On a generalization of the spectral Mantel's theorem

TL;DR

This paper extends Mantel's theorem using the clique tensor, providing a spectral bound on the number of cliques in a graph and linking it to classical extremal graph theory results.

Contribution

It introduces the clique tensor and generalizes the spectral Mantel's theorem, offering new bounds on clique counts based on spectral properties.

Findings

Derived a sharp upper bound on clique numbers via the clique tensor's spectral radius.

Extended spectral Mantel's theorem to broader graph classes.

Connected new spectral bounds with classical extremal graph results.

Abstract

Mantel's theorem is a classical result in extremal graph theory which implies that the maximum number of edges of a triangle-free graph of order $n$. In 1970, E. Nosal obtained a spectral version of Mantel's theorem which gave the maximum spectral radius of a triangle-free graph of order $n$. In this paper, the clique tensor of a graph $G$ is proposed and the spectral Mantel's theorem is extended via the clique tensor. Furthermore, a sharp upper bound of the number of cliques in $G$ via the spectral radius of the clique tensor is obtained. And we show that the results of this paper implies that a result of Erd\H{o}s [Magyar Tud. Akad. Mat. Kutat\'{o} Int. K\"{o}zl. 7 (1962)] under certain conditions.