On the Sα-matrix of graphs.

Let A (G) and D (G) be the adjacency matrix and the diagonal matrix of vertex degrees of a graph G , respectively. For any real α ∈ [ 0 , 1 ] , Nikiforov defined the matrix A α (G) as A α (G) = α D (G) + (1 − α) A (G). Let G ¯ be complement of G. Define S α (G) = A α (G ¯) − A α (G) = (1 − α) J + (α n − 1) I − 2 A α (G) , α ∈ [ 0 , 1 ] , where I and J are the identity matrix and the all-ones matrix, respectively. Since S 0 (G) = A (G ¯) − A (G) = J − I − 2 A (G) is the Seidel matrix of G , this new matrix is the generalization of Seidel matrix. Further, S α (G) = (1 − α) J + (α n − 1) I − 2 α D (G) − 2 (1 − α) A (G) is a subset of universal adjacency matrices defined by Haemers and Omidi. In this paper, S α -eigenvalues and S α -energy of G , respectively, are investigated, which not only extends Seidel matrix but also enriches the theory of universal adjacency matrices. [ABSTRACT FROM AUTHOR]