On a theorem of Nosal

TL;DR

This paper extends Nosal's theorem by showing that a spectral radius condition implies a lower bound on the number of triangles sharing an edge, confirming a conjecture and providing a simpler proof for related eigenvalue inequalities.

Contribution

It proves a new bound on the number of triangles sharing an edge based on spectral radius, settling a conjecture and simplifying existing eigenvalue inequality proofs.

Findings

Spectral radius condition implies a lower bound on shared triangles.

Confirmed a conjecture relating spectral radius and triangle structure.

Provided a simpler proof for eigenvalue inequalities in triangle-free graphs.

Abstract

Let $G$ be a graph with $m$ edges and spectral radius $\lambda_{1}$. Let $bk\left( G\right) $ stand for the maximal number of triangles with a common edge in $G$. In 1970 Nosal proved that if $\lambda_{1}^{2}>m,$ then $G$ contains a triangle. In this paper we show that the same premise implies that \[ bk\left( G\right) >\frac{1}{12}\sqrt[4]{m}. \] This result settles a conjecture of Zhai, Lin, and Shu. Write $\lambda_{2}$ for the second largest eigenvalue of $G$. Recently, Lin, Ning, and Wu showed that if $G$ is a triangle-free graph of order at least three, then \[ \lambda_{1}^{2}+\lambda_{2}^{2}\leq m, \] thereby settling the simplest case of a conjecture of Bollob\'{a}s and the author. We give a simpler proof of their result.