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A 2-edge-connected graph G is minimally 2-edge-connected if deleting arbitrary edge of G always leaves a graph which is not 2-edge-connected. In this paper, we completely characterize the (minimally) 2-edge-connected graphs with given size having the maximal signless Laplacian (Laplacian) index. [ABSTRACT FROM AUTHOR]

The A α -matrix of a graph G was defined by Nikiforov in 2017 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 A α -index of G. In this paper, we completely determine the extremal graphs with maximal A α -index among all graphs with size m , domination number γ and no isolated vertices for α ∈ [ 1 2 , 1). [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]

A c-cyclic graph is a connected graph with n vertices and n + c−1 edges. In this paper, we consider the problem that: in the class of graphs, each of which is the complement graph of a c-cyclic graph with n vertices, which graph has the largest A α -spectral radius. We shall show that, for 0 ≤ c ≤ n − 4 , the extremal graph must be a graph with an isolated vertex. This implies that the maximum degree of G is larger, then the A α -spectral radius of its complement graph is also larger. We also determine the structure of the extremal graph with largest A α -spectral radius among the class of graphs, each of which is the complement graph of a c-cyclic graph with n vertices and maximum degree n−1 for 0 ≤ c ≤ n − 3 and n−2 for 0 ≤ c ≤ n − 4 , respectively. [ABSTRACT FROM AUTHOR]