Gary R. W. Greaves

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20112026

most citedEquiangular lines in Euclidean spaces

96 citations · 298 across the 25 of their papers we have counts for

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25 papers · 1 filter

math.CO2026

A proof of Haemers' toughness conjecture

Gary Greaves, Haoran Zhu

We prove that if $Γ$ is a connected graph with minimum degree $δ$ and Laplacian eigenvalues $0 = μ_{1} < μ_{2} ⩽ \dots ⩽ μ_{n}$ , then the toughness of $Γ$ is bounded below by…

math.CO2026

Aldous property for full-flag Johnson graphs

Gary Greaves, Haoran Zhu

We show that the full-flag Johnson graph has spectral gap equal to that of its Schreier quotient arising from the point-stabiliser equitable partition. Our results confirm two conj…

math.CO2025

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

Gary R. W. Greaves, Jeven Syatriadi

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 coeff…

math.CO2024

Chromatic polynomials of signed graphs and dominating-vertex deletion formulae

Gary R. W. Greaves, Jeven Syatriadi, Charissa I. Utomo

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 variou…

math.CO2022★ 3 cited

Real equiangular lines in dimension 18 and the Jacobi identity for complementary subgraphs

Gary R. W. Greaves, Jeven Syatriadi

We show that the maximum cardinality of an equiangular line system in $R^{18}$ is at most $59$ . Our proof includes a novel application of the Jacobi identity for complement…

math.CO2021

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

Gary R. W. Greaves, Zoran Stanić

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 wi…

math.CO2021

Hermitian matrices of roots of unity and their characteristic polynomials

Gary R. W. Greaves, Chin Jian Woo

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 characteri…

math.CO2021

Equiangular lines in Euclidean spaces: dimensions 17 and 18

Gary R. W. Greaves, Jeven Syatriadi, Pavlo Yatsyna

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 construct…

math.CO2020★ 2 cited

Improving the Delsarte bound

Gary R. W. Greaves, Jack H. Koolen, Jongyook Park

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) vert…

math.CO2020

Frames over finite fields: Equiangular lines in orthogonal geometry

Gary R. W. Greaves, Joseph W. Iverson, John Jasper +1

We investigate equiangular lines in finite orthogonal geometries, focusing specifically on equiangular tight frames (ETFs). In parallel with the known correspondence between real E…

math.CO2020

Equiangular lines in low dimensional Euclidean spaces

Gary R. W. Greaves, Jeven Syatriadi, Pavlo Yatsyna

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…

math.CO2018

Another construction of edge-regular graphs with regular cliques

Gary R. W. Greaves, J. H. Koolen

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…

math.CO2018

Reed-Solomon codes over small fields with constrained generator matrices

Gary Greaves, Jeven Syatriadi

We give constructions of some special cases of $[n, k]$ Reed-Solomon codes over finite fields of size at least $n$ and $n + 1$ whose generator matrices have constrained support. Furth…

math.CO2018★ 20 cited

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

Gary R. W. Greaves, Pavlo Yatsyna

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 matri…

math.CO2018

On the Smith normal form of a skew-symmetric D-optimal design of order $n \equiv 2 (mod 4)$

Gary Greaves, Sho Suda

We show that the Smith normal form of a skew-symmetric D-optimal design of order $n \equiv 2 (mod 4)$ is determined by its order. Furthermore, we show that the Smith normal form of…

math.CO2017

Edge-regular graphs with regular cliques

Gary R. W. Greaves, Jack H. Koolen

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.CO2017

On a lower bound for the Laplacian eigenvalues of a graph

Gary R. W. Greaves, Akihiro Munemasa, Anni Peng

If $μ_{m}$ and $d_{m}$ denote, respectively, the $m$ -th largest Laplacian eigenvalue and the $m$ -th largest vertex degree of a graph, then $μ_{m} ⩾ d_{m} - m + 2$ . This inequality was…

math.CO2016★ 39 cited

Equiangular line systems and switching classes containing regular graphs

Gary R. W. Greaves

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…

math.CO2016

On the clique number of a strongly regular graph

Gary R. W. Greaves, Leonard H. Soicher

We determine new upper bounds for the clique numbers of strongly regular graphs in terms of their parameters. These bounds improve on the Delsarte bound for infinitely many feasibl…

math.CO2016★ 9 cited

Interlacing families and the Hermitian spectral norm of digraphs

Gary Greaves, Bojan Mohar, O Suil

It is proved that for any finite connected graph $G$ , there exists an orientation of $G$ such that the spectral radius of the corresponding Hermitian adjacency matrix is smaller or…

math.CO2016★ 10 cited

Symmetric and skew-symmetric ${0, \pm 1}$ -matrices with large determinants

Gary Greaves, Sho Suda

We show that the existence of ${\pm 1}$ -matrices having largest possible determinant is equivalent to the existence of certain tournament matrices. In particular, we prove a rece…

math.CO2015★ 22 cited

Graphs with three eigenvalues and second largest eigenvalue at most 1

Xi-Ming Cheng, Gary R. W. Greaves, Jack H. Koolen

We classify the connected graphs with precisely three distinct eigenvalues and second largest eigenvalue at most 1.

math.CO2014★ 39 cited

Biregular graphs with three eigenvalues

Xi-Ming Cheng, Alexander L. Gavrilyuk, Gary R. W. Greaves +1

We consider nonregular graphs having precisely three distinct eigenvalues. The focus is mainly on the case of graphs having two distinct valencies and our results include construct…

math.CO2014★ 96 cited

Equiangular lines in Euclidean spaces

G. Greaves, J. H. Koolen, A. Munemasa +1

We obtain several new results contributing to the theory of real equiangular line systems. Among other things, we present a new general lower bound on the maximum number of equiang…

A proof of Haemers' toughness conjecture

Aldous property for full-flag Johnson graphs

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

Chromatic polynomials of signed graphs and dominating-vertex deletion formulae

Real equiangular lines in dimension 18 and the Jacobi identity for complementary subgraphs

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

Hermitian matrices of roots of unity and their characteristic polynomials

Equiangular lines in Euclidean spaces: dimensions 17 and 18

Improving the Delsarte bound

Frames over finite fields: Equiangular lines in orthogonal geometry

Equiangular lines in low dimensional Euclidean spaces

Another construction of edge-regular graphs with regular cliques

Reed-Solomon codes over small fields with constrained generator matrices

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

On the Smith normal form of a skew-symmetric D-optimal design of order n≡2(mod4)

Edge-regular graphs with regular cliques

On a lower bound for the Laplacian eigenvalues of a graph

Equiangular line systems and switching classes containing regular graphs

On the clique number of a strongly regular graph

Interlacing families and the Hermitian spectral norm of digraphs

Symmetric and skew-symmetric {0,±1}-matrices with large determinants

Graphs with three eigenvalues and second largest eigenvalue at most 1

Biregular graphs with three eigenvalues

Equiangular lines in Euclidean spaces

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

On the Smith normal form of a skew-symmetric D-optimal design of order $n \equiv 2 (mod 4)$

Symmetric and skew-symmetric ${0, \pm 1}$ -matrices with large determinants