A note on spectral radius and degree deviation in graphs
The study of spectral radius of irregular graphs is an interesting topic. Let \(G\) be a graph on \(n\) vertices with \(m\) edges, and \(d_1, d_2, \ldots, d_n\) be the degrees of the vertices in \(G\). Set \[ s(G) = \sum_{1\leq i\leq n}\left|d_i - \frac{2m}{n}\right|, \] which is called the degree deviation. Let \(\lambda_1(G)\) denote the largest eigenvalue of the adjacency matrix of \(G\). \textit{V. Nikiforov} [Linear Algebra Appl. 414, No. 1, 347--360 (2006; Zbl 1092.05046)] proved that \[ \frac{s^2(G)}{2n^2\sqrt{2m}}\leq \lambda_1(G)- \frac{2m}{n} \le \sqrt{s(G)}, \] and conjectured that \[ \lambda_1(G)- \frac{2m}{n} \le \sqrt{\frac{1}{2}s(G)}, \] if \(n\) and \(m\) are large enough. In this paper, the author prove that \[ \lambda_1(G)- \frac{2m}{n} \le \sqrt{\frac{9}{10}s(G)}, \] which improves the result in [loc. cit.]. Moreover, the author confirm that the above conjecture is valid if \(G\) is bipartite.
scientific article; zbMATH DE number 7358427
scientific article