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

## 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: