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arXiv · 1612.07085

A generalization of a theorem of Hoffman

Jack H. Koolen·Qianqian Yang·Jae Young Yang
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Abstract

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 $\lambda \leq -2$, if a graph with smallest eigenvalue at least $\lambda$ 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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Jack H. Koolen, Qianqian Yang, Jae Young Yang. 2016-12-21. A generalization of a theorem of Hoffman. https://arxiv.org/abs/1612.07085

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