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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]

A 3-connected graph G is minimally 3-connected if it becomes not 3-connected after deleting any edge of G. In this paper, we give an upper bound on the signless Laplacian spectral radius of minimally 3-connected graphs with m edges, and completely characterize the corresponding extremal graph in the case when m is even. As a corollary, we determine the unique graph with the maximal signless Laplacian spectral radius among all Halin graphs with m edges. [ABSTRACT FROM AUTHOR]

In this paper, we study the signless Laplacian spectral Turán type problem in terms of the size, obtain sharp upper bounds on the signless spectral radius of C 3 ( C 4 )-free graphs with given size and minimum degree δ ≥ 2 , and characterize the corresponding extremal graphs completely. [ABSTRACT FROM AUTHOR]

Let G be a simple undirected graph. For real number α ∈ [0, 1], Nikiforov defined the Aα-matrix of G as Aα(G) = αD(G) + (1 − α)A(G), where A(G) and D(G) are the adjacency matrix and the degree diagonal matrix of G respectively. In this paper, we obtain a sharp upper bound on the largest eigenvalue ρα(G) of Aα(G) for α ∈ [1/2, 1). Employing this upper bound, we prove that 'For connected G1 and G2 with n vertices and m edges, if the maximum degree Δ(G1) ≥ 2α(1 − α)(2m − n + 1) + 2α and Δ(G1) > Δ(G2), then ρ α (G 1) > ρ α (G 2) '. Let λα(G) denote the least eigenvalue of Aα(G). For α ∈ (1/2, 1), we prove that 'For two connected G1 and G2, if the minimum degree δ (G 1) ≤ 1 1 − α − 2 and δ(G1) < δ(G2), then λα(G1) < λα(G2)'. [ABSTRACT FROM AUTHOR]

A 2-connected graph G is minimally 2-connected if deleting any arbitrary chosen edge of G always leaves a graph which is not 2-connected. In this paper, we give sharp upper bounds for the Laplacian spectral radius of (minimally) 2-connected graphs with given size, and determine the corresponding extremal graphs. [ABSTRACT FROM AUTHOR]

Let g (G) denote the girth of a graph G. In this paper, we determine the unique graph with the maximum least Q-eigenvalue among all unicyclic graphs of order n = 6k (k ≥ 8) with perfect matchings. For the cases when n = 6k + 2 and n = 6k + 4, we prove that g (G) = 3 if G is a graph with the maximum least Q-eigenvalue, and provide a conjecture and a problem on the sharp upper bound of the least Q-eigenvalue. [ABSTRACT FROM AUTHOR]

Let G be a simple undirected graph. For any real number α ∈ [ 0 , 1 ] , Nikiforov defined the A α -matrix of G as A α (G) = α D (G) + (1 - α) A (G) , where A(G) and D(G) are the adjacency matrix and the degree diagonal matrix of G respectively. The largest eigenvalue of A α (G) is called the A α -spectral radius of G. In this paper, we give sharp upper bounds on the A α -spectral radius of C 4 -free graphs and Halin graphs for α ∈ [ 1 / 2 , 1) respectively. [ABSTRACT FROM AUTHOR]