio-frequency energy harvesting, and wearable antennas Qian and Itoh (1998);
Guo  _et al._ (2016); Zhu  _et al._ (2021). An important factor in antenna
design is the operating frequency-bandwidth about the resonance, and this is
typically quantified by defining the quality (Q) factor of an antenna Collin
and Rothschild (1964). Depending on the application it may be desirable to
have a low Q or a high Q.

A number of definitions for Q are found in literature that try to best
approximate the exact bandwidth of an antennaCollin and Rothschild (1964);
Fante (1969); Rhodes (1976); Yaghjian and Best (2005); Gustafsson and Nordebo
(2006). The two most common conventional definitions of Q-factor that have
been in use are: one, the ratio of energy stored to energy radiated or
dissipated; and, two, the ratio of center (resonance) frequency to frequency-
bandwidth.

Figure 1: $\displaystyle Q_{c}$ (circle marker) vs. $\displaystyle
Q_{\mathrm{new}}$ (cross marker) comparison for Lorentz models with varying
values of the damping coefficient $\displaystyle\gamma$ and resonant frequency
$\displaystyle\omega_{0}$. In the limits $\displaystyle\omega_{0}/\gamma>>1$
and $\displaystyle\omega_{0}/\Delta\omega>>1$, we see good agreement between
$\displaystyle Q_{c}$ and $\displaystyle Q_{\mathrm{new}}$. The inset shows
the discrepancy between $\displaystyle Q_{c}$ and $\displaystyle
Q_{\mathrm{new}}$ near the origin where the approximation no longer holds.

Using the first definition of Q-factor, there are several works that have
proposed limitations or bounds on the Q-factor of antennas. ChuChu (1948)
first set a lower limit on Q-factor for small antennas. StuartStuart (2008)
proposed bandwidth limitations on small antennas with negative permittivity
materials, while lower bounds on small antennas were given by Yaghjian et
al.Yaghjian and Stuart (2010). Gustafsson et al.Gustafsson and Nordebo (2012)
considered optimizing current densities to numerically obtain physical bounds
on antennas of arbitrary shape and size, and also provided shape dependent
bounds in a series of papersGustafsson, Cismasu, and Jonsson (2012);
Gustafsson  _et al._ (2015); Jonsson and Gustafsson (2015). More recently,
lower bounds on Q-factors of small-size, high-permittivity, dielectric
resonators were obtained by Pascale et al.Pascale  _et al._ (2023).

In this work, we focus on the problem of finding bounds on the Q-factor in
lossy two-phase quasi-static metamaterial resonators, and on identifying
microgeometries that achieve those bounds. The use of metamaterials enriches
the design space of resonators. To our knowledge, bounds on Q in this case
have not been studied so far. We consider two-phase metamaterials where the 2
pure phases are isotropic, dielectric materials, with complex relative
permittivity $\displaystyle\epsilon_{1}(\omega)$ of one phase given, for
example, by a Drude model or a Lorentz model, and the relative permittivity
$\displaystyle\epsilon_{2}(\omega)=\epsilon_{2}$ of the other phase is taken
to be a real constant. The effective permittivity of the metamaterial is some
complex valued function of the permittivities of the pure phases
$\displaystyle\varepsilon_{\mathrm{eff}}(\omega)=\varepsilon_{\mathrm{eff}}(\epsilon_{1}(\omega),\epsilon_{2})$.
Without loss of generality, we can rescale the dimensions and set the
permittivity of the constant phase asMilton (1981a),
$\displaystyle\epsilon_{2}=1$. Considering the ambiguity associated with
defining the energy stored in a lossy materialGustafsson and Jonsson (2015);
Schab  _et al._ (2018), we adopt the second definition for Q, i.e.,
$\displaystyle Q=\omega_{R}/\Delta\omega$, where $\displaystyle\omega_{R}$ is
the resonance frequency and $\displaystyle\Delta\omega$ is the bandwidth at
half-height of the resonance value, and refer to this definition as the
conventional Q-factor, $\displaystyle Q_{c}$. However, to proceed with finding
the bounds on Q in two-phase metamaterials it is desirable to have an
expression for Q in terms of the material parameters at the resonant
frequency.

Assume that the response of a material permittivity is given by a Lorentz
model,

$\varepsilon=1+\frac{\omega_{p}^{2}}{\omega_{0}^{2}-\omega^{2}-\imath\omega\gamma},$
(1)

where $\displaystyle\omega_{p}$, $\displaystyle\omega_{0}$, and
$\displaystyle\gamma$ are the plasma frequency, natural frequency, and damping
coefficient respectively. We show analytically, that in the limit
$\displaystyle\omega_{0}/\gamma>>1$ (see supplementary information (SI)
Section S1 and Figure S1 for details),
$\displaystyle\gamma\approx\Delta\omega$, i.e. the bandwidth is narrow
compared to the resonance frequency, and the resonant frequency
$\displaystyle\omega_{R}$ approaches the natural frequency
$\displaystyle\omega_{0}$ ($\displaystyle\omega_{R}\approx\omega_{0}$). Then,
the expression for $\displaystyle Q_{c}$ at resonance can be approximated
alternatively by defining a new Q-factor ($\displaystyle Q_{\mathrm{new}}$) in
terms of the material parameters:

$Q_{c}\approx
Q_{\mathrm{new}}=-\frac{\omega_{0}}{2}\left[\operatorname{Im}\Big{(}\varepsilon(\omega_{0})\Big{)}\right]^{-1}\operatorname{Re}\Big{(}\frac{\
\mathrm{d}\varepsilon(\omega)}{\
\mathrm{d}\omega}\Big{)}\Big{|}_{\omega=\omega_{0}}.$ (2)

Here $\displaystyle\operatorname{Re}(\cdot)$ and
$\displaystyle\operatorname{Im}(\cdot)$ denote the real and imaginary parts.
This expression (2) is similar to ones found in the literature for antennas
with known impedanceRhodes (1976); Yaghjian and Best (2005). Figure 1
illustrates the good agreement between $\displaystyle Q_{c}$ and
$\displaystyle Q_{\mathrm{new}}$ for different Lorentz models with varying
values of $\displaystyle\gamma$ and $\displaystyle\omega_{0}$ even for
moderately large values of $\displaystyle\omega_{0}/\gamma$. The zoomed inset
shows that close to origin, where $\displaystyle\omega_{0}$ is comparable to
$\displaystyle\gamma$ and we are well outside the validity of the
approximation, $\displaystyle Q_{c}$ and $\displaystyle Q_{\mathrm{new}}$
differ significantly.

The main idea to obtain bounds on $\displaystyle Q_{\mathrm{new}}$ in two-
phase metamaterials at a given resonance frequency ($\displaystyle\omega_{R}$)
of the composite, is to correlate, at $\displaystyle\omega=\omega_{R}$, the
quantities $\displaystyle\operatorname{Re}\Big{(}\frac{\
\mathrm{d}\varepsilon_{\mathrm{eff}}(\omega)}{\ \mathrm{d}\omega}\Big{)}$ and
$\displaystyle\operatorname{Im}\Big{(}\varepsilon_{\mathrm{eff}}(\omega)\Big{)}$
when $\displaystyle\operatorname{Im}\Big{(}\frac{\
\mathrm{d}\varepsilon_{\mathrm{eff}}(\omega)}{\ \mathrm{d}\omega}\Big{)}=0$,
the latter being a necessary condition for resonance at
$\displaystyle\omega_{R}$. Note that the effective permittivity of the
metamaterial is a function of its pure phases,
$\displaystyle\epsilon_{1}(\omega)$ and $\displaystyle\epsilon_{2}=1$. The
derivative with respect to frequency of the effective permittivity can be
rewritten as,

$\frac{\ \mathrm{d}\varepsilon_{\mathrm{eff}}(\omega)}{\
\mathrm{d}\omega}\Big{|}_{\omega=\omega_{R}}=\left.\bigg{(}\frac{\partial\varepsilon_{\mathrm{eff}}(\epsilon_{1}(\omega),1)}{\partial\epsilon_{1}}\frac{\
\mathrm{d}\epsilon_{1}(\omega)}{\
\mathrm{d}\omega}\bigg{)}\right|_{\omega=\omega_{R}},$ (3)

where $\displaystyle\frac{\ \mathrm{d}\epsilon_{1}(\omega)}{\
\mathrm{d}\omega}$ is a known quantity as $\displaystyle\epsilon_{1}(\omega)$
is known.

Figure 2: $\displaystyle Q_{\mathrm{new}}$ vs.
$\displaystyle\operatorname{Im}(\varepsilon_{\mathrm{eff}})$ for two-phase
metamaterial quasi-static resonators with $\displaystyle\epsilon_{1}(\omega)$
given by the Drude model. Bounds on $\displaystyle Q_{\mathrm{new}}$ (solid
black curve) are shown as the value of
$\displaystyle\operatorname{Im}(\varepsilon_{\mathrm{eff}}(\omega_{R}))$ at
resonance frequency $\displaystyle\omega_{R}=3$ is varied within the range of
$\displaystyle\varepsilon_{\mathrm{eff}}(\omega_{R})$. Metamaterial geometries
achieving all points on these bounds are shown: Coated ellipsoids (CE1: yellow
squares) attain values on the upper bound for $\displaystyle Q_{\mathrm{new}}$
as the eccentricities and volume fractions of the core phase are varied;
laminate geometry (L: green triangle) attains the point on extreme right;
points on the lower bound are achieved by doubly coated ellipsoids (DCE1: red
circles). Specific points on the bounds are highlighted as A, B, C, D, and E
and more information on the microstructure geometries at these points is given
in Figure 3

This reduces the problem to correlating the quantities
$\displaystyle\partial\varepsilon_{\mathrm{eff}}(\epsilon_{1}(\omega),1)/\partial\epsilon_{1}$
and
$\displaystyle\operatorname{Im}\Big{(}\varepsilon_{\mathrm{eff}}(\epsilon_{1}(\omega),1)\Big{)}$
at $\displaystyle\omega=\omega_{R}$. To obtain this correlation we formulate
the following problem: Given a fixed realizable value of
$\displaystyle\operatorname{Im}\left(\varepsilon_{\mathrm{eff}}(\epsilon_{1}(\omega),1)\right)$
at $\displaystyle\omega=\omega_{R}$, what are the bounds on the values of
$\displaystyle\operatorname{Re}\Big{(}\partial\varepsilon_{\mathrm{eff}}(\epsilon_{1}(\omega),1)/\partial\epsilon_{1}\Big{)}$,
with the constraint that
$\displaystyle\left.\operatorname{Im}\bigg{(}\frac{\partial\varepsilon_{\mathrm{eff}}(\epsilon_{1}(\omega),1)}{\partial\omega}\bigg{)}\right|_{\omega=\omega_{R}}=0$,
which is associated with resonance at $\displaystyle\omega=\omega_{R}$? We
solve this problem numerically by using the bounds of Milton et al.Milton
(1980, 1981a, 1981b, 2002) (see, Chapter 27 in MiltonMilton (2002)), and more
details on the method can be found in SI Section S2. In fact, in a broader
mathematical context outside the theory of composites, there is a long history
of such bounds: see, for example, Krein and NudelmanKrein and Nudelman (1977).
From the bounds, we then easily obtain constraints on $\displaystyle
Q_{\mathrm{new}}$ for any possible value of
$\displaystyle\operatorname{Im}\Big{(}\varepsilon_{\mathrm{eff}}(\epsilon_{1}(\omega_{R}),1)\Big{)}$.

First, we present our results for 2-phase composites with
$\displaystyle\epsilon_{1}(\omega)$ given by the Drude model,

$\epsilon_{1}(\omega)=1+\omega_{p}^{2}/(\omega^{2}+\imath\omega\gamma).$ (4)

with, $\displaystyle\omega_{p}=5$, $\displaystyle\gamma=0.1$. Plots for bounds
on the effective complex permittivity
$\displaystyle\varepsilon_{\mathrm{eff}}(\omega_{R})$ and for the
corresponding range of $\displaystyle\ \mathrm{d}\varepsilon_{\mathrm{eff}}/\
\mathrm{d}\omega$ in this case are shown in the SI (Section S2). Figure 2
shows the wedge-shaped bounds on $\displaystyle Q_{\mathrm{new}}$ (shown by
the solid black curve) as the resonance value of
$\displaystyle\operatorname{Im}(\varepsilon_{\mathrm{eff}})$ varies within the
range prescribed by $\displaystyle\varepsilon_{\mathrm{eff}}(\omega_{R})$. To
obtain our bounds on $\displaystyle Q_{\mathrm{new}}$, we allow for values of
$\displaystyle\operatorname{Re}\Big{(}\
\mathrm{d}{\varepsilon_{\mathrm{eff}}(\omega)}/\ \mathrm{d}{\omega}\Big{)}$ at
$\displaystyle\omega=\omega_{R}$ that are positive. Due to this and from (2),
we see that negative values of $\displaystyle Q_{\mathrm{new}}$ are allowed by
the bounds as seen in Figure 2. Negative Q-factor values are irrelevant for
our study, and as such they must be ignored when they occur in our bounds.

Figure 3: Optimal two-phase metamaterial microstructure designs: Pure phase
with permittivity given by Drude model is shown in pink and the pure phase
with constant permittivity is shown in gray. a) Coated ellipsoid (CE1)
assemblage corresponding to point A in Figure 2. b) Laminate geometry (L),
corresponding to point B in Figure 2. c) Doubly coated ellipsoid (DCE1)
assemblage (Point C in Figure 2). In each of the designs, optimal
$\displaystyle Q_{\mathrm{new}}$ is obtained in the direction of the applied
electric field (seen in blue).

Further, we also find the optimal metamaterial microstructures that achieve
these bounds. Specifically, we find that points on the upper horizontal bound
are achieved by assemblages of confocal coated ellipsoids (CE1) with the core
phase given by $\displaystyle\epsilon_{1}(\omega)$ (4). The effective
permittivity of the coated ellipsoid (CE1) has two free parametersMilton
(2002); the depolarization factor and the volume fraction. The depolarization
factor is used to tune the resonance and the volume fraction can be varied
within limits to trace points on the upper bound. In Figure 2 the yellow
square markers denote the values from CE1. Simple laminate geometries (L)
attain only the extreme right point on the bounds (green triangle), when the
layers of the laminate are normal to the direction of the electric field.
Since, volume fraction is the only free parameter in the complex effective
permittivity of laminates, it is varied to fix the pole and we get only one
point. In the case when electric field is parallel to the layers, the
effective permittivity of the laminate is an arithmetic mean of
$\displaystyle\epsilon_{1}(\omega)$ and $\displaystyle\epsilon_{2}=1$, and
consequently there is no resonance observed at any frequency. The lower bound
in Figure 2 is traced by assemblages of highly sensitive doubly coated
ellipsoids (DCE1, shown by red circles) where the innermost and outermost
phases are the same and given by $\displaystyle\epsilon_{1}(\omega)$. The
inner coated ellipsoid and outer coated ellipsoid are not restricted to have
the same eccentricities, and hence are more general than confocal doubly
coated ellipsoids. Doing so provides us with 4 parameters; the depolarization
factor of the inner coated ellipsoid, depolarization factor of the outer
coated ellipsoid, volume fraction of the Drude phase, and volume fraction of
the core phase. These parameters can be varied so that the resulting effective
permittivity function
$\displaystyle\varepsilon_{\mathrm{eff}}(\epsilon_{1},1)$ matches any formula
that defines the bounds (see, Section 18.5 in MiltonMilton (2002)). As such,
the assemblages of doubly coated ellipsoids necessarily attain values on the
bound.

Figure 4: Plots of $\displaystyle\varepsilon_{\mathrm{eff}}$ vs.
$\displaystyle\omega$, associated with the optimal metamaterial designs. a)
$\displaystyle\operatorname{Im}(\varepsilon_{\mathrm{eff}})$ vs.
$\displaystyle\omega$, and b)
$\displaystyle\operatorname{Re}(\varepsilon_{\mathrm{eff}})$ vs.
$\displaystyle\omega$, for the points A (coated ellipsoid, CE1, shown here by
yellow line), B (laminate, L, shown by green line), and C (doubly coated
ellipsoid, DCE1, shown by red line) from Figure 2. c), and d) show plots for
$\displaystyle\operatorname{Im}(\varepsilon_{\mathrm{eff}})$ vs.
$\displaystyle\omega$, and
$\displaystyle\operatorname{Re}(\varepsilon_{\mathrm{eff}})$ vs.
$\displaystyle\omega$, respectively, for three different doubly coated
ellipsoid geometries that achieve the points C, D, and E on the bounds in
Figure 2. Note, the behavior of
$\displaystyle\operatorname{Im}(\varepsilon_{\mathrm{eff}})$ in c), where we
can see a small local minimum (blue and cyan colored curves) at
$\displaystyle\omega_{R}=3$.

In Figure 3, we present schematic drawings of the optimal metamaterial designs
corresponding to three specific points A, B, and C on the bounds in Figure 2.
In each of the subfigures, $\displaystyle\epsilon_{1}(\omega)$ is shown in
pink color, and the constant phase $\displaystyle\epsilon_{2}=1$ is shown in
gray color. Point A corresponds to a coated ellipsoid assemblage depicted in
Figure 3a. Point B corresponds to a material with laminate geometry which is
shown in Figure 3b. Most interesting, is Point C that corresponds to a
limiting case of doubly coated ellipsoid geometry as described before, with
the following parameters: depolarization factors of the inner and outer coated
ellipsoid are $\displaystyle 0.5614$ and $\displaystyle 1$, respectively, the
volume fraction of the outermost phase $\displaystyle=0.4374$, and volume
fraction of the core phase $\displaystyle=0.0002363$. Thus, the ellipsoidal
inclusions only occupy an extremely small volume fraction, but are significant
due to resonance effects. The schematic drawing seen in Figure 3c, depicts
this laminate geometry with the constant phase (outermost layer of DCE1)
forming one of the laminates, and the second laminate being formed from a very
dilute assemblage of coated ellipsoids (inner coated ellipsoid of DCE1). For
each of these designs, $\displaystyle Q_{\mathrm{new}}$ values attaining the
bounds are obtained when the electric field is applied in the direction shown
by the blue arrows.

Next, we obtain the response of
$\displaystyle\varepsilon_{\mathrm{eff}}(\omega)$ with respect to
$\displaystyle\omega$ for the specific geometries indicated by the points A
(CE1), B (L), C (DCE1), D (DCE1), and E (DCE1) in Figure 2. Figures 4a-4b,
show the plots for
$\displaystyle\operatorname{Im}(\varepsilon_{\mathrm{eff}})$ vs.
$\displaystyle\omega$ and
$\displaystyle\operatorname{Re}(\varepsilon_{\mathrm{eff}})$ vs.
$\displaystyle\omega$, respectively, for three different geometries given by
points A, B, and C. We observe that despite the large difference in the
$\displaystyle\operatorname{Im}(\varepsilon_{\mathrm{eff}})$ values of points
A (CE1, seen in yellow) and B (L, seen in green), they have the same
$\displaystyle Q_{\mathrm{new}}$. Figures 4c-d show a similar comparison, but
for three similar doubly coated ellipsoid geometries that attain points C, D,
and E on the lower bound. We make an interesting observation here with regards
to the values of $\displaystyle\operatorname{Im}(\varepsilon_{\mathrm{eff}})$
at the resonant frequency $\displaystyle\omega=\omega_{R}$. As we move from
point C towards point E, we observe the values of
$\displaystyle\operatorname{Im}(\varepsilon_{\mathrm{eff}})$ at resonance
frequency undergo a transition and exhibit a small local minimum at the center
frequency of the bandwidth (shown by blue and cyan colored curves in Figure
4c).

Figure 5: Bounds on $\displaystyle Q_{\mathrm{new}}$ for 3-d, isotropic, and
fixed volume fraction 2-phase quasi-static metamaterial resonators at the
resonance frequency $\displaystyle\omega_{R}=3$ with
$\displaystyle\epsilon_{1}(\omega)$ given by the Drude model. a)
$\displaystyle Q_{\mathrm{new}}$ bounds for volume fractions $\displaystyle
f_{1}=0.1$ (blue line), $\displaystyle f_{1}=0.25$ (dashed green line),
$\displaystyle f_{1}=0.5$ (red line), $\displaystyle f_{1}=0.75$ (yellow line)
are superposed for comparison with the volume fraction independent bounds
(solid black curve). b) $\displaystyle Q_{\mathrm{new}}$ values obtained from
Schulgasser laminates of doubly coated ellipsoids with the outer coated
ellipsoid forming a prolate spheroid are plotted (green circles). The solid
red line in b) denotes the bounds for $\displaystyle f_{1}=0.5$. Some of the
doubly coated ellipsoid geometries are found to be almost optimal

Similarly, bounds on $\displaystyle Q_{\mathrm{new}}$ can be obtained for the
case when the permittivity of the pure phase
$\displaystyle\epsilon_{1}(\omega)$ is given by a Lorentz model. In the SI, we
present plots for $\displaystyle Q_{\mathrm{new}}$ vs.
$\displaystyle\operatorname{Im}(\varepsilon_{\mathrm{eff}})$ for 3 different
frequencies; one, at the resonance frequency of the pure phase, two, near the
resonance of the pure phase, and three at a frequency away from the resonance
of the pure phase (see, Section S4 in SI). These results show that the region
enclosed by the $\displaystyle Q_{\mathrm{new}}$ bounds can be non-convex.

We now consider the problem of obtaining bounds on $\displaystyle
Q_{\mathrm{new}}$, when the 2-phase metamaterials are three-dimensional (3-d),
isotropic, and have fixed volume fraction $\displaystyle f_{1}$ for the pure
phase $\displaystyle\epsilon_{1}(\omega)$, which is again given by the Drude
model, and the second phase is chosen to be $\displaystyle\epsilon_{2}=1$. To
obtain bounds that include the volume fraction and the fact that the
microstructure is isotropic, we let $\displaystyle f_{2}=1-f_{1}$, be the
volume fraction of phase 2 and introduce the functionMilton and Golden (1985),

$\widetilde{\varepsilon}_{\mathrm{eff}}(\epsilon_{1},\epsilon_{2})=-\frac{1}{2}f_{2}\epsilon_{1}-\frac{1}{2}f_{1}\epsilon_{2}+\frac{f_{1}f_{2}(\epsilon_{1}-\epsilon_{2})^{2}}{2[f_{1}\epsilon_{1}+f_{2}\epsilon_{2}-\varepsilon_{\mathrm{eff}}]},$
(5)

(related to the so called Y transformMilton (1981a). See, Chapters 19 and 20
in MiltonMilton (2002)) that has basically the same analytic properties, and
hence is subject to basically the same bounds, as
$\displaystyle\varepsilon_{\mathrm{eff}}(\epsilon_{1},\epsilon_{2})$. In the
isotropic case bounds on $\displaystyle\varepsilon_{\mathrm{eff}}$ with known
volume fraction were first obtained by Bergman and MiltonMilton (1980);
Bergman (1980); Milton (1981a, b), and were recently improved by Kern, Miller
and MiltonKern, Miller, and Milton (2020). We seek bounds that also involve
$\displaystyle\frac{\ \mathrm{d}\varepsilon_{\mathrm{eff}}(\omega)}{\
\mathrm{d}\omega}$ (see SI Section S4 for more explanation).

Bounds on $\displaystyle Q_{\mathrm{new}}$ for all possible values of
$\displaystyle\operatorname{Im}(\varepsilon_{\mathrm{eff}})$ are obtained for
four different volume fractions, $\displaystyle f_{1}=0.1,0.25,0.5,$ and
$\displaystyle 0.75$, for a resonance frequency of
$\displaystyle\omega_{R}=3$. Figure 5a shows all the 3-d, isotropic, fixed
volume fraction bounds superposed on top of the bounds shown in Figure 2 for
comparison, as they are all evaluated at the same frequency
$\displaystyle\omega_{R}=3$. The plots show that the area of the region
occupied by the bounds does not monotonically increase as the volume fraction
is increased, which is clear since the bounds for $\displaystyle f_{1}=0.5$
(solid red curve) occupies a larger region than the bounds for $\displaystyle
f_{1}=0.75$ (solid yellow curve), and the bounds for $\displaystyle
f_{1}=0.25$ (dashed green curve).

Here too, we find some optimal isotropic, fixed volume fraction metamaterial
designs that attain certain points on the bounds by using the
SchulgasserSchulgasser (1977) lamination technique to construct the isotropic
effective permittivities. Given an anistropic effective tensor
$\displaystyle\boldsymbol{\varepsilon}_{\mathrm{eff}}$, he showed that one
could obtain an isotropic material with permittivity
$\displaystyle\mathrm{Tr(\boldsymbol{\varepsilon}_{\mathrm{eff}})}/3$.

The almost optimal geometries are Schulgasser laminates of assemblages of
doubly coated ellipsoids with the outer coated ellipsoid forming a prolate
spheroid, while there is no special form of the inner coated ellipsoid (see SI
for parameter values). Figure 5b, shows the $\displaystyle Q_{\mathrm{new}}$
values (green dots) obtained by doubly coated ellipsoids with outer coated
ellipsoid taken to be prolate spheroids for a volume fraction of
$\displaystyle f_{1}=0.5$, with some microstructures apparently attaining
points on the bound.

Similar, bounds on $\displaystyle Q_{\mathrm{new}}$ can be obtained for 3-d,
isotropic, fixed volume fraction materials using Lorentz model for
$\displaystyle\epsilon_{1}(\omega)$. The associated results are presented in
the supplementary information. Note that, when one of pure phases is
Lorentzian, $\displaystyle\varepsilon_{\mathrm{eff}}$ may have resonances due
to both resonance of the pure Lorentzian phase, and resonances due to the
composite microgeometry. Instead, if the pure phase is given by the Drude
model (4), then $\displaystyle\varepsilon_{\mathrm{eff}}$ has resonances due
to microstructure geometry only.

In conclusion, our work provides limits on the quality of resonances that can
be achieved in 2-phase quasi-static metamaterial resonators. Such resonances
are important in nano-photonics and optics. Optimal metamaterial
microstructures have been identified that achieve these limits. It will be
interesting to see then, how these theoretical and numerical results compare
against any experiments done to validate the results in the quasi-static
regime or invalidate them at higher frequencies. Since, this work provides
bounds on $\displaystyle Q_{\mathrm{new}}$ for isotropic metamaterials too, it
can be applied to design directional and omni-directional resonance responses
in metamaterials.

See supplementary information document for additional details related to this
work.

The authors are grateful to the National Science Foundation for support
through grant DMS-2107926. We thank Mats Gustafsson for suggesting a
connection between Q-factor and the derivative
$\displaystyle\operatorname{Re}\left(\frac{\
\mathrm{d}\varepsilon_{\mathrm{eff}}(\omega)}{\ \mathrm{d}\omega}\right)$ at
resonance. This formed the basis for our formula for $\displaystyle
Q_{\mathrm{new}}$.

## Author Declarations

### Conflict of Interest

The authors have no conflicts to disclose.

### Authors’ Contribution

K. J. Deshmukh and G.W. Milton contributed equally on this paper.

## Data Availability

Aside from the data available within the article, any other data are available
from the corresponding author upon request.

## References

## References

  * Aden and Kerker (1951) A. L. Aden and M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” Journal of Applied Physics 22, 1242–1246 (1951).
  * Averitt, Sarkar, and Halas (1997) R. Averitt, D. Sarkar, and N. Halas, “Plasmon resonance shifts of Au-coated $\displaystyle Au_{2}S$ nanoshells: insight into multicomponent nanoparticle growth,” Physical Review Letters 78, 4217 (1997).
  * Rastinehad  _et al._ (2019) A. R. Rastinehad, H. Anastos, E. Wajswol, J. S. Winoker, J. P. Sfakianos, S. K. Doppalapudi, M. R. Carrick, C. J. Knauer, B. Taouli, S. C. Lewis, _et al._ , “Gold nanoshell-localized photothermal ablation of prostate tumors in a clinical pilot device study,” Proceedings of the National Academy of Sciences 116, 18590–18596 (2019).
  * Gustafsson and Nordebo (2012) M. Gustafsson and S. Nordebo, “Optimal antenna currents for Q, superdirectivity, and radiation patterns using convex optimization,” IEEE Transactions on Antennas and Propagation 61, 1109–1118 (2012).
  * Gustafsson, Capek, and Schab (2019) M. Gustafsson, M. Capek, and K. Schab, “Tradeoff between antenna efficiency and Q-factor,” IEEE Transactions on Antennas and Propagation 67, 2482–2493 (2019).
  * Qian and Itoh (1998) Y. Qian and T. Itoh, “Progress in active integrated antennas and their applications,” IEEE Transactions on Microwave Theory and Techniques 46, 1891–1900 (1998).
  * Guo  _et al._ (2016) W. Guo, S. Zhou, Y. Chen, S. Wang, X. Chu, and Z. Niu, “Simultaneous information and energy flow for IoT relay systems with crowd harvesting,” IEEE Communications Magazine 54, 143–149 (2016).
  * Zhu  _et al._ (2021) J. Zhu, S. Zhang, N. Yi, C. Song, D. Qiu, Z. Hu, B. Li, C. Xing, H. Yang, Q. Wang, _et al._ , “Strain-insensitive hierarchically structured stretchable microstrip antennas for robust wireless communication,” Nano-Micro Letters 13, 1–12 (2021).
  * Collin and Rothschild (1964) R. Collin and S. Rothschild, “Evaluation of antenna Q,” IEEE Transactions on Antennas and Propagation 12, 23–27 (1964).
  * Fante (1969) R. Fante, “Quality factor of general ideal antennas,” IEEE Transactions on Antennas and Propagation 17, 151–155 (1969).
  * Rhodes (1976) D. R. Rhodes, “Observable stored energies of electromagnetic systems,” Journal of the Franklin Institute 302, 225–237 (1976).
  * Yaghjian and Best (2005) A. D. Yaghjian and S. R. Best, “Impedance, bandwidth, and Q of antennas,” IEEE Transactions on Antennas and Propagation 53, 1298–1324 (2005).
  * Gustafsson and Nordebo (2006) M. Gustafsson and S. Nordebo, “Bandwidth, Q factor, and resonance models of antennas,” Progress In Electromagnetics Research 62, 1–20 (2006).
  * Chu (1948) L. J. Chu, “Physical limitations of omni-directional antennas,” Journal of applied physics 19, 1163–1175 (1948).
  * Stuart (2008) H. R. Stuart, “Bandwidth limitations in small antennas composed of negative permittivity materials and metamaterials,” XXIX General Assembly of the International Union of Radio Science (URSI), Chicago, IL (2008).
  * Yaghjian and Stuart (2010) A. D. Yaghjian and H. R. Stuart, “Lower bounds on the Q of electrically small dipole antennas,” IEEE Transactions on Antennas and Propagation 58, 3114–3121 (2010).
  * Gustafsson, Cismasu, and Jonsson (2012) M. Gustafsson, M. Cismasu, and B. L. G. Jonsson, “Physical bounds and optimal currents on antennas,” IEEE transactions on antennas and propagation 60, 2672–2681 (2012).
  * Gustafsson  _et al._ (2015) M. Gustafsson, D. Tayli, M. Cismasu, and Z. Chen, “Physical bounds of antennas,” Handbook of Antenna Technologies , 1–32 (2015).
  * Jonsson and Gustafsson (2015) B. L. G. Jonsson and M. Gustafsson, “Stored energies in electric and magnetic current densities for small antennas,” Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 471, 20140897 (2015).
  * Pascale  _et al._ (2023) M. Pascale, S. A. Mann, D. C. Tzarouchis, G. Miano, A. Alù, and C. Forestiere, “Lower bounds to the Q factor of electrically small resonators through quasistatic modal expansion,” IEEE Transactions on Antennas and Propagation (2023).
  * Milton (1981a) G. W. Milton, “Bounds on the complex permittivity of a two-component composite material,” Journal of Applied Physics 52, 5286–5293 (1981a).
  * Gustafsson and Jonsson (2015) M. Gustafsson and B. L. G. Jonsson, “Stored electromagnetic energy and antenna Q,” Progress in Electromagnetics Research 150, 13–27 (2015).
  * Schab  _et al._ (2018) K. Schab, L. Jelinek, M. Capek, C. Ehrenborg, D. Tayli, G. A. Vandenbosch, and M. Gustafsson, “Energy stored by radiating systems,” IEEE Access 6, 10553–10568 (2018).
  * Milton (1980) G. W. Milton, “Bounds on the complex dielectric constant of a composite material,” Applied Physics Letters 37, 300–302 (1980).
  * Milton (1981b) G. W. Milton, “Bounds on the transport and optical properties of a two-component composite material,” Journal of Applied Physics 52, 5294–5304 (1981b).
  * Milton (2002) G. W. Milton, “The theory of composites. 2002,” Cambridge Monographs on Applied and Computational Mathematics (2002).
  * Krein and Nudelman (1977) M. G. Krein and A. A. Nudelman, “The Markov moment problem and extremal problems,” Translations of Mathematical Monographs 50 (1977).
  * Milton and Golden (1985) G. Milton and K. Golden, “Thermal conduction in composites,” Thermal conductivity 18 , 571–582 (1985).
  * Bergman (1980) D. J. Bergman, “Exactly solvable microscopic geometries and rigorous bounds for the complex dielectric constant of a two-component composite material.” Phys. Rev. Lett. 45, 148–148 (1980).
  * Kern, Miller, and Milton (2020) C. Kern, O. D. Miller, and G. W. Milton, “Tight bounds on the effective complex permittivity of isotropic composites and related problems,” Physical Review Applied 14, 054068 (2020).
  * Schulgasser (1977) K. Schulgasser, “Bounds on the conductivity of statistically isotropic polycrystals,” Journal of Physics C: Solid State Physics 10, 407 (1977).