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arXiv · 1612.02643

The $p$-spectral radius of the Laplacian

Elizandro Max Borba·Sebastian Richter·Eliseu Fritscher·Carlos Hoppen
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Abstract

The $p$-spectral radius of a graph $G=(V,E)$ with adjacency matrix $A$ is defined as $\lambda^{(p)}(G)=\max \{x^TAx : \|x\|_p=1 \}$. This parameter shows remarkable connections with graph invariants, and has been used to generalize some extremal problems. In this work, we extend this approach to the Laplacian matrix $L$, and define the $p$-spectral radius of the Laplacian as $\mu^{(p)}(G)=\max \{x^TLx : \|x\|_p=1 \}$. We show that $\mu^{(p)}(G)$ relates to invariants such as maximum degree and size of a maximum cut. We also show properties of $\mu^{(p)}(G)$ as a function of $p$, and a upper bound on $\max_{G \colon |V(G)|=n} \mu^{(p)}(G)$ in terms of $n=|V|$ for $p\ge 2$, which is attained if $n$ is even.

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Elizandro Max Borba, Sebastian Richter, Eliseu Fritscher, Carlos Hoppen. 2016-12-08. The $p$-spectral radius of the Laplacian. https://arxiv.org/abs/1612.02643

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