Elizandro Max Borba, Eliseu Fritscher, Carlos Hoppen, Sebastian Richter: The p-spectral radius of the Laplacian matrix

Author(s):

Elizandro Max Borba
Eliseu Fritscher
Carlos Hoppen
Sebastian Richter

Title:

Elizandro Max Borba, Eliseu Fritscher, Carlos Hoppen, Sebastian Richter: The p-spectral radius of the Laplacian matrix

Electronic source:

application/pdf

Preprint series:

Technische Universität Chemnitz, Fakultät für Mathematik (Germany). Preprint 01, 2017

Mathematics Subject Classification:

    05C50 []
    90C26 []

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 $p> 2$, which is attained if $n$ is even.

Keywords:

Laplacian Matrix, p-spectral radius

Language:

English

Publication time:

2/2017