Universal Distance Spectra of Join of Graphs

Let G be a simple undirected graph of order n. In this paper, we introduce a new distance matrix called the universal distance matrix of G, denoted as UD (G) and it is defined as U-D (G) = alpha Tr (G) + beta D (G) + gamma J + delta I, where Tr (G) is the diagonal matrix whose elements are the vertex transmissions, and D (G) is the distance matrix of G. Here J is the all -ones matrix, and I is the identity matrix and alpha, beta, gamma, delta is an element of R and beta not equal 0. This unified definition enables us to derive the spectra of different matrices associated with the distance matrix of graphs. The set of eigenvalues of the universal distance matrix namely, {rho(1), rho(2), ... , rho(n)} is known as the universal distance spectrum of G. As a consequence, by taking appropriate values for alpha, beta, gamma, delta is an element of R and beta not equal; 0, we obtain the eigenvalues of distance matrix, distance Laplacian matrix, distance signless Laplacian matrix, generalized distance matrix, distance Seidal matrix and distance matrices of graph complements. In this paper, we obtain the universal distance spectra of regular graph, join of two regular graphs, joined union of three regular graphs, generalized joined union of n disjoint graphs with one arbitrary graph H using the Schur complement of a block matrix.