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On a theorem of Nosal (2104.12171v1)
Published 25 Apr 2021 in math.CO
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.