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Jack H. Koolen

Publications and source records attributed to Jack H. Koolen.

At least 55 records · Page 3Linked to original sources
arXiv2019

Open problems in the spectral theory of signed graphs

Signed graphs are graphs whose edges get a sign $+1$ or $-1$ (the signature). Signed graphs can be studied by means of graph matrices extended to signed graphs in a natural way. Recently, the spectra of signed graphs have attracted much attention from graph spectra specialists. One motivation is that the spectral theory of signed graphs elegantly generalizes the spectral theories of unsigned graphs. On the other hand, unsigned graphs do not disappear completely, since their role can be taken by the special case of balanced signed graphs. Therefore, spectral problems defined and studied for unsigned graphs can be considered in terms of signed graphs, and sometimes such generalization shows nice properties which cannot be appreciated in terms of (unsigned) graphs. Here, we survey some general results on the adjacency spectra of signed graphs, and we consider some spectral problems which are inspired from the spectral theory of (unsigned) graphs.

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arXiv2019

Some Remarks on Systems of Equiangular Lines

In this note, we study the maximum number $N_α(d)$ of a system of equiangular lines in $\mathbb{R}^d$ with cosine $α$, where $\frac{1}α$ is not an odd positive integer. This note is motivated by a remark in a $2018$ paper by Balla, Dräxler, Keevash and Sudakov.

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arXiv2019

A spectral version of the Moore problem for bipartite regular graphs

Let $b(k,θ)$ be the maximum order of a connected bipartite $k$-regular graph whose second largest eigenvalue is at most $θ$. In this paper, we obtain a general upper bound for $b(k,θ)$ for any $0\leq θ< 2\sqrt{k-1}$. Our bound gives the exact value of $b(k,θ)$ whenever there exists a bipartite distance-regular graph of degree $k$, second largest eigenvalue $θ$, diameter $d$ and girth $g$ such that $g\geq 2d-2$. For certain values of $d$, there are infinitely many such graphs of various valencies $k$. However, for $d=11$ or $d\geq 15$, we prove that there are no bipartite distance-regular graphs with $g\geq 2d-2$.

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arXiv2018

Addressing Johnson graphs, complete multipartite graphs, odd cycles and other graphs

Graham and Pollak showed that the vertices of any graph $G$ can be addressed with $N$-tuples of three symbols, such that the distance between any two vertices may be easily determined from their addresses. An addressing is optimal if its length $N$ is minimum possible. In this paper, we determine an addressing of length $k(n-k)$ for the Johnson graphs $J(n,k)$ and we show that our addressing is optimal when $k=1$ or when $k=2, n=4,5,6$, but not when $n=6$ and $k=3$. We study the addressing problem as well as a variation of it in which the alphabet used has more than three symbols, for other graphs such as complete multipartite graphs and odd cycles. We also present computations describing the distribution of the minimum length of addressings for connected graphs with up to $10$ vertices. Motivated by these computations we settle a problem of Graham, showing that most graphs on $n$ vertices have an addressing of length at most $n-(2-o(1))\log_2 n$.

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arXiv2018

There does not exist a distance-regular graph with intersection array $\{80, 54,12; 1, 6, 60\}$

In this paper we will show that there does not exist a distance-regular graph $Γ$ with intersection array $\{80, 54,12; 1, 6, 60\}$. We first show that a local graph $Δ$ of $Γ$ does not contain a coclique with 5 vertices, and then we prove that the graph $Γ$ is geometric by showing that $Δ$ consists of 4 disjoint cliques with 20 vertices. Then we apply a result of Koolen and Bang to the graph $Γ$, and we could obtain that there is no such a distance-regular graph.

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arXiv2018

On the order of regular graphs with fixed second largest eigenvalue

Let $v(k, λ)$ be the maximum number of vertices of a connected $k$-regular graph with second largest eigenvalue at most $λ$. The Alon-Boppana Theorem implies that $v(k, λ)$ is finite when $k > \frac{λ^2 + 4}{4}$. In this paper, we show that for fixed $λ\geq1$, there exists a constant $C(λ)$ such that $2k+2 \leq v(k, λ) \leq 2k + C(λ)$ when $k > \frac{λ^2 + 4}{4}$.

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arXiv2018

A generalization of a theorem of Hoffman

In 1977, Hoffman gave a characterization of graphs with smallest eigenvalue at least $-2$. In this paper we generalize this result to graphs with smaller smallest eigenvalue. For the proof, we use a combinatorial object named Hoffman graph, introduced by Woo and Neumaier in 1995. Our result says that for every $λ\leq -2$, if a graph with smallest eigenvalue at least $λ$ satisfies some local conditions, then it is highly structured. We apply our result to graphs which are cospectral with the Hamming graph $H(3,q)$, the Johnson graph $J(v, 3)$ and the $2$-clique extension of grids, respectively.

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arXiv2018

On the integrability of strongly regular graphs

Koolen et al. showed that if a connected graph with smallest eigenvalue at least $-3$ has large minimal valency, then it is $2$-integrable. In this paper, we will prove that a lower bound for the minimal valency is 166.

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arXiv2018

A spectral characterization of the s-clique extension of the square grid graphs

In this paper we show that for integers $s\geq2$, $t\geq1$, any co-edge-regular graph which is cospectral with the $s$-clique extension of the $t\times t$-grid is the $s$-clique extension of the $t\times t$-grid, if $t$ is large enough. Gavrilyuk and Koolen used a weaker version of this result to show that the Grassmann graph $J_q(2D,D)$ is characterized by its intersection array as a distance-regular graph, if $D$ is large enough.

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arXiv2018

On a characterization of the Grassmann graphs

In 1995, Metsch showed that the Grassmann graph $J_q(n,D)$ of diameter $D\geq 3$ is characterized by its intersection numbers with the following possible exceptions: (-) $n=2D$ or $n=2D+1$, $q\geq 2$; (-) $n=2D+2$ and $q\in \{2,3\}$; (-) $n=2D+3$ and $q=2$. In 2005, Van Dam and Koolen constructed the twisted Grassmann graphs with the same intersection numbers as the Grassmann graphs $J_q(2D+1,D)$, for any prime power $q$ and diameter $D\geq 2$, but they are not isomorphic. We show that the Grassmann graph $J_q(2D,D)$ is characterized by its intersection numbers provided that the diameter $D$ is large enough.

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arXiv2018

A characterization of the graphs of bilinear $(d\times d)$-forms over $\mathbb{F}_2$

The bilinear forms graph denoted here by $Bil_q(e\times d)$ is a graph defined on the set of $(e\times d)$-matrices ($e\geq d$) over $\mathbb{F}_q$ with two matrices being adjacent if and only if the rank of their difference equals $1$. In 1999, K. Metsch showed that the bilinear forms graph $Bil_q(e\times d)$ is characterized by its intersection array if one of the following holds: (-) $q=2$ and $e\geq d+4$, (-) $q\geq 3$ and $e\geq d+3$. Thus, the following cases have been left unsettled: (-) $q=2$ and $e\in \{d,d+1,d+2,d+3\}$, (-) $q\geq 3$ and $e\in \{d,d+1,d+2\}$. In this work, we show that the graph of bilinear $(d\times d)$-forms over the binary field, where $d\geq 3$, is characterized by its intersection array. In doing so, we also classify locally grid graphs whose $μ$-graphs are hexagons and the intersection numbers $b_i,c_i$ are well-defined for all $i=0,1,2$.

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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.

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arXiv2017

Partially metric association schemes with a multiplicity three

An association scheme is called partially metric if it has a connected relation whose distance-two relation is also a relation of the scheme. In this paper we determine the symmetric partially metric association schemes with a multiplicity three. Besides the association schemes related to regular complete $4$-partite graphs, we obtain the association schemes related to the Platonic solids, the bipartite double scheme of the dodecahedron, and three association schemes that are related to well-known $2$-arc-transitive covers of the cube: the Möbius-Kantor graph, the Nauru graph, and the Foster graph F048A. In order to obtain this result, we also determine the symmetric association schemes with a multiplicity three and a connected relation with valency three. Moreover, we construct an infinite family of cubic arc-transitive $2$-walk-regular graphs with an eigenvalue with multiplicity three that give rise to non-commutative association schemes with a symmetric relation of valency three and an eigenvalue with multiplicity three.

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arXiv2017

Max-cut and extendability of matchings in distance-regular graphs

Let $G$ be a distance-regular graph of order $v$ and size $e$. In this paper, we show that the max-cut in $G$ is at most $e(1-1/g)$, where $g$ is the odd girth of $G$. This result implies that the independence number of $G$ is at most $\frac{v}{2}(1-1/g)$. We use this fact to also study the extendability of matchings in distance-regular graphs. A graph $G$ of even order $v$ is called $t$-extendable if it contains a perfect matching, $t<v/2$ and any matching of $t$ edges is contained in some perfect matching. The extendability of $G$ is the maximum $t$ such that $G$ is $t$-extendable. We generalize previous results on strongly regular graphs and show that all distance-regular graphs with diameter $D\geq 3$ are $2$-extendable. We also obtain various lower bounds for the extendability of distance-regular graphs of valency $k$ that depend on $k$, $λ$ and $μ$, where $λ$ is the number of common neighbors of any two adjacent vertices and $μ$ is the number of common neighbors of any two vertices in distance two.

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arXiv2016

Graphs with two main and two plain eigenvalues

In this paper, we introduce the concepts of the plain eigenvalue, the main-plain index and the refined spectrum of graphs. We focus on the graphs with two main and two plain eigenvalues and give some characterizations of them.

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