Bounds of Eigenvalues of a Nontrivial Bipartite Graph

Abstract

Let $G$ be a simple graph with $n$ vertices and $m$ edges, and let ${\lambda }_1$ and ${\lambda }_2$ denote the largest and second largest eigenvalues of $G$. For a nontrivial bipartite graph $G$, we prove that:
(i) ${\lambda }_1\le \sqrt{m–\frac{3-\sqrt{5}}{2}}$, where equality holds if and only if $G\cong P_4$;
(ii) If $G≆P_n$, then ${\lambda }_1\le \sqrt{m–(\frac{5-\sqrt{17}}{2})}$, where equality holds if and only if $G\cong K_{3,3}–e$;
(iii) If $G$ is connected, then ${\lambda }_2\le \sqrt{m–4{\cos }^2(\frac{\pi }{n+1})}$, where equality holds if and only if $G\cong P_{n,2}\le n\le 5$;
(iv) ${\lambda }_2\ge \frac{\sqrt{5}-1}{2}$, where equality holds if and only if $G\cong P_4$;
(v) If $G$ is connected and $G≆P_n$, then ${\lambda }_2\ge \frac{5-\sqrt{17}}{2}$, where equality holds if and only if $G\cong K_{3,3}–e$.