Bollob\'as-Nikiforov Conjecture for graphs with not so many triangles

TL;DR

This paper verifies a generalized eigenvalue inequality conjecture for graphs with relatively few triangles, including planar, book-free, and cycle-free graphs, extending previous results to broader graph families.

Contribution

It proves the conjecture for graphs with at most O(m^{1.5- ext{epsilon}}) triangles, broadening the classes of graphs where the conjecture holds.

Findings

The conjecture holds for graphs with O(m^{1.5- ext{epsilon}}) triangles.

It confirms the conjecture for planar, book-free, and cycle-free graphs.

The result extends previous verifications to graphs with sparse triangle counts.

Abstract

Bollob\'as and Nikiforov conjectured that for any graph $G \neq K_n$ with $m$ edges \[ \lambda_1^2+\lambda_2^2\le \bigg( 1-\frac{1}{\omega(G)}\bigg)2m\] where $\lambda_1$ and $\lambda_2$ denote the two largest eigenvalues of the adjacency matrix $A(G)$, and $\omega$ denotes the clique number of $G$. This conjecture was recently verified for triangle-free graphs by Lin, Ning and Wu and for regular graphs by Zhang. Elphick, Wocjan and Linz proposed a generalization of this conjecture. In this note, we verify this generalized conjecture for the family of graphs on $m$ edges, which contain at most $O(m^{1.5-\varepsilon})$ triangles for some $\varepsilon > 0$. In particular, we show that the conjecture is true for planar graphs, book-free graphs and cycle-free graphs.