# Clive ElphickUniversity of Birmingham · School of Mathematics

Clive Elphick

MA (Cantab) MSc PhD

## About

35

Publications

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554

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Introduction

I did a PhD in spectral graph theory 35 years ago and have returned to the subject after a career in industry.
Much of my work focuses on spectral bounds for chromatic numbers of a graph, including the fractional chromatic number and the quantum chromatic number.

**Skills and Expertise**

## Publications

Publications (35)

The positive and negative square energies of a graph, $s^+(G)$ and $s^-(G)$, are the sums of squares of the positive and negative eigenvalues of the adjacency matrix, respectively. The first results on square energies revealed symmetry between $s^+(G)$ and $s^-(G)$. This paper reviews examples of asymmetry between these parameters, for example usin...

A well known upper bound for the independence number $\alpha(G)$ of a graph $G$, due to Cvetkovi ́c, is that \begin{equation*}\alpha(G) \le n^0 + \min\{n^+ , n^-\}\end{equation*}where $(n^+, n^0, n^-)$ is the inertia of $G$. We prove that this bound is also an upper bound for the quantum independence number $\alpha_q$(G), where $\alpha_q(G) \ge \al...

Let $\mu_1 \ge \ldots \ge \mu_n$ denote the eigenvalues of a graph $G$ with $m$ edges and clique number $\omega(G)$. Nikiforov proved a spectral version of Tur\'an's theorem that \[ \mu_1^2 \le \frac{2m(\omega - 1)}{\omega}, \] and Bollob\'as and Nikiforov conjectured that for $G \not = K_n$ \[ \mu_1^2 + \mu_2^2 \le \frac{2m(\omega - 1)}{\omega}. \...

Hoffman proved that a graph $G$ with eigenvalues $\mu_1 \geqslant \cdots \geqslant \mu_n$ and chromatic number $\chi(G)$ satisfies: \[\chi \geqslant 1 + \kappa\] where $\kappa$ is the smallest integer such that \[\mu_1 + \sum_{i=1}^{\kappa} \mu_{n+1-i} \leqslant 0.\] We strengthen this well known result by proving that $\chi(G)$ can be replaced by...

Let χ(G) denote the chromatic number of a graph and χv(G) denote the vector chromatic number. For all graphs χv(G) ≤ χ(G) and for some graphs χv(G) ≪ χ(G). Galtman proved that Hoffman’s well-known lower bound for χ(G) is in fact a lower bound for χv(G). We prove that two more spectral lower bounds for χ(G) are also lower bounds for χv(G). We then u...

The quantum chromatic number, χq(G), of a graph G was originally defined as the minimal number of colors necessary in a quantum protocol in which two provers that cannot communicate with each other but share an entangled state can convince an interrogator with certainty that they have a coloring of the graph. We use an equivalent purely combinatori...

A well known upper bound for the independence number $\alpha(G)$ of a graph $G$, due to Cvetkovi\'{c}, is that \begin{equation*} \alpha(G) \le n^0 + \min\{n^+ , n^-\} \end{equation*} where $(n^+, n^0, n^-)$ is the inertia of $G$. We prove that this bound is also an upper bound for the quantum independence number $\alpha_q$(G), where $\alpha_q(G) \g...

Hoffman proved that a graph $G$ with eigenvalues $\mu_1 \ge \ldots \ge \mu_n$ and chromatic number $\chi(G)$ satisfies: \[ \chi \ge 1 + \kappa \] where $\kappa$ is the smallest integer such that \[ \mu_1 + \sum_{i=1}^{\kappa} \mu_{n+1-i} \le 0. \] We strengthen this well known result by proving that $\chi(G)$ can be replaced by the quantum chromati...

The orthogonal rank of a graph $G=(V,E)$ is the smallest dimension $\xi$ such that there exist non-zero column vectors $x_v\in\mathbb{C}^\xi$ for $v\in V$ satisfying the orthogonality condition $x_v^\dagger x_w=0$ for all $vw\in E$. We prove that many spectral lower bounds for the chromatic number, $\chi$, are also lower bounds for $\xi$. This resu...

Let $\chi(G)$ denote the chromatic number of a graph and $\chi_v(G)$ denote the vector chromatic number. For all graphs $\chi_v(G) \le \chi(G)$ and for some graphs $\chi_v(G) \ll \chi(G)$. Bilu proved that Hoffman's well known lower bound for $\chi(G)$ is in fact a lower bound for $\chi_v(G)$. We prove that two more spectral lower bounds for $\chi(...

A well known upper bound for the independence number $\alpha(G)$ of a graph $G$, is that \[ \alpha(G) \le n^0 + \min\{n^+ , n^-\}, \] where $(n^+, n^0, n^-)$ is the inertia of $G$. We prove that this bound is also an upper bound for the quantum independence number $\alpha_q$(G), where $\alpha_q(G) \ge \alpha(G)$. We investigate implications of this...

The orthogonal rank of a graph $G=(V,E)$ is the smallest dimension $\xi$ such that there exist non-zero column vectors $x_v\in\mathbb{C}^\xi$ for $v\in V$ satisfying the orthogonality condition $x_v^* x_w=0$ for all $vw\in E$. We show that \[ 1 + \max\left(\frac{n^+}{n^-}, \frac{n^-}{n^+} \right) \le \xi, \] where $n^+$ and $n^-$ denote the number...

The quantum chromatic number, $\chi_q(G)$, of a graph $G$ was originally defined as the minimal number of colors necessary in a quantum protocol in which two provers that cannot communicate with each other but share an entangled state can convince an interrogator with certainty that they have a coloring of the graph. We use an equivalent purely com...

It is well known that $n/(n - \mu)$, where $\mu$ is the spectral radius of a graph with $n$ vertices, is a lower bound for the clique number. We conjecture that $\mu$ can be replaced in this bound with $\sqrt{s^+}$, where $s^+$ is the sum of the squares of the positive eigenvalues. We prove this conjecture for various classes of graphs, including t...

Let $(n^+, n^0, n^-)$ denote the inertia of a graph $G$ with $n$ vertices. Nordhaus-Gaddum bounds are known for inertia, except for an upper bound for $n^-$. We conjecture that for any graph % \[ n^-(G) + n^-(\overline{G}) \le 1.5(n - 1), \] % and prove this bound for various classes of graphs. We consider the relationship between this bound and th...

Terpai [22] proved the Nordhaus-Gaddum bound that $\mu(G) + \mu(\overline{G}) \le 4n/3 - 1$, where $\mu(G)$ is the spectral radius of a graph $G$ with $n$ vertices. Let $s^+$ denote the sum of the squares of the positive eigenvalues of $G$. We prove that $\sqrt{s^{+}(G)} + \sqrt{s^+(\overline{G})} < \sqrt{2}n$ and conjecture that $\sqrt{s^{+}(G)} +...

Let $\chi(G$) and $\chi_f(G)$ denote the chromatic and fractional chromatic numbers of a graph $G$, and let $(n^+ , n^0 , n^-)$ denote the inertia of $G$. We prove that:\[1 + \max\left(\frac{n^+}{n^-} , \frac{n^-}{n^+}\right) \le \chi(G)\] and conjecture that \[ 1 + \max\left(\frac{n^+}{n^-} , \frac{n^-}{n^+}\right) \le \chi_f(G).\] We investigate...

Dvo\v{r}\'ak \emph{et al.} introduced a variant of the Randi\'c index of a
graph $G$, denoted by $R'(G)$, where $R'(G)=\sum_{uv\in E(G)}\frac 1
{\max\{d(u), d(v)\}}$, and $d(u)$ denotes the degree of a vertex $u$ in $G$.
The coloring number $col(G)$ of a graph $G$ is the smallest number $k$ for
which there exists a linear ordering of the vertices o...

We review bounds for the general Randi\'c index, $R_{\alpha} = \sum_{ij \in
E} (d_i d_j)^\alpha$, and use the power mean inequality to prove, for example,
that $R_\alpha \ge m\lambda^{2\alpha}$ for $\alpha < 0$, where $\lambda$ is the
spectral radius of a graph. This enables us to strengthen various known lower
and upper bounds for $R_\alpha$ and t...

This paper gives an errata to the paper "New measure of graph irregularity", Electronic Journal of Graph Theory and Applications {\bf 2}(1) (2014), 52-65.

Abdo et al. demonstrated [MATCH Commun. Math. Comput. Chem. 72 (2014) 741-751] that there exist connected graphs for whichμ2 (G) ≈M2 (G) /m where μ(G) is the spectral radius of a graph G, M2(G) is the second Zagreb index and m the number of edges. We use and extend this approximation to investigate opportunities to convert results from spectral gra...

The best degree-based upper bound for the spectral radius is due to Liu and
Weng. This paper begins by demonstrating that a (forgotten) upper bound for the
spectral radius dating from 1983 is equivalent to their much more recent bound.
This bound is then used to compare lower bounds for the clique number. A series
of sharp upper bounds for the sign...

A well known upper bound for the spectral radius of a graph, due to Hong, is
that $\mu_1^2 \le 2m - n + 1$. It is conjectured that for connected graphs $n -
1 \le s^+ \le 2m - n + 1$, where $s^+$ denotes the sum of the squares of the
positive eigenvalues. The conjecture is proved for various classes of graphs,
including bipartite, complete $q$-part...

Three vertex-degree-based graph invariants are presented, that earlier have been considered in the chemical and/or mathematical literature, but that evaded the attention of most mathematical chemists. These are the reciprocal Randíc index (RR), the reduced second Zagreb index RM2, and the reduced reciprocal Randíc index (RRR). If d1, d2,.., dn are...

In this paper, we define and compare three new measures of graph irregularity. We use these measures to tighten upper bounds for the chromatic number and the Colin de Verdiere parameter. We also strengthen the concise Turan theorem for irregular graphs and investigate to what extent Turan's theorem can be similarly strengthened for generalized r-pa...

One of the best known results in spectral graph theory is the following lower
bound on the chromatic number due to Alan Hoffman, where mu_1 and mu_n are
respectively the maximum and minimum eigenvalues of the adjacency matrix: q >=
1 + mu_1 / -mu_n.
Vladimir Nikiforov has proved an eigenvalue inequality for Hermitian
matrices, which enabled him to...

The purpose of this article is to improve existing lower bounds on the chromatic number χ. Let μ[subscript 1],…,μ[subscript n] be the eigenvalues of the adjacency matrix sorted in non-increasing order.
First, we prove the lower bound χ ≥ 1 + max[subscript m]{∑[m over i=1]μ[subscript i]/ − ∑[m over i=1]μ[subscript n−i+1]} for m = 1,…,n − 1. This gen...

School timetabling problems containing multiple period lessons are formulated in terms of the coloring of composite graphs. Several approximate coloring algorithms are proposed and compared empirically.

G is any simple graph with m edges and n vertices where these vertices have vertex degrees d(1)≥d(2)≥⋯≥d(n), respectively. General algorithms for the exact calculation of χ(G), the chromatic number of G, are almost always of ‘branch and bound’ type; this type of algorithm requires an easily constructed lower bound for χ(G). In this paper it is show...

This note presents an alternative proof of the necessary and sufficient conditions, due to De Werra, for the existence of a solution to an extended version of the simple timetabling problem. A generalized canonical requirements matrix is defined and an algorithm for the solution of such problems is proposed.RésuméOn dérive d’une façon alternative u...

A conjecture concerning a continuous formulation of timetabling problems is discussed and an alternative discrete formulation is proposed.

A conjecture concerning a continuous formulation of timetabling problems is discussed and an alternative discrete formulation is proposed.