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

Abstract

The A α -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 α ( G ) = αD ( G ) + ( 1 − α ) A ( G ) with α ∈ [ 0 , 1 ] . The maximum modulus among all A α -eigenvalues is called the A α -spectral radius. In this paper, we order the connected graphs with size m and diameter (at least) d from the second to the ( ⌊ d 2 ⌋ + 1 ) th regarding to the A α -spectral radius for α ∈ [ 1 2 , 1 ) . As by-products, we identify the first ⌊ d 2 ⌋ largest trees of order n and diameter (at least) d in terms of their A α -spectral radii, and characterize the unique graph with at least one cycle having the largest A α -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.