Some extremal problems on [formula omitted]-spectral radius of graphs with given size.

Nikiforov defined the A α -matrix of a graph G as A α (G) = α D (G) + (1 − α) A (G) , where α ∈ [ 0 , 1 ] , D (G) and A (G) are the diagonal matrix of degrees and the adjacency matrix respectively. The largest eigenvalue of A α (G) is called the A α -spectral radius of G , denoted by ρ α (G). In this paper, we first give an upper bound on ρ α (G) of a connected graph G with fixed size m ≥ 3 k and maximum degree Δ ≤ 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 ρ α (G 1) > ρ α (G 2) if Δ (G 1) > Δ (G 2) and Δ (G 1) ≥ 2 m 3 + 1. As an application, we determine the graph with the maximal A α -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. [ABSTRACT FROM AUTHOR]