#
Observations on the Lovász θ-Function, Graph Capacity, Eigenvalues, and Strong Products †

^{1}

^{2}

^{†}

## Abstract

**:**

## 1. Introduction

- (1)
- A known upper bound on the Lovász $\theta $-function of a regular graph is expressed in terms of the smallest eigenvalue of its adjacency matrix [6]. A key result in this work provides a lower bound on the Lovász $\theta $-function of a regular graph, which is expressed in terms of the second-largest eigenvalue of its adjacency matrix. New sufficient conditions for equalities in these bounds are also obtained (Proposition 1).
- (2)
- A simple and closed-form expression of the Lovász $\theta $-function is derived for all strongly regular graphs (Corollary 1).
- (3)
- Eigenvalue inequalities are derived, which relate the smallest and second-largest eigenvalues of a regular graph. They hold with equality if and only if the graph is strongly regular (Corollaries 2 and 3).
- (4)
- The Shannon capacity of several strongly regular graphs is determined (Section 3.5).
- (5)
- Bounds on parameters of regular graphs, and in particular of Ramanujan graphs, are derived (Corollaries 4–6).
- (6)
- Bounds on the smallest and the second-largest eigenvalues of strong products of regular graphs are derived, which are expressed in terms of calculable parameters of its factors (Proposition 2).
- (7)
- A new lower bound on the second-largest eigenvalue of a k-fold strong power of a regular graph is compared to the Alon–Boppana bound. Under a certain condition, the former bound shows an improvement in its exponential growth rate as a function of k (Section 3.3).
- (8)
- Every non-complete and non-empty connected regular graph, whose Lovász $\theta $-function is below a certain value, is proved to have the property that almost all its strong powers are highly non-Ramanujan (Proposition 3).
- (9)
- Lower bounds on the chromatic number of strong products of graphs are expressed in terms of the order and Lovász $\theta $-function of each factor (Proposition 4). Their utility is exemplified, while also leading to exact chromatic numbers in some cases.

## 2. Preliminaries

- (a)
- ${\mathsf{K}}_{n}$ denotes the complete graph with $n\in \mathbb{N}$ vertices, where every pair of distinct vertices are adjacent; hence, ${\mathsf{K}}_{1}$ is an empty graph with a single vertex.
- (b)
- ${\mathsf{K}}_{n,m}$ denotes the complete bipartite graph, which is a bipartite graph consisting of a vertex set that is a disjoint union of two finite sets ${\mathcal{V}}_{1}$ and ${\mathcal{V}}_{2}$ of cardinalities n and m, respectively, and a set of edges that are all the possible connections of a vertex in ${\mathcal{V}}_{1}$ and a vertex in ${\mathcal{V}}_{2}$.
- (c)
- ${\mathsf{P}}_{n}$ denotes an $(n-1)$-length path with $n\in \mathbb{N}$, which is a graph with n vertices that forms a path of length $n-1$; in particular, ${\mathsf{P}}_{1}={\mathsf{K}}_{1}$.
- (d)
- ${\mathsf{C}}_{n}$ denotes an n-length cycle, which is a graph with $n\ge 3$ vertices that forms a cycle of length n.
- (e)
- $\mathsf{K}(m,r)$ denotes the Kneser graph with integers $1\le r\le m$. It has $n=\left(\genfrac{}{}{0pt}{}{m}{r}\right)$ vertices, represented by all r-subsets of $\left[m\right]$. Two vertices are adjacent in that graph if they are represented by disjoint r-subsets. The graph $\mathsf{K}(m,r)$, provided that it has more than one vertex, is a connected graph if and only if either $m>2r$ or $(m,r)=(2,1)$.

- (a)
- The complete d-regular graph ${\mathsf{K}}_{d+1}$, with $d\ge 2$, whose eigenvalues are equal to d with multiplicity 1, and $-1$ with multiplicity d;
- (b)
- The complete bipartite graph ${\mathsf{K}}_{d,d}$, with $d\ge 2$, is a d-regular graph whose two nonzero eigenvalues are $\pm d$ (each of multiplicity 1), and its other $2d-2$ eigenvalues are zeros.
- (c)
- The Petersen graph, which is isomorphic to the Kneser graph $\mathsf{K}(5,2)$, is a Ramanujan graph since it is 3-regular with the distinct eigenvalues 3, $-1$, and $-2$.

- Every pair of adjacent vertices have exactly $\lambda $ common neighbors;
- Every pair of distinct and non-adjacent vertices have exactly $\mu $ common neighbors.

- (a)
- The complement of a strongly regular graph is also strongly regular. More explicitly, the complement of $\mathsf{srg}(n,d,\lambda ,\mu )$ is $\mathsf{srg}(n,n-d-1,n-2d+\mu -2,n-2d+\lambda )$.
- (b)
- The four parameters of a strongly regular graph $\mathsf{srg}(n,d,\lambda ,\mu )$ satisfy the relation$$\begin{array}{c}\hfill (n-d-1)\phantom{\rule{0.166667em}{0ex}}\mu =d\phantom{\rule{0.166667em}{0ex}}(d-\lambda -1).\end{array}$$
- (c)
- A strongly regular graph $\mathsf{srg}(n,d,\lambda ,\mu )$ has at most three distinct eigenvalues. If it is connected, then ${\lambda}_{1}\left(\mathsf{G}\right)=d$ (multiplicity 1), and the other two distinct eigenvalues are$$\begin{array}{c}\hfill {p}_{1,2}={\textstyle \frac{1}{2}}\phantom{\rule{0.166667em}{0ex}}\left[\lambda -\mu \pm \sqrt{{(\lambda -\mu )}^{2}+4(d-\mu )}\phantom{\rule{0.166667em}{0ex}}\right],\end{array}$$$$\begin{array}{c}\hfill {m}_{1,2}=\frac{1}{2}\left[n-1\mp \frac{2d+(n-1)(\lambda -\mu )}{\sqrt{{(\lambda -\mu )}^{2}+4(d-\mu )}}\right].\end{array}$$
- (d)
- A connected regular graph with exactly three distinct eigenvalues is strongly regular.
- (e)
- A strongly regular graph $\mathsf{srg}(n,d,\lambda ,\mu )$, with $\mu >0$, is a connected graph whose diameter is equal to 2. This holds since two non-adjacent vertices have $\mu >0$ common neighbors, so the distance between any pair of non-adjacent vertices is equal to 2. This can be also explained by spectral graph theory since the diameter of a connected graph is strictly smaller than the number of its distinct eigenvalues (see Theorem 4.4.1 of [40]). In light of that, the above claim about the diameter holds for all graphs that are connected and strongly regular since these graphs only have three distinct eigenvalues.
- (f)
- If $\mu =0$, the strongly regular graph is disconnected, and it is a disjoint union of equal-sized complete graphs (i.e., a disjoint union of cliques of the same size). A disjoint union of an arbitrary number $\ell \ge 2$ of equal-sized complete graphs, ${\mathsf{K}}_{d+1}$, has the parameters $\mathsf{srg}\left(\right(d+1)\ell ,d,d-1,0)$. In that case, $d={p}_{1}$ (see (10)), so the largest and second-largest eigenvalues coincide (by (11), that common eigenvalue has multiplicity ${m}_{1}+1=\ell $ in the graph spectrum). A strongly regular graph $\mathsf{G}$ is called primitive if both $\mathsf{G}$ and its complement $\overline{\mathsf{G}}$ are connected graphs. Otherwise, $\mathsf{G}$ is called imprimitive. An imprimitive graph is, therefore, either a disjoint union of equal-sized complete graphs or its complement, which is a non-empty complete multipartite graph. A strongly regular graph $\mathsf{G}$ is imprimitive if and only if 0 or $-1$ is an eigenvalue of $\mathsf{G}$.
- (g)
- Let $\mathsf{G}$ be a primitive strongly regular graph $\mathsf{srg}(n,d,\lambda ,\mu )$ with the largest eigenvalue d (multiplicity 1), second-largest eigenvalue $r={p}_{1}$ (multiplicity ${m}_{1}$), and smallest eigenvalue $s={p}_{2}$ (multiplicity ${m}_{2}$). By (3) and (4), the complement $\overline{\mathsf{G}}$ is a primitive strongly regular graph, having the largest eigenvalue $n-d-1$ (multiplicity 1), second-largest eigenvalue $-1-s$ (multiplicity ${m}_{2}$), and smallest eigenvalue $-1-r$ (multiplicity ${m}_{1}$). Each of these primitive strongly regular graphs has three distinct eigenvalues.

**Definition**

**1.**

- (a)
- ${u}_{1}={v}_{1}$ and $\{{u}_{2},{v}_{2}\}\in \mathsf{E}\left({\mathsf{G}}_{2}\right)$,
- (b)
- $\{{u}_{1},{v}_{1}\}\in \mathsf{E}\left({\mathsf{G}}_{1}\right)$ and ${u}_{2}={v}_{2}$,
- (c)
- $\{{u}_{1},{v}_{1}\}\in \mathsf{E}\left({\mathsf{G}}_{1}\right)$ and $\{{u}_{2},{v}_{2}\}\in \mathsf{E}\left({\mathsf{G}}_{2}\right)$.

**Definition**

**2.**

**Remark**

**1.**

**Definition**

**3.**

- (a)
- The sandwich theorem ([5,61], Lemma 3.2.4 of [62], and Theorem 11.1 of [63]) is stated in the two equivalent forms$$\begin{array}{ccc}\hfill & & \alpha \left(\mathsf{G}\right)\le \theta \left(\mathsf{G}\right)\le \chi \left(\overline{\mathsf{G}}\right),\hfill \end{array}$$$$\begin{array}{ccc}\hfill & & \omega \left(\mathsf{G}\right)\le \theta \left(\overline{\mathsf{G}}\right)\le \chi \left(\mathsf{G}\right).\hfill \end{array}$$
- (b)
- Theorem 7 of [6]: The Lovász $\theta $-function factorizes for the strong product of graphs, i.e.,$$\begin{array}{c}\hfill \theta ({\mathsf{G}}_{1}\u22a0{\mathsf{G}}_{2})=\theta \left({\mathsf{G}}_{1}\right)\phantom{\rule{0.166667em}{0ex}}\theta \left({\mathsf{G}}_{2}\right).\end{array}$$
- (c)
- (d)
- Theorem 9 of [6]: Let $\mathsf{G}$ be a d-regular graph of order n. Then,$$\begin{array}{c}\hfill \theta \left(\mathsf{G}\right)\le -\frac{n\phantom{\rule{0.166667em}{0ex}}{\lambda}_{n}\left(\mathsf{G}\right)}{d-{\lambda}_{n}\left(\mathsf{G}\right)},\end{array}$$
- (e)
- Two simple observations relating the Lovász $\theta $-functions of a graph and its subgraphs:
- If $\mathsf{F}$ is a spanning subgraph of a graph $\mathsf{G}$, then $\theta \left(\mathsf{F}\right)\ge \theta \left(\mathsf{G}\right)$.
- If $\mathsf{F}$ is an induced subgraph of a graph $\mathsf{G}$, then $\theta \left(\mathsf{F}\right)\le \theta \left(\mathsf{G}\right)$.

- (f)
- Theorem 2 of [14]: Although unrelated to the analysis in this paper, another interesting property of the Lovász $\theta $-function is given by the identity$$\begin{array}{c}\hfill \underset{\mathsf{H}}{sup}\frac{\alpha (\mathsf{G}\u22a0\mathsf{H})}{\theta (\mathsf{G}\u22a0\mathsf{H})}=1,\end{array}$$

## 3. Theorems, Discussions and Examples

#### 3.1. Bounds on Lovász $\theta $-Function, and an Exact Result for Strongly Regular Graphs

- (a)
- It forms a counterpart of a bound by Lovász (Theorem 9 of [6]), providing a lower bound on $\theta \left(\mathsf{G}\right)$ and an upper bound on $\theta \left(\overline{\mathsf{G}}\right)$ that are both expressed in terms of the second-largest eigenvalue of the adjacency matrix of $\mathsf{G}$.
- (b)
- It asserts that these two pairs of upper and lower bounds on $\theta \left(\mathsf{G}\right)$ and $\theta \left(\overline{\mathsf{G}}\right)$ are tight for the family of strongly regular graphs. This gives a simple closed-form expression of the Lovász $\theta $-function of a strongly regular graph $\mathsf{srg}(n,d,\lambda ,\mu )$ (and the complement graph) as a function of its four parameters.
- (c)
- Further sufficient conditions for the tightness of these bounds are provided.

**Proposition**

**1.**

- (a)
- $$\begin{array}{c}\hfill \frac{n-d+{\lambda}_{2}\left(\mathsf{G}\right)}{1+{\lambda}_{2}\left(\mathsf{G}\right)}\le \theta \left(\mathsf{G}\right)\le -\frac{n{\lambda}_{n}\left(\mathsf{G}\right)}{d-{\lambda}_{n}\left(\mathsf{G}\right)}.\end{array}$$
- Equality holds in the leftmost inequality of (24) if $\overline{\mathsf{G}}$ is both vertex-transitive and edge-transitive, or if $\mathsf{G}$ is a strongly regular graph;
- Equality holds in the rightmost inequality of (24) if $\mathsf{G}$ is edge-transitive, or if $\mathsf{G}$ is a strongly regular graph.

- (b)
- $$\begin{array}{c}\hfill 1-\frac{d}{{\lambda}_{n}\left(\mathsf{G}\right)}\le \theta \left(\overline{\mathsf{G}}\right)\le \frac{n\left(1+{\lambda}_{2}\left(\mathsf{G}\right)\right)}{n-d+{\lambda}_{2}\left(\mathsf{G}\right)}.\end{array}$$
- Equality holds in the leftmost inequality of (25) if $\mathsf{G}$ is both vertex-transitive and edge-transitive, or if $\mathsf{G}$ is a strongly regular graph;
- Equality holds in the rightmost inequality of (25) if $\overline{\mathsf{G}}$ is edge-transitive, or if $\mathsf{G}$ is a strongly regular graph.

**Proof.**

**Remark**

**2.**

- Let ${\mathcal{G}}_{1}$ be the family of graphs $\mathsf{G}$ such that $\overline{\mathsf{G}}$ is both vertex-transitive and edge-transitive;
- Let ${\mathcal{G}}_{2}$ be the family of regular and edge-transitive graphs;
- Let ${\mathcal{G}}_{3}$ be the family of graphs $\mathsf{G}$ such that $\overline{\mathsf{G}}$ is regular and edge-transitive;
- Let ${\mathcal{G}}_{4}$ be the family of graphs that are both vertex-transitive and edge-transitive;
- Let ${\mathcal{G}}_{5}$ be the family of the strongly regular graphs.

- (a)
- The Cameron graph is a strongly regular graph $\mathsf{srg}(231,30,9,3)$ (see Section 10.54 of [60]). Its complement is vertex-transitive (hence, regular), but not edge-transitive. This shows that ${\mathcal{G}}_{5}\u2288{\mathcal{G}}_{3}$, so also ${\mathcal{G}}_{5}\u2288{\mathcal{G}}_{1}$.
- (b)
- The complement of the Cameron graph is a strongly regular graph $\mathsf{srg}(231,200,172,180)$; it is vertex-transitive (hence, regular), but not edge-transitive. This shows that ${\mathcal{G}}_{5}\u2288{\mathcal{G}}_{2}$, so also ${\mathcal{G}}_{5}\u2288{\mathcal{G}}_{4}$.
- (c)
- The Foster graph is 3-regular on 90 vertices (see page 305 of [60]), which is vertex-transitive and edge-transitive, but it is not strongly regular. This shows that ${\mathcal{G}}_{4}\u2288{\mathcal{G}}_{5}$, so also ${\mathcal{G}}_{2}\u2288{\mathcal{G}}_{5}$.
- (d)
- The complement of the Foster graph is an 86-regular graph on 90 vertices, whose complement (i.e., the Foster graph) is vertex-transitive and edge-transitive, but it is not strongly regular. This shows that ${\mathcal{G}}_{1}\u2288{\mathcal{G}}_{5}$, so also ${\mathcal{G}}_{3}\u2288{\mathcal{G}}_{5}$.

**Corollary**

**1.**

**Proof.**

**Remark**

**3.**

**Remark**

**4.**

#### 3.2. Eigenvalue Inequalities, Strongly Regular Graphs, and Ramanujan Graphs

- (a)
- Derivation of inequalities that relate the second-largest and smallest eigenvalues of a regular graph. These inequalities are proved to hold with equality if and only if the graph is strongly regular.
- (b)
- Derivation of bounds on parameters of Ramanujan graphs.
- (c)
- A more general result is presented for a sequence of regular graphs whose degrees scale sub-linearly with the orders of these graphs, and their orders tend to infinity.

**Corollary**

**2.**

**Proof.**

**Corollary**

**3.**

- (a)
- $$\begin{array}{c}\hfill {g}_{n}\left(\mathsf{G}\right)\le -1\le {g}_{2}\left(\mathsf{G}\right),\end{array}$$
- (b)
- If $\mathsf{G}$ is a strongly regular graph, then the number of distinct values in the sequence ${\left\{{g}_{\ell}\left(\mathsf{G}\right)\right\}}_{\ell =1}^{n}$ is either 2 or 3, and
- it is equal to 2 if the multiplicities of the second-largest and smallest eigenvalues of $\mathsf{G}$ are identical in the subsequence $({\lambda}_{2}\left(\mathsf{G}\right),\dots ,{\lambda}_{n}\left(\mathsf{G}\right))$;
- it is otherwise equal to 3.

- (c)
- If $\mathsf{G}$ is self-complementary, then$$\begin{array}{ccc}\hfill & & {\lambda}_{2}\left(\mathsf{G}\right)\ge {\textstyle \frac{1}{2}}\left(\sqrt{n}-1\right),\hfill \end{array}$$$$\begin{array}{ccc}\hfill & & {\lambda}_{n}\left(\mathsf{G}\right)\le -{\textstyle \frac{1}{2}}\left(\sqrt{n}+1\right).\hfill \end{array}$$
- (d)

**Proof.**

**Example**

**1.**

**Example**

**2.**

**Remark**

**5.**

- ($\mathsf{G}$ is edge transitive) ⇒ (every edge in $\mathsf{G}$ is contained in the same number of triangles) ⇔ (every pair of adjacent vertices in $\mathsf{G}$ has the same number of common neighbors);
- ($\overline{\mathsf{G}}$ is edge transitive) ⇒ (for every edge $\{u,v\}\in \mathsf{E}\left(\overline{\mathsf{G}}\right)$, the same number of vertices are not adjacent in $\overline{\mathsf{G}}$ to either u or v) ⇔ (every pair of non-adjacent vertices in $\mathsf{G}$ has the same number of common neighbors);
- $\mathsf{G}$ is regular (by assumption);

**Example**

**3.**

**Remark**

**6.**

**Remark**

**7.**

- (a)
- A connected, edge-transitive and strongly regular graph is vertex-transitive (Lemma 1.3 of [71]).
- (b)
- A vertex-transitive and edge-transitive graph containing a regular clique is strongly regular (Corollary 2.4 of [72]). (A clique $\mathcal{C}$ is called regular if every vertex not in $\mathcal{C}$ is adjacent to the same positive number of vertices in $\mathcal{C}$).

**Corollary**

**4.**

**Proof.**

**Corollary**

**5.**

**Proof.**

**Corollary**

**6.**

**Proof.**

**Remark**

**8.**

#### 3.3. Bounds on Eigenvalues of Strong Products of Regular Graphs

- (1)
- (2)
- The Witsenhausen rate [13] in the zero-error source coding problem, with perfect side information at the receiver, is expressed in a dual form to (22), where the independence numbers of k-fold strong powers of a graph (with $k\in \mathbb{N}$) are replaced by their chromatic numbers, and the supremum over k is replaced by an infimum (see Section 3 of [3]);
- (3)
- There exists a polynomial-time algorithm that finds the unique prime factorization of any connected graph under the operation of strong graph multiplication [12].

**Proposition**

**2.**

- (a)
- Unless all ${\mathsf{G}}_{\ell}$ (with $\ell \in \left[k\right]$) are complete graphs, then$$\begin{array}{cc}\hfill {\lambda}_{2}({\mathsf{G}}_{1}\u22a0\dots \u22a0{\mathsf{G}}_{k})\phantom{\rule{1.em}{0ex}}& \ge \frac{{\displaystyle \prod _{\ell =1}^{k}}{n}_{\ell}-{\displaystyle \prod _{\ell =1}^{k}}(1+{d}_{\ell})}{{\displaystyle \prod _{\ell =1}^{k}}\theta \left({\mathsf{G}}_{\ell}\right)-1}-1\hfill \end{array}$$$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& \ge \frac{{\displaystyle \prod _{\ell =1}^{k}}{n}_{\ell}-{\displaystyle \prod _{\ell =1}^{k}}(1+{d}_{\ell})}{{\displaystyle \prod _{\ell =1}^{k}}\left(-{\displaystyle \frac{{n}_{\ell}\phantom{\rule{0.166667em}{0ex}}{\lambda}_{min}\left({\mathsf{G}}_{\ell}\right)}{{d}_{\ell}-{\lambda}_{min}\left({\mathsf{G}}_{\ell}\right)}}\right)-1}-1,\hfill \end{array}$$
- (b)
- Unless all ${\mathsf{G}}_{\ell}$ (with $\ell \in \left[k\right]$) are empty graphs, then$$\begin{array}{c}\hfill {\lambda}_{min}({\mathsf{G}}_{1}\u22a0\dots \u22a0{\mathsf{G}}_{k})\le -\frac{{\displaystyle \prod _{\ell =1}^{k}}(1+{d}_{\ell})-1}{{\displaystyle \prod _{\ell =1}^{k}}\left({\displaystyle \frac{{n}_{\ell}}{\theta \left({\mathsf{G}}_{\ell}\right)}}\right)-1}\phantom{\rule{0.166667em}{0ex}}.\end{array}$$$$\begin{array}{c}\hfill {\lambda}_{min}({\mathsf{G}}_{1}\u22a0\dots \u22a0{\mathsf{G}}_{k})\le -\frac{{\displaystyle \prod _{\ell =1}^{k}}(1+{d}_{\ell})-1}{{\displaystyle \prod _{\ell =1}^{k}}\left(1-{\displaystyle \frac{{d}_{\ell}}{{\lambda}_{min}\left({\mathsf{G}}_{\ell}\right)}}\right)-1}.\end{array}$$

**Proof.**

**Remark**

**9.**

**Corollary**

**7.**

**Proof.**

**Example**

**4.**

**Proposition**

**3.**

**Proof.**

**Remark**

**10.**

**Remark**

**11.**

**Example**

**5.**

**Remark**

**12.**

#### 3.4. Lower Bounds on the Chromatic Numbers of Strong Products

**Proposition**

**4.**

- (a)
- Let ${\mathsf{G}}_{1},\dots ,{\mathsf{G}}_{k}$ be k simple graphs, $\left|\mathsf{V}\left({\mathsf{G}}_{\ell}\right)\right|={n}_{\ell}$ for $\ell \in \left[k\right]$, and $\mathsf{G}={\mathsf{G}}_{1}\u22a0\dots \u22a0{\mathsf{G}}_{k}$. Then,$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& \chi \left(\mathsf{G}\right)\ge \u2308\prod _{\ell =1}^{k}\frac{{n}_{\ell}}{\theta \left({\mathsf{G}}_{\ell}\right)}\u2309,\hfill \end{array}$$$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& \chi \left(\overline{\mathsf{G}}\right)\ge \u2308\prod _{\ell =1}^{k}\theta \left({\mathsf{G}}_{\ell}\right)\u2309.\hfill \end{array}$$
- (b)
- Let ${\mathsf{G}}_{1},\dots ,{\mathsf{G}}_{k}$ be regular graphs, where ${\mathsf{G}}_{\ell}$ is ${d}_{\ell}$-regular of order ${n}_{\ell}$ for all $\ell \in \left[k\right]$. Then,$$\begin{array}{cc}\hfill \chi \left(\mathsf{G}\right)& \ge \u2308\prod _{\ell =1}^{k}\frac{{n}_{\ell}}{\theta \left({\mathsf{G}}_{\ell}\right)}\u2309\hfill \end{array}$$$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& \ge \u2308\prod _{\ell =1}^{k}\left(1-\frac{{d}_{\ell}}{{\lambda}_{min}\left({\mathsf{G}}_{\ell}\right)}\right)\u2309,\hfill \end{array}$$and inequality (68) holds with equality if each ${\mathsf{G}}_{\ell}$ is either edge-transitive or strongly regular.
- (c)
- If, for all $\ell \in \left[k\right]$, ${\mathsf{G}}_{\ell}$ is ${d}_{\ell}$-regular, and it is either edge-transitive or strongly regular, then$$\begin{array}{c}\hfill \prod _{\ell =1}^{k}\left(1-\frac{{d}_{\ell}}{{\lambda}_{min}\left({\mathsf{G}}_{\ell}\right)}\right)\ge 1-\frac{d\left(\mathsf{G}\right)}{{\lambda}_{min}\left(\mathsf{G}\right)},\end{array}$$where$$\begin{array}{c}\hfill d\left(\mathsf{G}\right)=\prod _{\ell =1}^{k}(1+{d}_{\ell})-1\end{array}$$is the valency of the regular graph $\mathsf{G}={\mathsf{G}}_{1}\u22a0\dots \u22a0{\mathsf{G}}_{k}$, and ${\lambda}_{min}\left(\mathsf{G}\right)$ is its smallest eigenvalue.
- (d)
- Let ${\mathsf{G}}_{1},\dots ,{\mathsf{G}}_{k}$ be regular graphs, where ${\mathsf{G}}_{\ell}$ is ${d}_{\ell}$-regular of order ${n}_{\ell}$ for all $\ell \in \left[k\right]$.
- (1)
- If, for all $\ell \in \left[k\right]$, the graph ${\mathsf{G}}_{\ell}$ is either vertex-transitive or strongly regular, then the lower bound on $\chi \left(\mathsf{G}\right)$ in the right-hand side of (65) is larger than or equal to the lower bound $\prod _{\ell =1}^{k}}\omega \left({\mathsf{G}}_{\ell}\right)$.
- (2)
- If, for all $\ell \in \left[k\right]$, the graph ${\mathsf{G}}_{\ell}$ is either (i) both vertex-transitive and edge-transitive, or (ii) strongly regular, then the lower bound on $\chi \left(\mathsf{G}\right)$ in the right-hand side of (68) is larger than or equal to the lower bound $\prod _{\ell =1}^{k}}\omega \left({\mathsf{G}}_{\ell}\right)$.

- (e)
- Let, for all $\ell \in \left[k\right]$, the graph ${\mathsf{G}}_{\ell}$ be ${d}_{\ell}$-regular on ${n}_{\ell}$ vertices, and suppose that it is either edge-transitive or strongly regular. Then,$$\begin{array}{c}\hfill \chi \left(\overline{\mathsf{G}}\right)\ge \u2308\prod _{\ell =1}^{k}\left(-\frac{{n}_{\ell}\phantom{\rule{0.166667em}{0ex}}{\lambda}_{min}\left({\mathsf{G}}_{\ell}\right)}{{d}_{\ell}-{\lambda}_{min}\left({\mathsf{G}}_{\ell}\right)}\right)\u2309.\end{array}$$
- (f)
- Let, for all $\ell \in \left[k\right]$, ${\mathsf{G}}_{\ell}$ be a self-complementary graph on ${n}_{\ell}$ vertices that is either vertex-transitive or strongly regular. Let $n\triangleq {\displaystyle \prod _{\ell =1}^{k}}{n}_{\ell}$ be the order of $\mathsf{G}={\mathsf{G}}_{1}\u22a0\dots \u22a0{\mathsf{G}}_{n}$. Then,$$\begin{array}{ccc}\hfill & & \chi \left(\mathsf{G}\right)\ge \u2308\sqrt{n}\phantom{\rule{0.166667em}{0ex}}\u2309,\hfill \end{array}$$$$\begin{array}{ccc}\hfill & & \chi \left(\overline{\mathsf{G}}\right)\ge \u2308\sqrt{n}\phantom{\rule{0.166667em}{0ex}}\u2309.\hfill \end{array}$$

**Proof.**

**Remark**

**13.**

**Remark**

**14.**

**Remark**

**15.**

**Remark**

**16.**

**Corollary**

**8.**

**Proof.**

**Example**

**6.**

**Example**

**7.**

**Example**

**8.**

**Example**

**9.**

**Example**

**10.**

**Example**

**11.**

#### 3.5. The Shannon Capacity of Strongly Regular Graphs

**Example**

**12.**

**Example**

**13.**

**Example**

**14.**

**Example**

**15.**

**Example**

**16.**

**Example**

**17.**

**Example**

**18.**

**Example**

**19.**

**Example**

**20.**

**Remark**

**17.**

**Example**

**21.**

## 4. Proofs

#### 4.1. Proofs for Section 3.1

#### 4.1.1. Proof of Proposition 1

#### 4.1.2. Proof of Corollary 1

#### 4.2. Proofs for Section 3.2

#### 4.2.1. Proof of Corollary 2

- (a)
- If the d-regular graph $\mathsf{G}$ is connected, then ${\lambda}_{1}\left(\mathsf{G}\right)=d>{\lambda}_{2}\left(\mathsf{G}\right)$. By assumption, $\mathsf{G}$ is also non-complete and non-empty graph, so ${\lambda}_{2}\left(\mathsf{G}\right)>{\lambda}_{n}\left(\mathsf{G}\right)$. The connected regular graph $\mathsf{G}$ thus has exactly three distinct eigenvalues, so it is strongly regular.
- (b)
- If the d-regular graph $\mathsf{G}$ is disconnected, then ${\lambda}_{1}\left(\mathsf{G}\right)=d={\lambda}_{2}\left(\mathsf{G}\right)$. If, by assumption, inequality (30) holds with equality, then ${\lambda}_{n}\left(\mathsf{G}\right)=-1$. This means that $\mathsf{G}$ is a disjoint union of equal-sized complete graphs ${\mathsf{K}}_{d+1}$, so it is an imprimitive strongly regular graph (i.e., there are no common neighbors of any pair of non-adjacent vertices in $\mathsf{G}$).

#### 4.2.2. Proof of Corollary 3

- (1)
- For a d-regular graph on n vertices, ${\lambda}_{1}\left(\mathsf{G}\right)=d$, and ${\lambda}_{1}\left(\overline{\mathsf{G}}\right)=n-d-1$. Substituting these eigenvalues into (32) gives that ${g}_{1}\left(\mathsf{G}\right)=n-d-2\ge 0$ ($\mathsf{G}$ is non-complete, so $d\le n-2$).
- (2)
- The graph $\mathsf{G}$ and its complement $\overline{\mathsf{G}}$ are both strongly regular, so each one of them has at most three distinct eigenvalues.
- (3)
- For a strongly regular graph, by Item (a), ${g}_{2}\left(\mathsf{G}\right)=-1={g}_{n}\left(\mathsf{G}\right)$, and ${g}_{\ell}\left(\mathsf{G}\right)=-1$ if either (i) ${\lambda}_{\ell}\left(\mathsf{G}\right)={\lambda}_{2}\left(\mathsf{G}\right)$ and ${\lambda}_{\ell}\left(\overline{\mathsf{G}}\right)={\lambda}_{2}\left(\overline{\mathsf{G}}\right)$, or (ii) ${\lambda}_{\ell}\left(\mathsf{G}\right)={\lambda}_{n}\left(\mathsf{G}\right)$ and ${\lambda}_{\ell}\left(\overline{\mathsf{G}}\right)={\lambda}_{n}\left(\overline{\mathsf{G}}\right)$.
- (4)
- By assumption, $\mathsf{G}$ is a strongly regular graph, which implies that so is $\overline{\mathsf{G}}$. Due to their regularity, ${\lambda}_{1}\left(\mathsf{G}\right)\ge {\lambda}_{2}\left(\mathsf{G}\right)$, and ${\lambda}_{1}\left(\overline{\mathsf{G}}\right)\ge {\lambda}_{2}\left(\overline{\mathsf{G}}\right)$ with equalities, respectively, if and only if $\mathsf{G}$ or $\overline{\mathsf{G}}$ are disconnected graphs. The sequence ${\left\{{g}_{\ell}\left(\mathsf{G}\right)\right\}}_{\ell =1}^{n}$ gets an additional (third) distinct value if and only if the multiplicities of the smallest and the second-largest eigenvalues of $\mathsf{G}$ in the subsequence $({\lambda}_{2}\left(\mathsf{G}\right),\dots ,{\lambda}_{n}\left(\mathsf{G}\right))$ are distinct. Indeed, in the latter case, only one of the following two options is possible: (iii) ${\lambda}_{\ell}\left(\mathsf{G}\right)={\lambda}_{2}\left(\mathsf{G}\right)$ and ${\lambda}_{\ell}\left(\overline{\mathsf{G}}\right)={\lambda}_{n}\left(\overline{\mathsf{G}}\right)$, or (iv) ${\lambda}_{\ell}\left(\mathsf{G}\right)={\lambda}_{n}\left(\mathsf{G}\right)$ and ${\lambda}_{\ell}\left(\overline{\mathsf{G}}\right)={\lambda}_{2}\left(\overline{\mathsf{G}}\right)$. This holds since, by (4), the multiplicity of the second-largest eigenvalue of $\mathsf{G}$ is equal to the multiplicity of the smallest eigenvalue of $\overline{\mathsf{G}}$, and similarly, the multiplicity of the smallest eigenvalue of $\mathsf{G}$ is equal to the multiplicity of the second-largest eigenvalue of $\overline{\mathsf{G}}$. It therefore follows that the third distinct value (as above) is attained by the sequence ${\left\{{g}_{\ell}\right\}}_{\ell =1}^{n}$ a number of times that is equal to the absolute value of the difference between the multiplicities of the second-largest and the smallest eigenvalues of $\mathsf{G}$ in the subsequence $({\lambda}_{2}\left(\mathsf{G}\right),\dots ,{\lambda}_{n}\left(\mathsf{G}\right))$ (provided that the latter two multiplicities are distinct).

#### 4.2.3. Proof of Corollary 4

#### 4.2.4. Proof of Corollary 5

#### 4.2.5. Proof of Corollary 6

#### 4.3. Proofs for Section 3.3

#### 4.3.1. Proof of Proposition 2

- (a)
- By the leftmost inequality in (24), unless $\mathsf{G}={\mathsf{K}}_{n}$,$$\begin{array}{c}\hfill \theta \left(\mathsf{G}\right)\ge \frac{n\left(\mathsf{G}\right)-d\left(\mathsf{G}\right)+{\lambda}_{2}\left(\mathsf{G}\right)}{1+{\lambda}_{2}\left(\mathsf{G}\right)},\end{array}$$$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& n\left(\mathsf{G}\right)=\prod _{\ell =1}^{k}{n}_{\ell},\hfill \end{array}$$$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& d\left(\mathsf{G}\right)=\prod _{\ell =1}^{k}(1+{d}_{\ell})-1,\hfill \end{array}$$$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& \theta \left(\mathsf{G}\right)=\prod _{\ell =1}^{k}\theta \left({\mathsf{G}}_{\ell}\right).\hfill \end{array}$$Indeed, equality (161) holds since the cardinality of a Cartesian product of finite sets is equal to the product of the cardinalities of each set; equality (162) can be justified by first verifying the special case of a strong product of two regular graphs, and then proceeding by a mathematical induction on k. Finally, equality (163) holds by (17) (see Theorem 7 of [6]). Combining the bound in (160) with equalities (161)–(163) gives$$\begin{array}{c}\hfill \prod _{\ell =1}^{k}\theta \left({\mathsf{G}}_{\ell}\right)\ge 1+\frac{{\displaystyle \prod _{\ell =1}^{k}}{n}_{\ell}-{\displaystyle \prod _{\ell =1}^{k}}(1+{d}_{\ell})}{1+{\lambda}_{2}\left(\mathsf{G}\right)}.\end{array}$$Unless all ${\mathsf{G}}_{\ell}$ (with $\ell \in \left[k\right]$) are complete graphs, the left-hand side of (164) is strictly larger than 1, and then rearrangement of the terms in (164) gives the lower bound on ${\lambda}_{2}\left(\mathsf{G}\right)$ in (52). Next, the possible loosening of the lower bound in the right-hand side of (52) to the lower bound in the right-hand side of (53) holds by (19) (see Theorem 9 of [6]). Inequality (53) holds with equality if each regular factor ${\mathsf{G}}_{\ell}$ is either edge-transitive (by Theorem 9 of [6]) or strongly regular (by Item (a) of Proposition 1).
- (b)
- Combining (19) with equalities (161)–(163) gives, with $n=n\left(\mathsf{G}\right)$ and $d=d\left(\mathsf{G}\right)$,$$\begin{array}{cc}\hfill \prod _{\ell =1}^{k}\theta \left({\mathsf{G}}_{\ell}\right)& =\theta \left(\mathsf{G}\right)\hfill \end{array}$$$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& \le -\frac{n{\lambda}_{n}\left(\mathsf{G}\right)}{d-{\lambda}_{n}\left(\mathsf{G}\right)}\hfill \end{array}$$$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& =-\frac{{\displaystyle \prod _{\ell =1}^{k}}{n}_{\ell}\xb7{\lambda}_{n}\left(\mathsf{G}\right)}{{\displaystyle \prod _{\ell =1}^{k}}(1+{d}_{\ell})-1-{\lambda}_{n}\left(\mathsf{G}\right)}.\hfill \end{array}$$Unless all ${\mathsf{G}}_{\ell}$ (with $\ell \in \left[k\right]$) are empty graphs, the denominator in the right-hand side of (167) is strictly positive. This gives (54) after rearrangement of terms. Finally, the transition from (54) to (55) is justified if$$\begin{array}{c}\hfill \theta \left({\mathsf{G}}_{\ell}\right)=-\frac{{n}_{\ell}{\lambda}_{min}\left({\mathsf{G}}_{\ell}\right)}{{d}_{\ell}-{\lambda}_{min}\left({\mathsf{G}}_{\ell}\right)},\phantom{\rule{2.em}{0ex}}\forall \phantom{\rule{0.166667em}{0ex}}\ell \in \left[k\right].\end{array}$$As above (the end of the proof of Item (a)), the condition in (168) holds if the regular graph ${\mathsf{G}}_{\ell}$ is either edge-transitive or strongly regular.

#### 4.3.2. Proof of Corollary 7

#### 4.3.3. Proof of Proposition 3

#### 4.4. Proofs for Section 3.4

#### 4.4.1. Proof of Proposition 4

- (a)
- Let ${\mathsf{G}}_{1},\dots ,{\mathsf{G}}_{k}$ be k simple, finite and undirected graphs, $\left|\mathsf{V}\left({\mathsf{G}}_{\ell}\right)\right|={n}_{\ell}$ for $\ell \in \left[k\right]$, and let $\mathsf{G}={\mathsf{G}}_{1}\u22a0\dots \u22a0{\mathsf{G}}_{k}$. We provide two alternative simple proofs of (65).First proof:$$\begin{array}{cc}\hfill \chi \left(\mathsf{G}\right)& \ge \theta \left(\overline{\mathsf{G}}\right)\hfill \end{array}$$$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& \ge \frac{\left|\mathsf{V}\right(\mathsf{G}\left)\right|}{\theta \left(\mathsf{G}\right)}\hfill \end{array}$$$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& =\prod _{\ell =1}^{k}\frac{\left|\mathsf{V}\right({\mathsf{G}}_{\ell}\left)\right|}{\theta \left({\mathsf{G}}_{\ell}\right)},\hfill \end{array}$$Second proof:$$\begin{array}{cc}\hfill \chi \left(\mathsf{G}\right)& \ge \frac{\left|\mathsf{V}\right(\mathsf{G}\left)\right|}{\alpha \left(\mathsf{G}\right)}\hfill \end{array}$$$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& \ge \frac{\left|\mathsf{V}\right(\mathsf{G}\left)\right|}{\theta \left(\mathsf{G}\right)}\hfill \end{array}$$$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& =\prod _{\ell =1}^{k}\frac{\left|\mathsf{V}\right({\mathsf{G}}_{\ell}\left)\right|}{\theta \left({\mathsf{G}}_{\ell}\right)},\hfill \end{array}$$We next prove (66).$$\begin{array}{cc}\hfill \chi \left(\overline{\mathsf{G}}\right)& \ge \theta \left(\mathsf{G}\right)\hfill \end{array}$$$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& =\prod _{\ell =1}^{k}\theta \left({\mathsf{G}}_{\ell}\right),\hfill \end{array}$$
- (b)
- Let ${\mathsf{G}}_{1},\dots ,{\mathsf{G}}_{k}$ be regular graphs, where ${\mathsf{G}}_{\ell}$ is ${d}_{\ell}$-regular of order ${n}_{\ell}$ for all $\ell \in \left[k\right]$. Inequality (67) is (65). Inequality (68) follows from (19) and (67). Furthermore, by Item (a) in Proposition 1, inequality (68) holds with equality if each regular graph ${\mathsf{G}}_{\ell}$, for $\ell \in \left[k\right]$, is either edge-transitive or strongly regular.
- (c)
- By (182), with $|\mathsf{V}\left({\mathsf{G}}_{\ell}\right)|={n}_{\ell}$,$$\begin{array}{c}\hfill \frac{\left|\mathsf{V}\right(\mathsf{G}\left)\right|}{\theta \left(\mathsf{G}\right)}=\prod _{\ell =1}^{k}\frac{{n}_{\ell}}{\theta \left({\mathsf{G}}_{\ell}\right)}.\end{array}$$Suppose that, for all $\ell \in \left[k\right]$, ${\mathsf{G}}_{\ell}$ is ${d}_{\ell}$-regular, and it is also either edge-transitive or strongly regular. By Item (a) in Proposition 1, for all $\ell \in \left[k\right]$,$$\begin{array}{c}\hfill \theta \left({\mathsf{G}}_{\ell}\right)=-\frac{{n}_{\ell}\phantom{\rule{0.166667em}{0ex}}{\lambda}_{min}\left({\mathsf{G}}_{\ell}\right)}{{d}_{\ell}-{\lambda}_{min}\left({\mathsf{G}}_{\ell}\right)}.\end{array}$$Combining (188) and (189) gives$$\begin{array}{c}\hfill \frac{\left|\mathsf{V}\right(\mathsf{G}\left)\right|}{\theta \left(\mathsf{G}\right)}=\prod _{\ell =1}^{k}\left(1-\frac{{d}_{\ell}}{{\lambda}_{min}\left({\mathsf{G}}_{\ell}\right)}\right).\end{array}$$On the other hand, since $\mathsf{G}={\mathsf{G}}_{1}\u22a0\dots \u22a0{\mathsf{G}}_{\ell}$ is d-regular, with $d\triangleq d\left(\mathsf{G}\right)$ as given in (70), it follows from (19) that$$\begin{array}{c}\hfill \theta \left(\mathsf{G}\right)\le -\frac{\left|\mathsf{V}\left(\mathsf{G}\right)\right|\phantom{\rule{0.277778em}{0ex}}{\lambda}_{min}\left(\mathsf{G}\right)}{d\left(\mathsf{G}\right)-{\lambda}_{min}\left(\mathsf{G}\right)}.\end{array}$$It should be noted, in regard to (191), that even if all ${\mathsf{G}}_{\ell}$’s are regular and edge-transitive graphs, their strong product $\mathsf{G}$ is not necessarily edge-transitive. In fact, $\mathsf{G}$ is not edge-transitive, unless all the k factors ${\left\{{\mathsf{G}}_{\ell}\right\}}_{\ell =1}^{k}$ are complete graphs (see Theorem 3.1 of [28]). For this reason, (191) does not hold in general with equality (see Theorem 9 of [6]). Finally, combing (190) and (191) gives inequality (69).
- (d)
- Let ${\mathsf{G}}_{1},\dots ,{\mathsf{G}}_{k}$ be regular graphs, where ${\mathsf{G}}_{\ell}$ is ${d}_{\ell}$-regular on ${n}_{\ell}$ vertices for all $\ell \in \left[k\right]$. Then, under the assumptions of Item (d),
- (1)
- $$\prod _{\ell =1}^{k}\frac{\left|\mathsf{V}\right({\mathsf{G}}_{\ell}\left)\right|}{\theta \left({\mathsf{G}}_{\ell}\right)}=\prod _{\ell =1}^{k}\theta \left(\overline{{\mathsf{G}}_{\ell}}\right)$$$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& \ge \prod _{\ell =1}^{k}\omega \left({\mathsf{G}}_{\ell}\right)\hfill \end{array}$$
- (2)
- $$\prod _{\ell =1}^{k}\left(1-\frac{{d}_{\ell}}{{\lambda}_{min}\left({\mathsf{G}}_{\ell}\right)}\right)=\prod _{\ell =1}^{k}\frac{\left|\mathsf{V}\right({\mathsf{G}}_{\ell}\left)\right|}{\theta \left({\mathsf{G}}_{\ell}\right)}$$$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& \ge \prod _{\ell =1}^{k}\omega \left({\mathsf{G}}_{\ell}\right)\hfill \end{array}$$

To summarize, it shows that under proper assumptions, the lower bound on the chromatic number of $\mathsf{G}$ in the right-hand side of (65), or even its loosened bound in the right-hand side of (68), are larger than or equal to the lower bound $\prod _{\ell =1}^{k}}\omega \left({\mathsf{G}}_{\ell}\right)$. - (e)
- By (66),$$\begin{array}{c}\hfill \chi \left(\overline{\mathsf{G}}\right)\ge \prod _{\ell =1}^{k}\theta \left({\mathsf{G}}_{\ell}\right).\end{array}$$Let, for all $\ell \in \left[k\right]$, the graph ${\mathsf{G}}_{\ell}$ be ${d}_{\ell}$-regular on ${n}_{\ell}$ vertices, and suppose that it is either edge-transitive or strongly regular. Then, by Item (a) of Proposition 1,$$\begin{array}{c}\hfill \theta \left({\mathsf{G}}_{\ell}\right)=-\frac{{n}_{\ell}\phantom{\rule{0.166667em}{0ex}}{\lambda}_{min}\left({\mathsf{G}}_{\ell}\right)}{{d}_{\ell}-{\lambda}_{min}\left({\mathsf{G}}_{\ell}\right)},\phantom{\rule{1.em}{0ex}}\forall \phantom{\rule{0.166667em}{0ex}}\ell \in \left[k\right].\end{array}$$
- (f)
- By the assumption that ${\mathsf{G}}_{1},\dots ,{\mathsf{G}}_{k}$ are self-complementary,$$\begin{array}{c}\hfill \theta \left({\mathsf{G}}_{\ell}\right)=\theta \left({\overline{\mathsf{G}}}_{\ell}\right),\phantom{\rule{1.em}{0ex}}\forall \phantom{\rule{0.166667em}{0ex}}\ell \in \left[k\right].\end{array}$$Furthermore, by the assumption that for all $\ell \in \left[k\right]$, ${\mathsf{G}}_{\ell}$ is a graph on ${n}_{\ell}$ vertices that is either vertex-transitive or strongly regular,$$\begin{array}{c}\hfill \theta \left({\mathsf{G}}_{\ell}\right)\phantom{\rule{0.166667em}{0ex}}\theta \left({\overline{\mathsf{G}}}_{\ell}\right)={n}_{\ell},\phantom{\rule{1.em}{0ex}}\forall \phantom{\rule{0.166667em}{0ex}}\ell \in \left[k\right].\end{array}$$Combining (198) and (199) gives$$\begin{array}{c}\hfill \theta \left({\mathsf{G}}_{\ell}\right)=\sqrt{{n}_{\ell}},\phantom{\rule{1.em}{0ex}}\forall \phantom{\rule{0.166667em}{0ex}}\ell \in \left[k\right].\end{array}$$Consequently, by (65) and (200),$$\begin{array}{cc}\hfill \chi \left(\mathsf{G}\right)& \ge \u2308\prod _{\ell =1}^{k}\frac{{n}_{\ell}}{\theta \left({\mathsf{G}}_{\ell}\right)}\u2309\hfill \end{array}$$$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& =\u2308\prod _{\ell =1}^{k}\sqrt{{n}_{\ell}}\phantom{\rule{0.166667em}{0ex}}\u2309\hfill \end{array}$$$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& =\u2308\sqrt{n}\phantom{\rule{0.166667em}{0ex}}\u2309,\hfill \end{array}$$$$\begin{array}{cc}\hfill \chi \left(\overline{\mathsf{G}}\right)& \ge \u2308\prod _{\ell =1}^{k}\theta \left({\mathsf{G}}_{\ell}\right)\u2309\hfill \end{array}$$$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& =\u2308\prod _{\ell =1}^{k}\sqrt{{n}_{\ell}}\phantom{\rule{0.166667em}{0ex}}\u2309\hfill \end{array}$$$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& =\u2308\sqrt{n}\phantom{\rule{0.166667em}{0ex}}\u2309.\hfill \end{array}$$

#### 4.4.2. Proof of Corollary 8

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**MDPI and ACS Style**

Sason, I.
Observations on the Lovász *θ*-Function, Graph Capacity, Eigenvalues, and Strong Products *Entropy* **2023**, *25*, 104.
https://doi.org/10.3390/e25010104

**AMA Style**

Sason I.
Observations on the Lovász *θ*-Function, Graph Capacity, Eigenvalues, and Strong Products *Entropy*. 2023; 25(1):104.
https://doi.org/10.3390/e25010104

**Chicago/Turabian Style**

Sason, Igal.
2023. "Observations on the Lovász *θ*-Function, Graph Capacity, Eigenvalues, and Strong Products *Entropy* 25, no. 1: 104.
https://doi.org/10.3390/e25010104