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Gary R. W. Greaves

Publications and source records attributed to Gary R. W. Greaves.

At least 19 recordsLinked to original sources
arXiv2026

Spectral gap of biased adjacent-transposition chains

We establish a sharp lower bound on the spectral gap of the biased adjacent-transposition Markov chain on the symmetric group. As a consequence, we resolve a longstanding conjecture of Fill, proving that among all regular probability vectors, the minimum spectral gap of the transition matrix is attained by the uniform probability vector. We also characterise the regular probability vectors attaining the minimum spectral gap and determine the exact multiplicity of the corresponding second-largest eigenvalue. Our proof relies on a novel algebraic decomposition of the transition matrix into elementary orthogonal projections.

math.PR↗
arXiv2026

A note on some spectral properties of generalised pancake graphs

We prove that the spectral gap of generalised pancake graphs is strictly less than 2 and strictly less than 1 for burnt pancake graphs. In addition, we establish lower bounds on the multiplicities of certain integer eigenvalues of generalised pancake graphs. Together, these results settle two recent conjectures of Blanco and Buehrle.

math.CO↗
arXiv2025

Neumaier graphs from cyclotomy with small coherent rank

Using cyclotomy, we construct a new infinite family of Neumaier graphs that includes infinitely many strongly regular graphs. Notably, this family conjecturally contains infinitely many graphs with coherent rank $6$. Our construction also provides the first known examples that answer a question posed by Evans, Goryainov, and Panasenko regarding the existence of Neumaier graphs whose nexus is not a power of $2$. In addition, we show that a construction of Greaves and Koolen yields an infinite family of Neumaier graphs with coherent rank $6$.

math.CO↗
arXiv2025

Real-rooted integer polynomial enumeration algorithms and interlacing polynomials via linear programming

We extend the algorithms of Robinson, Smyth, and McKee--Smyth to enumerate all real-rooted integer polynomials of a fixed degree, where the first few (at least three) leading coefficients are specified. Additionally, we introduce new linear programming algorithms to enumerate all feasible interlacing polynomials of a given polynomial that comes from a certain family of real-rooted integer polynomials. These algorithms are further specialised for the study of real equiangular lines, incorporating additional number-theoretic constraints to restrict the enumeration. Our improvements significantly enhance the efficiency of the methods presented in previous work by the authors.

math.CO↗
arXiv2024

Chromatic polynomials of signed graphs and dominating-vertex deletion formulae

We exhibit non-switching-isomorphic signed graphs that share a common underlying graph and common chromatic polynomials, thereby answering a question posed by Zaslavsky. For various joins of all-positive or all-negative signed complete graphs, we derive a closed-form expression for their chromatic polynomials. As a generalisation of the chromatic polynomials for a signed graph, we introduce a new pair of bivariate chromatic polynomials. We establish recursive dominating-vertex deletion formulae for these bivariate chromatic polynomials. Finally, we show that for certain families of signed threshold graphs, isomorphism is equivalent to the equality of bivariate chromatic polynomials.

math.CO↗
arXiv2024

Bootstrapping cascaded random matrix models: correlations in permutations of matrix products

Random matrix theory is a useful tool in the study of the physics of multiple scattering systems, often striking a balance between computation speed and physical rigour. Propagation of waves through thick disordered media, as arises in for example optical scattering or electron transport, typically necessitates cascading of multiple random matrices drawn from an underlying ensemble for thin media, greatly increasing computational burden. Here we propose a dual pool based bootstrapping approach to speed up statistical studies of scattering in thick random media. We examine how potential matrix reuse in a pool based approach can impact statistical estimates of population averages. Specifically, we discuss how both bias and additional variance in the sample mean estimator are introduced through bootstrapping. In the diffusive scattering regime, the extra estimator variance is shown to originate from samples in which cascaded transfer matrices are permuted matrix products. Through analysis of the combinatorics and cycle structure of permutations we quantify the resulting correlations. Proofs of several analytic formulae enumerating the frequency with which correlations of different strengths occur are derived. Extension to the ballistic regime is briefly considered.

math-ph↗
arXiv2023

Hermitian matrices of roots of unity and their characteristic polynomials

We investigate spectral conditions on Hermitian matrices of roots of unity. Our main results are conjecturally sharp upper bounds on the number of residue classes of the characteristic polynomial of such matrices modulo ideals generated by powers of $(1-ζ)$, where $ζ$ is a root of unity. We also prove a generalisation of a classical result of Harary and Schwenk about a relation for traces of powers of a graph-adjacency matrix, which is a crucial ingredient for the proofs of our main results.

math.CO↗
arXiv2023

Equiangular lines in Euclidean spaces: dimensions 17 and 18

We show that the maximum cardinality of an equiangular line system in 17 dimensions is 48, thereby solving a longstanding open problem. Furthermore, by giving an explicit construction, we improve the lower bound on the maximum cardinality of an equiangular line system in 18 dimensions to 57.

math.CO↗
arXiv2021

Frames over finite fields: Equiangular lines in orthogonal geometry

We investigate equiangular lines in finite orthogonal geometries, focusing specifically on equiangular tight frames (ETFs). In parallel with the known correspondence between real ETFs and strongly regular graphs (SRGs) that satisfy certain parameter constraints, we prove that ETFs in finite orthogonal geometries are closely aligned with a modular generalization of SRGs. The constraints in our finite field setting are weaker, and all but~18 known SRG parameters on $v \leq 1300$ vertices satisfy at least one of them. Applying our results to triangular graphs, we deduce that Gerzon's bound is attained in finite orthogonal geometries of infinitely many dimensions. We also demonstrate connections with real ETFs, and derive necessary conditions for ETFs in finite orthogonal geometries. As an application, we show that Gerzon's bound cannot be attained in a finite orthogonal geometry of dimension~5.

math.CO↗
arXiv2021

Signed $(0,2)$-graphs with few eigenvalues and a symmetric spectrum

We investigate properties of signed graphs that have few distinct eigenvalues together with a symmetric spectrum. Our main contribution is to determine all signed $(0,2)$-graphs with vertex degree at most $6$ that have precisely two distinct eigenvalues $\pm λ$. Next, we consider to what extent induced subgraphs of signed graph with two distinct eigenvalues $\pm λ$ are determined by their spectra. Lastly, we classify signed $(0,2)$-graphs that have a symmetric spectrum with three distinct eigenvalues and give a partial classification for those with four distinct eigenvalues.

math.CO↗
arXiv2021

Frames over finite fields: Basic theory and equiangular lines in unitary geometry

We introduce the study of frames and equiangular lines in classical geometries over finite fields. After developing the basic theory, we give several examples and demonstrate finite field analogs of equiangular tight frames (ETFs) produced by modular difference sets, and by translation and modulation operators. Using the latter, we prove that Gerzon's bound is attained in each unitary geometry of dimension $d = 2^{2l+1}$ over the field $\mathbb{F}_{3^2}$. We also investigate interactions between complex ETFs and those in finite unitary geometries, and we show that every complex ETF implies the existence of ETFs with the same size over infinitely many finite fields.

math.MG↗
arXiv2020

Improving the Delsarte bound

In this paper, we study the order of a maximal clique in an amply regular graph with a fixed smallest eigenvalue by considering a vertex that is adjacent to some (but not all) vertices of the maximal clique. As a consequence, we show that if a strongly regular graph contains a Delsarte clique, then the parameter $μ$ is either small or large. Furthermore, we obtain a cubic polynomial that assures that a maximal clique in an amply regular graph is either small or large (under certain assumptions). Combining this cubic polynomial with the claw-bound, we rule out an infinite family of feasible parameters $(v,k,λ,μ)$ for strongly regular graphs. Lastly, we provide tables of parameters $(v,k,λ,μ)$ for nonexistent strongly regular graphs with smallest eigenvalue $-4, -5, -6$ or $-7$.

math.CO↗
arXiv2020

Equiangular lines in low dimensional Euclidean spaces

We show that the maximum cardinality of an equiangular line system in 14 and 16 dimensions is 28 and 40, respectively, thereby solving a longstanding open problem. We also improve the upper bounds on the cardinality of equiangular line systems in 19 and 20 dimensions to 74 and 94, respectively.

math.CO↗
arXiv2019

On equiangular lines in 17 dimensions and the characteristic polynomial of a Seidel matrix

For $e$ a positive integer, we find restrictions modulo $2^e$ on the coefficients of the characteristic polynomial $χ_S(x)$ of a Seidel matrix $S$. We show that, for a Seidel matrix of order $n$ even (resp. odd), there are at most $2^{\binom{e-2}{2}}$ (resp. $2^{\binom{e-2}{2}+1}$) possibilities for the congruence class of $χ_S(x)$ modulo $2^e\mathbb Z[x]$. As an application of these results, we obtain an improvement to the upper bound for the number of equiangular lines in $\mathbb R^{17}$, that is, we reduce the known upper bound from $50$ to $49$.

math.CO↗
arXiv2018

Another construction of edge-regular graphs with regular cliques

We exhibit a new construction of edge-regular graphs with regular cliques that are not strongly regular. The infinite family of graphs resulting from this construction includes an edge-regular graph with parameters $(24,8,2)$. We also show that edge-regular graphs with $1$-regular cliques that are not strongly regular must have at least $24$ vertices.

math.CO↗
arXiv2018

Edge-regular graphs with regular cliques

We exhibit infinitely many examples of edge-regular graphs that have regular cliques and that are not strongly regular. This answers a question of Neumaier from 1981.

math.CO↗
arXiv2017

Equiangular line systems and switching classes containing regular graphs

We develop the theory of equiangular lines in Euclidean spaces. Our focus is on the question of when a Seidel matrix having precisely three distinct eigenvalues has a regular graph in its switching class. We make some progress towards an answer to this question by finding some necessary conditions and some sufficient conditions. Furthermore, we show that the cardinality of an equiangular line system in $18$ dimensional Euclidean space is at most $60$.

math.CO↗