J. Koolen

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20242026

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12 papers · 1 filter

math.CO2026

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

Yu Ning, Jack H. Koolen, Xiande Zhang

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,…

math.CO2026

On the characterization of geometric distance-regular graphs

Chenhui Lv, Jack H. Koolen

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 $D \geq 3$ …

math.CO2026

Towards a classification of $1$ -homogeneous distance-regular graphs with positive intersection number $a_{1}$

Jack H. Koolen, Mamoon Abdullah, Brhane Gebremichel +1

Let $I^$ be a graph with diameter at least two. Then $I^$ is said to be $1$ -homogeneous (in the sense of Nomura) whenever for every pair of adjacent vertices $x$ and $y$ in $I^$ , t…

math.CO2026

An improved bound for strongly regular graphs with smallest eigenvalue $- m$

Jack Koolen, Chenhui Lv, Greg Markowsky +1

In 1979, Neumaier gave a bound on $I^»$ in terms of $m$ and $I^¼$ , where $- m$ is the smallest eigenvalue of a primitive strongly regular graph, unless the graph in question belongs t…

math.CO2025

A Bose-Laskar-Hoffman theory for $I^¼$ -bounded graphs with fixed smallest eigenvalue

Jack H. Koolen, Hong-Jun Ge, Chenhui Lv +1

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…

math.CO2025

Bounding the parameter $I^^{2}$ of a distance-regular graph with classical parameters

Chenhui Lv, Jack H. Koolen

Let $I^$ be a distance-regular graph with classical parameters $(D, b, I^\pm, I^^{2})$ satisfying $b \geq 2$ and $D \geq 3$ . Let $r = 1 + b + b^{2} + \dots + b^{D - 1}$ . In 1999, K. Metsch showed that t…

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

On the characterization of geometric distance-regular graphs

Towards a classification of 1-homogeneous distance-regular graphs with positive intersection number a1​

An improved bound for strongly regular graphs with smallest eigenvalue −m

A Bose-Laskar-Hoffman theory for I^¼-bounded graphs with fixed smallest eigenvalue

Bounding the parameter I^2 of a distance-regular graph with classical parameters

Towards a classification of $1$ -homogeneous distance-regular graphs with positive intersection number $a_{1}$

An improved bound for strongly regular graphs with smallest eigenvalue $- m$

A Bose-Laskar-Hoffman theory for $I^¼$ -bounded graphs with fixed smallest eigenvalue

Bounding the parameter $I^^{2}$ of a distance-regular graph with classical parameters