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Jack H. Koolen

Publications and source records attributed to Jack H. Koolen.

At least 19 recordsLinked to original sources
arXiv2026

A structure theory for signed graphs with fixed smallest eigenvalue

In this paper, we give a structure theory for signed graphs with fixed smallest eigenvalue. As a consequence, we prove that for every $λ\in(-1-\sqrt{2},-2]$, if a connected signed graph has smallest eigenvalue at least $λ$ and sufficiently large minimum valency, then its smallest eigenvalue is at least $-2$ and it is $1$-integrable. Thus, every such signed graph admits a representation by a family of $\{0,\pm1\}$-vectors of squared norm $2$ and may therefore be viewed as a natural signed generalization of generalized line graphs.

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arXiv2026

Comparing classes of highly symmetric graphs: From $2$-arc-transitive to $2$-distance-transitive

A $2$-distance-transitive graph is a vertex-transitive graph whose vertex stabilizer is transitive on both the first- and second-step neighborhoods. This concept simultaneously generalizes both distance-transitive graphs and $2$-arc-transitive graphs. In this paper, we first determine the vertex-quasiprimitive types of $2$-distance-transitive graphs of odd order, partially answering a question posed by A. Devillers, M. Giudici, C. H. Li and C. E. Praeger in 2012. We then prove that a $2$-distance-transitive graph of valency $p+1$, where $p$ is a prime, is $2$-arc-transitive if and only if it has girth at least $4$. We also show that every locally-primitive $2$-distance-transitive graph of valency at most $8$ is $2$-arc-transitive, with the icosahedron as the unique exception. Finally, we prove that if $Γ$ is a $G$-locally-primitive, $(G,2)$-distance-transitive graph of valency at least $3$ and $G$ is soluble, then either $Γ\cong \K_{p,p}$ for some prime $p$, or the order of $Γ$ is not square-free.

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arXiv2026

Signed graphs with fixed smallest eigenvalue at least $-3$ and their lattices

In this paper, we consider connected signed graphs with smallest eigenvalue at least $-3-\varepsilon$ for a small positive constant $\varepsilon$. We prove that if such a signed graph has sufficiently large minimum valency, then its smallest eigenvalue is at least $-3$, and the lattice associated with it, which is generated by squared norm $3$ vectors, is a sublattice of a direct sum of the standard lattice $\mathbb{Z}^n$ and copies of the root lattice $E_8$. Moreover, there exist infinitely many connected signed graphs with smallest eigenvalue at least $-3$ containing it as a proper induced subgraph. Furthermore, we discuss signed graphs with smallest eigenvalue $-3$ arising from rootless irreducible unimodular lattices.

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arXiv2026

Co-edge-regular graphs with four eigenvalues and unbounded coherent rank

In the regular three-eigenvalue setting, spectral complexity and coherent-algebraic complexity coincide: a connected regular graph has exactly three distinct eigenvalues if and only if it is strongly regular, its coherent rank is three. Although examples of regular graphs with four distinct eigenvalues and coherent rank larger than four are known, it was unknown whether coherent rank is uniformly bounded among regular graphs with four distinct eigenvalues. We show that no such bound exists, even under the additional assumption of co-edge-regularity. For every prime power \(q\), we construct infinitely many co-edge-regular graphs with exactly four distinct eigenvalues, smallest eigenvalue \(-2q-1\), and coherent rank at least \(q+4\). Consequently, coherent rank is unbounded among co-edge-regular graphs with exactly four distinct eigenvalues.

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arXiv2026

On the Smallest Eigenvalues and Quantum Chromatic Numbers of Hamming Graphs and Generalizations

The smallest eigenvalues of (distance-j) Hamming graphs with distance parameter j at least half the length were completely determined by Brouwer et al. (2018). In the present work, we address the complementary regime, namely distances j strictly less than half the length, and derive asymptotic lower bounds on the smallest eigenvalue of binary Hamming graphs. For certain natural generalizations, specifically Cayley graphs defined over quaternary vector spaces, we asymptotically determine the smallest eigenvalue as well. As an application, we obtain lower bounds on the quantum chromatic number of these graphs. In particular, for the aforementioned Cayley graphs over quaternary vectors, our lower bounds for the quantum chromatic number coincide with known upper bounds.

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arXiv2026

Amply regular graphs with $μ$ close to half the valency and group divisible designs

In this paper, we classify connected amply regular graphs with diameter $d \geq 4$ and parameters $(v, k, λ, μ)$ satisfying $μ= \frac{k-1}{2}$, where $k\geq 5$ is odd. We prove that such a graph must be exactly one of the following: the $5$-cube, the graph $\K_2 \square Λ$, where $Λ$ is the unique bipartite $(0,2)$-graph on $14$ vertices, or the point--block incidence graph of a group divisible design with the dual property, namely a $GDDDP\left(2, k+1;\, k;\, 0, \frac{k-1}{2}\right)$. For the last family, we give equivalent characterizations in terms of bipartite $Q$-regular graphs and relation graphs of symmetric association schemes with five classes. Furthermore, we present constructions of such amply regular graphs, yielding infinite families of examples derived from Paley graphs, Peisert graphs, and Paley digraphs.

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arXiv2026

Almost amorphic association schemes

An association scheme is called amorphic if every possible fusion of relations gives rise to another association scheme. In earlier work, we showed that if an association scheme has at most one relation that is neither strongly regular of Latin square type nor strongly regular of negative Latin square type, then it must be amorphic. We now construct non-amorphic $d$-class association schemes in which precisely two relations are not strongly regular of Latin square type or strongly regular of negative Latin square type, for any $d \geq 4$. We also raise the question whether different types of strongly regular graphs can coexist in an association scheme. Among some other results, we show that if one of the relations is a lattice graph, then any other strongly regular relation in the scheme must be of Latin square type.

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arXiv2026

On the characterization of geometric distance-regular graphs

In 2010, Koolen and Bang proposed the following conjecture: For a fixed integer $m \geq 2$, any geometric distance-regular graph with smallest eigenvalue $-m$, diameter $D \geq 3$ and $c_2 \geq 2$ is either a Johnson graph, a Grassmann graph, a Hamming graph, a bilinear forms graph, or the number of vertices is bounded above by a function of $m$. In this paper, we obtain some partial results towards this conjecture.

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arXiv2026

Towards a classification of $1$-homogeneous distance-regular graphs with positive intersection number $a_1$

Let $Γ$ be a graph with diameter at least two. Then $Γ$ is said to be $1$-homogeneous (in the sense of Nomura) whenever for every pair of adjacent vertices $x$ and $y$ in $Γ$, the distance partition of the vertex set of $Γ$ with respect to both $x$ and $y$ is equitable, and the parameters corresponding to equitable partitions are independent of the choice of $x$ and $y$. Assume that $Γ$ is $1$-homogeneous distance-regular with intersection number $a_1>0$ and diameter $D\geqslant 5$. Define $b=b_1/(θ_1+1)$, where $b_1$ is the intersection number and $θ_1$ is the second largest eigenvalue of $Γ$. We show that if intersection number $c_2$ is at least $2$, then $b\geqslant 1$ and one of the following (i)--(vi) holds: (i) $Γ$ is a regular near $2D$-gon, (ii) $Γ$ is a Johnson graph $J(2D,D)$, (iii) $Γ$ is a halved $\ell$-cube with $\ell \in \{2D,2D+1\}$, (iv) $Γ$ is a folded Johnson graph $\bar{J}(4D,2D)$, (v) $Γ$ is a folded halved $4D$-cube, (vi) the valency of $Γ$ is bounded by a function of $b$. Using this result, we characterize $1$-homogeneous graphs with classical parameters and $a_1>0$, as well as tight distance-regular graphs.

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arXiv2025

A Bose-Laskar-Hoffman theory for $μ$-bounded graphs with fixed smallest eigenvalue

In 2018, by Ramsey and Hoffman theory, Koolen, Yang, and Yang presented a structural result on graphs with smallest eigenvalue at least $-3$ and large minimum degree. In this study, we depart from the conventional use of Ramsey theory and instead employ a novel approach that combines the Bose-Laskar type argument with Hoffman theory to derive structural insights into $μ$-bounded graphs with fixed smallest eigenvalue. Our method establishes a reasonable bound on the minimum degree. Note that local graphs of distance-regular graphs are $μ$-bounded. We apply these results to characterize the structure for any local graph of a distance-regular graph with classical parameters $(D,b,α,β)$. Consequently, we show that the parameter $α$ is bounded by a cubic polynomial in $b$ if $D \geq 9$ and $b \geq 2$. We also show that $α\leq 2$ if $b =2$ and $D \geq 12$.

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arXiv2025

Bounding the parameter $β$ of a distance-regular graph with classical parameters

Let $Γ$ be a distance-regular graph with classical parameters $(D, b, α, β)$ satisfying $b\geq 2$ and $D\geq 3$. Let $r=1+b+b^2+\cdots+b^{D-1}$. In 1999, K. Metsch showed that there exists a positive constant $C(α,b)$ only depending on $α$ and $b$, such that if $β\geq C(α, b)r^2$, then either $Γ$ is a Grassmann graph or a bilinear forms graph. In this work, we show that for $b\geq 2$ and $D\geq 3$, then there exists a constant $C_1(α, b)$ only depending on $α$ and $b$, such that if $β\geq C_1(α, b)r$, then either $Γ$ is a Grassmann graph, or a bilinear forms graph.

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arXiv2025

Edge-connectivity of graphs with non-negative Bakry-Émery curvature and amply regular graphs

We establish a sharp edge-connectivity estimate for graphs with non-negative Bakry-Émery curvature. This leads to a geometric criterion for the existence of a perfect matching. Precisely, we show that any regular graph with non-negative Bakry-Émery curvature and an even or infinite number of vertices has a perfect matching. Through a synthesis of combinatorial and curvature-related techniques, we determine the edge-connectivity of (possibly infinite) amply regular graphs.

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arXiv2025

On co-edge-regular graphs with 4 distinct eigenvalues

Tan et al. conjectured that connected co-edge-regular graphs with four distinct eigenvalues and fixed smallest eigenvalue, when having sufficiently large valency, belong to two different families of graphs. In this paper we construct two new infinite families of connected co-edge-regular graphs with four distinct eigenvalues and fixed smallest eigenvalue, thereby disproving their conjecture. Moreover, one of these constructions demonstrates that clique-extensions of Latin Square graphs are not determined by their spectrum.

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arXiv2025

Characterizations of amorphic schemes and fusions of pairs

An association scheme is called amorphic if every possible fusion of relations gives rise to a fusion scheme. We call a pair of relations fusing if fusing that pair gives rise to a fusion scheme. We define the fusing-relations graph on the set of relations, where a pair forms an edge if it fuses. We show that if the fusing-relations graph is connected but not a path, then the association scheme is amorphic. As a side result, we show that if an association scheme has at most one relation that is neither strongly regular of Latin square type nor strongly regular of negative Latin square type, then it is amorphic.

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arXiv2024

On the (non-)existence of tight distance-regular graphs: a local approach

Let $Γ$ denote a distance-regular graph with diameter $D\geq 3$. Jurišić and Vidali conjectured that if $Γ$ is tight with classical parameters $(D,b,α,β)$, $b\geq 2$, then $Γ$ is not locally the block graph of an orthogonal array nor the block graph of a Steiner system. In the present paper, we prove this conjecture and, furthermore, extend it from the following aspect. Assume that for every triple of vertices $x, y, z$ of $Γ$, where $x$ and $y$ are adjacent, and $z$ is at distance $2$ from both $x$ and $y$, the number of common neighbors of $x$, $y$, $z$ is constant. We then show that if $Γ$ is locally the block graph of an orthogonal array (resp. a Steiner system) with smallest eigenvalue $-m$, $m\geq 3$, then the intersection number $c_2$ is not equal to $m^2$ (resp. $m(m+1)$). Using this result, we prove that if a tight distance-regular graph $Γ$ is not locally the block graph of an orthogonal array or a Steiner system, then the valency (and hence diameter) of $Γ$ is bounded by a function in the parameter $b=b_1/(1+θ_1)$, where $b_1$ is the intersection number of $Γ$ and $θ_1$ is the second largest eigenvalue of $Γ$.

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