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arXiv · 1805.01056

A spectral version of the Moore problem for bipartite regular graphs

Sebastian M. Cioabă·Jack H. Koolen·Hiroshi Nozaki
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Abstract

Let $b(k,\theta)$ be the maximum order of a connected bipartite $k$-regular graph whose second largest eigenvalue is at most $\theta$. In this paper, we obtain a general upper bound for $b(k,\theta)$ for any $0\leq \theta< 2\sqrt{k-1}$. Our bound gives the exact value of $b(k,\theta)$ whenever there exists a bipartite distance-regular graph of degree $k$, second largest eigenvalue $\theta$, 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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Sebastian M. Cioabă, Jack H. Koolen, Hiroshi Nozaki. 2018-05-02. A spectral version of the Moore problem for bipartite regular graphs. https://arxiv.org/abs/1805.01056

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