On the third largest eigenvalue of a graph (Q1587896)

For a graph \(G\) with \(n\) vertices and \(m\) edges, let the \(\lambda_k(G)\) \((k=1,\ldots,n)\) be the eigenvalues of \(G\)'s incidence matrix, ordered nonincreasingly. Several bounds are derived for these eigenvalues in case \(\lambda_3(G)<0\), among which \(\lambda_k(G)\geq -1\) for all \(k=3,\ldots,\lfloor (n+1)/2\rfloor\), \(\lambda_k(G)\leq -1\) for all \(k\geq (n+3)/2\), \(\lambda_n(G)\geq -1-\sqrt{n(n-1)/2-m}\), and when \(G=T\) is a tree not part of two special one-parameter families, \(\lambda_3(T)\geq 2\cos(3\pi/7)\), with equality only for \(T=P_6\).