On the Sα-matrix of graphs

Abstract

Let [Formula: see text] and [Formula: see text] be the adjacency matrix and the diagonal matrix of vertex degrees of a graph [Formula: see text], respectively. For any real [Formula: see text], Nikiforov defined the matrix [Formula: see text] as [Formula: see text]. Let [Formula: see text] be complement of [Formula: see text]. Define [Formula: see text] where [Formula: see text] and [Formula: see text] are the identity matrix and the all-ones matrix, respectively. Since [Formula: see text] is the Seidel matrix of [Formula: see text], this new matrix is the generalization of Seidel matrix. Further, [Formula: see text] is a subset of universal adjacency matrices defined by Haemers and Omidi. In this paper, [Formula: see text]-eigenvalues and [Formula: see text]-energy of [Formula: see text] respectively, are investigated, which not only extends Seidel matrix but also enriches the theory of universal adjacency matrices.