Some extremal problems on Aα-spectral radius of graphs with given size

Nikiforov defined the A alpha-matrix of a graph G as A(alpha)(G) = alpha D(G) + (1 - alpha)A(G), where alpha is an element of [0, 1], D(G) and A(G) are the diagonal matrix of degrees and the adjacency matrix respectively. The largest eigenvalue of A(alpha)(G) is called the A alpha-spectral radius of G, denoted by rho(alpha)(G). In this paper, we first give an upper bound on rho(alpha)(G) of a connected graph G with fixed size m >= 3k and maximum degree triangle <= m - k, where k is a positive integer. For two connected graphs G(1) and G(2 )with size m >= 4, employing this upper bound, we prove that rho(alpha)(G(1)) > rho(alpha)(G(2)) if triangle(G(1)) > triangle(G(2)) and triangle(G(1)) >= (2m)/(3) + 1. As an application, we determine the graph with the maximal A alpha-spectral radius among all graphs with fixed size and girth. Our theorems generalize the recent results for the signless Laplacian spectral radius of a graph.