A NOTE ON THE LARGEST EIGENVALUE OF NON-REGULAR GRAPHS (vol 17, pg 54, 2008)

Let lambda(1)(G) be the largest eigenvalue of the adjacency matrix of graph G with n vertices and maximum degree Delta. Recently, Delta - lambda(1)(G) > Delta+1/n(3n+Delta-8) for a non-regular connected graph G was obtained in [B. L. Liu and G. Li, A note on the largest eigenvalue of non-regular graphs, Electron J. Linear Algebra, 17:54-61, 2008]. But unfortunately, a mistake was found in the cited preprint [T. Buyikoglu and J. Leydold, Largest eigenvalues of degree sequences], which led to an incorrect proof of the main result of [B. L. Liu and G. Li]. This paper presents a correct proof of the main result in [B. L. Liu and G. Li], which avoids the incorrect theorem in [T. Biiyiko. glu and J. Leydold].