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) \(\lambda_1 \leq \sqrt{m – \frac{3-\sqrt{5}}{2}}\), where equality holds if and only if \(G \cong P_4\);
(ii) If \(G \ncong P_n\), then \(\lambda_1 \leq \sqrt{{m} – (\frac{5-\sqrt{17}}{2})}\), where equality holds if and only if \(G \cong K_{3,3} – e\);
(iii) If \(G\) is connected, then \(\lambda_2 \leq \sqrt{{m} – 4{\cos}^2(\frac{\pi}{n+1})}\), where equality holds if and only if \(G \cong P_{n,2} \leq n \leq 5\);
(iv) \(\lambda_2 \geq \frac{\sqrt{5}-1}{2}\), where equality holds if and only if \(G \cong P_4\);
(v) If \(G\) is connected and \(G \ncong P_n\), then \(\lambda_2 \geq \frac{5-\sqrt{17}}{2}\), where equality holds if and only if \(G \cong K_{3,3} – e\).