Bollobás-Nikiforov Conjecture for graphs with not so many triangles

Hitesh Kumar; Shivaramakrishna Pragada

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arXiv · 2407.19341

Bollobás-Nikiforov Conjecture for graphs with not so many triangles

Hitesh Kumar·Shivaramakrishna Pragada

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Abstract

Bollobás and Nikiforov conjectured that for any graph $G \neq K_n$ with $m$ edges \[ λ_1^2+λ_2^2\le \bigg( 1-\frac{1}{ω(G)}\bigg)2m\] where $λ_1$ and $λ_2$ denote the two largest eigenvalues of the adjacency matrix $A(G)$, and $ω$ 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.

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BibTeX RIS

Hitesh Kumar, Shivaramakrishna Pragada. 2024-07-27. Bollobás-Nikiforov Conjecture for graphs with not so many triangles. https://arxiv.org/abs/2407.19341

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