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arXiv · 1507.00625

Maxima of the Q-index: graphs with no K_s,t

Maria Aguieiras A. de Freitas·Vladimir Nikiforov·Laura Patuzzi
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Abstract

This note presents a new spectral version of the graph Zarankiewicz problem: How large can be the maximum eigenvalue of the signless Laplacian of a graph of order $n$ that does not contain a specified complete bipartite subgraph. A conjecture is stated about general complete bipartite graphs, which is proved for infinitely many cases. More precisely, it is shown that if $G$ is a graph of order $n,$ with no subgraph isomorphic to $K_{2,s+1},$ then the largest eigenvalue $q(G)$ of the signless Laplacian of $G$ satisfies \[ q(G)\leq\frac{n+2s}{2}+\frac{1}{2}\sqrt{(n-2s)^{2}+8s}, \] with equality holding if and only if $G$ is a join of $K_{1}$ and an $s$-regular graph of order $n-1.$

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Maria Aguieiras A. de Freitas, Vladimir Nikiforov, Laura Patuzzi. 2015-07-02. Maxima of the Q-index: graphs with no K_s,t. https://arxiv.org/abs/1507.00625

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