Extreme eigenvalues of nonregular graphs (Q875951)

This paper deals with the greatest eigenvalue of a nonregular graph. Let \(G\) be a connected graph with vertex set \(\{1,2,\dots,n\}\), maximum degree \(\Delta\) and minimum degree \(\delta\). Let \(\lambda_1\) be the largest eigenvalue of the adjacency matrix \(A\) of \(G\). The authors obtain a lower bound of \(\Delta - \lambda_1\). Specifically, they prove for a connected nonregular graph with diameter \(D\), that \[ \Delta-\lambda_1 > \frac{n\Delta-2m}{n(D(n\Delta-2m)+1)}, \] where \(m\) is the number of edges; and hence \[ \Delta-\lambda_1 > \frac{n\Delta-2m}{n(D(n\Delta-2m)+1)}\geq \frac{1}{n(D+1)}. \]