The distance Seidel matrix of connected graphs

For a connected graph G, we present the concept of a new graph matrix related to its distance and Seidel matrix, called distance Seidel matrix [Formula: see text]. Suppose that the eigenvalues of [Formula: see text] be [Formula: see text] In this article, we establish a relationship between the distance Seidel eigenvalues of a graph with its distance and adjacency eigenvalues. We characterize all the connected graphs with [Formula: see text] Also, we determine different bounds for the distance Seidel spectral radius and distance Seidel energy. We analyze the effect of edge deletion on the distance Seidel energy of the complete bipartite graph. Moreover, we obtain the distance Seidel spectra of different graph operations such as join, cartesian product, lexicographic product, and unary operations like the double graph and extended double cover graph. We give various families of distance Seidel cospectral and distance Seidel integral graphs as an application.