Maxima of the $Q$-index of non-bipartite $C_{3}$-free graphs

TL;DR

This paper extends Mantel's theorem by exploring the maximum $Q$-index in non-bipartite, triangle-free graphs, providing spectral bounds and new insights into graph spectral extremal problems.

Contribution

It introduces $Q$-spectral bounds for non-bipartite, triangle-free graphs, advancing spectral extremal graph theory beyond previous results.

Findings

Established upper bounds for the $Q$-index in non-bipartite $C_3$-free graphs.

Connected spectral bounds to classical extremal graph theory results.

Provided new characterizations of extremal graphs based on the $Q$-index.

Abstract

A classic result in extremal graph theory, known as Mantel's theorem, states that every non-bipartite graph of order $n$ with size $m>\lfloor \frac{n^{2}}{4}\rfloor$ contains a triangle. Lin, Ning and Wu [Comb. Probab. Comput. 30 (2021) 258-270] proved a spectral version of Mantel's theorem for given order $n.$ Zhai and Shu [Discrete Math. 345 (2022) 112630] investigated a spectral version for fixed size $m.$ In this paper, we prove $Q$-spectral versions of Mantel's theorem.