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Willem H. Haemers

Publications and source records attributed to Willem H. Haemers.

At least 19 recordsLinked to original sources
arXiv2026

On perfect matchings, edge-colourings and eigenvalues of cubic graphs

We discuss the question whether the existence of perfect matchings in a cubic graph can be seen from the spectrum of its adjacency matrix. For regular graphs in general and for three edge-disjoint perfect matchings in a cubic graph (that is, an edge colouring with three colors) the answer is known to be negative. In the latter case, a few counter examples (found by computer) are known. Here we show that these counter examples can be extended to an infinite family by use of truncation. Thus we obtain infinitely many pairs of cospectral cubic graphs with different edge-chromatic number. For all these pairs both graphs have a perfect matching, and the mentioned question is still open. We obtain a few more results concerning spectral characterisations of cubic graphs, and point out a serious mistake in Section 3 of an earlier version of this paper.

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arXiv2025

Thin divisible designs graphs: an interplay between fixed-point free involutions of $(v,k,λ)$-graphs and symmetric weighing matrices

In this paper, we illustrate important aspects of the interplay between weighing matrices, $(v,k,λ)$-graphs with fixed-point free involutions, and signed graphs with an orthogonal adjacency matrix, which arises from thin divisible design graphs. In particular, we present two new recursive constructions of regular symmetric Hadamard matrices with constant diagonal (equivalently, two new recursive constructions of strongly regular graphs) and we find a fixed-point free involution in the symplectic graph $Sp(4,q)$, where $q$ is odd, which leads to orthogonal signings for an infinite family of antipodal distance-regular graphs of diameter 3.

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arXiv2025

The Laplacian matrix of weighted threshold graphs

Threshold graphs are generated from one node by repeatedly adding a node that links to all existing nodes or adding a node without links. In the weighted threshold graph, we add a new node in step $i$, which is linked to all existing nodes by a link of weight $w_i$. In this work, we consider the set ${\cal A}_N$ that contains all Laplacian matrices of weighted threshold graphs of order $N$. We show that ${\cal A}_N$ forms a commutative algebra. Using this, we find a common basis of eigenvectors for the matrices in ${\cal A}_N$. It follows that the eigenvalues of each matrix in ${\cal A}_N$ can be represented as a linear transformation of the link weights. In addition, we prove that, if there are just three or fewer different weights, two weighted threshold graphs with the same Laplacian spectrum must be isomorphic.

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arXiv2023

Cospectrality results for signed graphs with two eigenvalues unequal to $\pm 1$

Recently the collection $\cal G$ of all signed graphs for which the adjacency matrix has all but at most two eigenvalues equal to $\pm 1$ has been determined. Here we investigate $\cal G$ for cospectral pairs, and for signed graphs determined by their spectrum (up to switching). If the order is at most 20, the outcome is presented in a clear table. If the spectrum is symmetric we find all signed graphs in $\cal G$ determined by their spectrum, and we obtain all signed graphs cospectral with the bipartite double of the complete graph. In addition we determine all signed graphs cospectral with the Friendship graph $F_\ell$, and show that there is no connected signed graph cospectral but not switching equivalent with $F_\ell$.

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arXiv2021

Hoffman's ratio bound

Hoffman's ratio bound is an upper bound for the independence number of a regular graph in terms of the eigenvalues of the adjacency matrix. The bound has proved to be very useful and has been applied many times. Hoffman did not publish his result, and for a great number of users the emergence of Hoffman's bound is a black hole. With this note I hope to clarify the history of this bound and some of its generalizations.

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arXiv2021

On signed graphs with at most two eigenvalues unequal to $\pm 1$

We present the first steps towards the determination of the signed graphs for which the adjacency matrix has all but at most two eigenvalues equal to 1 or -1. Here we deal with the disconnected, the bipartite and the complete signed graphs. In addition, we present many examples which cannot be obtained from an unsigned graph or its negative by switching.

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arXiv2021

Graph toughness from Laplacian eigenvalues

The toughness $t(G)$ of a graph $G=(V,E)$ is defined as $t(G)=\min\{\frac{|S|}{c(G-S)}\}$, in which the minimum is taken over all $S\subset V$ such that $G-S$ is disconnected, where $c(G-S)$ denotes the number of components of $G-S$. We present two tight lower bounds for $t(G)$ in terms of the Laplacian eigenvalues and provide strong support for a conjecture for a better bound which, if true, implies both bounds, and improves and generalizes known bounds by Alon, Brouwer, and the first author. As applications, several new results on perfect matchings, factors and walks from Laplacian eigenvalues are obtained, which leads to a conjecture about Hamiltonicity and Laplacian eigenvalues.

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arXiv2021

Spectral symmetry in conference matrices

A conference matrix of order $n$ is an $n\times n$ matrix $C$ with diagonal entries $0$ and off-diagonal entries $\pm 1$ satisfying $CC^\top=(n-1)I$. If $C$ is symmetric, then $C$ has a symmetric spectrum $Σ$ (that is, $Σ=-Σ$) and eigenvalues $\pm\sqrt{n-1}$. We show that many principal submatrices of $C$ also have symmetric spectrum, which leads to examples of Seidel matrices of graphs (or, equivalently, adjacency matrices of complete signed graphs) with a symmetric spectrum. In addition, we show that some Seidel matrices with symmetric spectrum can be characterized by this construction.

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arXiv2021

Spectra of strongly Deza graphs

A Deza graph $G$ with parameters $(n,k,b,a)$ is a $k$-regular graph with $n$ vertices such that any two distinct vertices have $b$ or $a$ common neighbours. The children $G_A$ and $G_B$ of a Deza graph $G$ are defined on the vertex set of $G$ such that every two distinct vertices are adjacent in $G_A$ or $G_B$ if and only if they have $a$ or $b$ common neighbours, respectively. A strongly Deza graph is a Deza graph with strongly regular children. In this paper we give a spectral characterisation of strongly Deza graphs, show relationships between eigenvalues, and study strongly Deza graphs which are distance-regular.

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arXiv2020

The chromatic index of strongly regular graphs

We determine (partly by computer search) the chromatic index (edge-chromatic number) of many strongly regular graphs (SRGs), including the SRGs of degree $k \leq 18$ and their complements, the Latin square graphs and their complements, and the triangular graphs and their complements. Moreover, using a recent result of Ferber and Jain it is shown that an SRG of even order $n$, which is not the block graph of a Steiner 2-design or its complement, has chromatic index $k$, when $n$ is big enough. Except for the Petersen graph, all investigated connected SRGs of even order have chromatic index equal to their degree, i.e., they are class 1, and we conjecture that this is the case for all connected SRGs of even order.

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arXiv2020

Universal spectra of the disjoint union of regular graphs

A universal adjacency matrix of a graph $G$ with adjacency matrix $A$ is any matrix of the form $U = αA + βI + γJ + δD$ with $α\neq 0$, where $I$ is the identity matrix, $J$ is the all-ones matrix and $D$ is the diagonal matrix with the vertex degrees. In the case that $G$ is the disjoint union of regular graphs, we present an expression for the characteristic polynomials of the various universal adjacency matrices in terms of the characteristic polynomials of the adjacency matrices of the components. As a consequence we obtain a formula for the characteristic polynomial of the Seidel matrix of $G$, and the signless Laplacian of the complement of $G$ (i.e. the join of regular graphs).

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arXiv2020

On sign-symmetric signed graphs

A signed graph is said to be sign-symmetric if it is switching isomorphic to its negation. Bipartite signed graphs are trivially sign-symmetric. We give new constructions of non-bipartite sign-symmetric signed graphs. Sign-symmetric signed graphs have a symmetric spectrum but not the other way around. We present constructions of signed graphs with symmetric spectra which are not sign-symmetric. This, in particular answers a problem posed by Belardo, Cioabă, Koolen, and Wang (2018).

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arXiv2020

On NP-hard graph properties characterized by the spectrum

Properties of graphs that can be characterized by the spectrum of the adjacency matrix of the graph have been studied systematically recently. Motivated by the complexity of these properties, we show that there are such properties for which testing whether a graph has that property can be NP-hard (or belong to other computational complexity classes consisting of even harder problems). In addition, we discuss a possible spectral characterization of some well-known NP-hard problems. In particular, for every integer $k\geq 6$ we construct a pair of $k$-regular cospectral graphs, where one graph is Hamiltonian and the other one not.

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arXiv2019

On the spectral characterization of mixed extensions of $P_3$

A mixed extension of a graph $G$ is a graph $H$ obtained from $G$ by replacing each vertex of $G$ by a clique or a coclique, whilst two vertices in $H$ corresponding to distinct vertices $x$ and $y$ of $G$ are adjacent whenever $x$ and $y$ are adjacent in $G$. If $G$ is the path $P_3$, then $H$ has at most three adjacency eigenvalues unequal to $0$ and $-1$. Recently, the first author classified the graphs with the mentioned eigenvalue property. Using this classification we investigate mixed extension of $P_3$ on being determined by the adjacency spectrum. We present several cospectral families, and with the help of a computer we find all graphs on at most $25$ vertices that are cospectral with a mixed extension of $P_3$.

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arXiv2018

Deza graphs with parameters $(n,k,k-1,a)$ and $β=1$

A Deza graph with parameters $(n,k,b,a)$ is a $k$-regular graph with $n$ vertices in which any two vertices have $a$ or $b$ ($a\leq b$) common neighbours. A Deza graph is strictly Deza if it has diameter $2$, and is not strongly regular. In an earlier paper, the two last authors et el. characterized the strictly Deza graphs with $b=k-1$ and $β> 1$, where $β$ is the number of vertices with $b$ common neighbours with a given vertex. Here we deal with the case $β=1$, thus we complete the characterization of strictly Deza graphs with $b=k-1$. It follows that all Deza graphs with $b=k-1$ and $β=1$ can be made from special strongly regular graphs, and we present several examples of such strongly regular graphs. A divisible design graph is a special Deza graph, and a Deza graph with $β=1$ is a divisible design graph. The present characterization reveals an error in a paper on divisible design graphs by the second author et al. We discuss the cause and the consequences of this mistake and give the required errata.

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arXiv2018

Signed graphs cospectral with the path

A signed graph $Γ$ is said to be determined by its spectrum if every signed graph with the same spectrum as $Γ$ is switching isomorphic with $Γ$. Here it is proved that the path $P_n$, interpreted as a signed graph, is determined by its spectrum if and only if $n\equiv 0, 1$, or 2 (mod 4), unless $n\in\{8, 13, 14, 17, 29\}$, or $n=3$.

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