The $p$-spectral radius of the Laplacian

TL;DR

This paper introduces the $p$-spectral radius of the Laplacian matrix of a graph, exploring its properties, connections to graph invariants, and bounds, extending previous work on the adjacency matrix to Laplacians.

Contribution

It extends the concept of the $p$-spectral radius from adjacency matrices to Laplacian matrices, establishing new relationships with graph invariants and deriving bounds for this parameter.

Findings

$ ext{max}_G \mu^{(p)}(G)$ is bounded by a function of $n$ for $p \\ge 2$

$ ext{max}_G \\mu^{(p)}(G)$ is attained when $n$ is even

Properties of $\\mu^{(p)}(G)$ as a function of $p$ are characterized

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.