A tensor's spectral bound on the clique number

TL;DR

This paper extends spectral graph theory to hypergraph tensors, providing new bounds on clique numbers and a spectral stability theorem, enhancing understanding of graph structure through tensor analysis.

Contribution

It introduces a spectral bound for the clique number using the clique tensor's spectral radius and extends the Erdos-Simonovits stability theorem to this tensor framework.

Findings

The spectral bound improves upon Nikiforov's bound in certain graph classes.

A spectral version of the Erdos-Simonovits stability theorem is established.

The bounds offer tighter estimates for clique numbers in specific cases.

Abstract

In this paper, we study the spectral radius of the clique tensor A(G) associated with a graph G. This tensor is a higher-order extensions of the adjacency matrix of G. A lower bound of the clique number is given via the spectral radius of A(G). It is an extension of Nikiforov's spectral bound and tighter than the bound of Nikiforov in some classes of graphs. Furthermore, we obtain a spectral version of the Erdos-Simonovits stability theorem for clique tensors based on this bound.