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

The Aα-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. Aα(G)=αD(G)+(1−α)A(G) with α∈[0,1]. The maximum modulus among all Aα-eigenvalues is called the Aα-spectral radius. In this paper, we order the connected graphs with size m and diameter (at least) d from the second to the (⌊d2⌋+1)th regarding to the Aα-spectral radius for α∈[12,1). As by-products, we identify the first ⌊d2⌋ largest trees of order n and diameter (at least) d in terms of their Aα-spectral radii, and characterize the unique graph with at least one cycle having the largest Aα-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.