Ordering of graphs with fixed size and diameter by Aα-spectral radii

The A(alpha)-matrix of a graph G is defined as the convex linear combination of the adjacency matrix A(G) and the diagonal matrix of degrees D(G), i.e. A(alpha)(G)=alpha D(G)+(1-alpha)A(G)=alpha D(G)+(1-alpha)A(G) with alpha is an element of[0,1]. The maximum modulus among all A(alpha)-eigenvalues is called the A(alpha)-spectral radius. In this paper, we order the connected graphs with size m and diameter (at least) d from the second to the (left perpendiculard/2right perpendicular+1)th regarding to the A(alpha)-spectral radius for alpha is an element of[1/2,1). As by-products, we identify the first left perpendiculard/2 right perpendicular largest trees of order n and diameter (at least) d in terms of their A(alpha)-spectral radii, and characterize the unique graph with at least one cycle having the largest A(alpha)-spectral radius among graphs of size m and diameter (at least) d. Consequently, the corresponding results for signless Laplacian matrix can be deduced as well.