The Signless p-Laplacian Spectral Radius of Graphs with Given Degree Sequences

Abstract

In this paper, we consider the spectral radius of signless p-Laplacian of a graph, which is a generalization of the quadratic form of the signless Laplacian matrix for $p=2$. Let $\pi =(d_0,d_1,\ldots ,d_{n-1})$ be a non-increasing sequence of positive integers and ${\mathcal {G}}_{\pi }$ the set of graphs with degree sequence $\pi$. In this paper, we obtain some transformations for graphs in ${\mathcal {G}}_{\pi }$ that do not decrease the largest signless p-Laplacian eigenvalue of a graph. Furthermore, if $\sum _{i=0}^{n-1}d_i=2n$ and $d_2\ge 2$, then we identify the graph maximizing the signless p-Laplacian spectral radius among ${\mathcal {G}}_{\pi }$. As an application, we get the extremal graph maximizing the signless p-Laplacian spectral radius among all unicyclic graphs.