Eigenvalues of the resistance-distance matrix of complete multipartite graphs

Let G= (V, E) be a simple graph. The resistance distance between i, j∈ V, denoted by r_{i j}, is defined as the net effective resistance between nodes i and j in the corresponding electrical network constructed from G by replacing each edge of G with a resistor of 1 Ohm. The resistance-distance matrix of G, denoted by R(G) , is a | V| × | V| matrix whose diagonal entries are 0 and for i≠ j, whose ij-entry is r_{i j}. In this paper, we determine the eigenvalues of the resistance-distance matrix of complete multipartite graphs. Also, we give some lower and upper bounds on the largest eigenvalue of the resistance-distance matrix of complete multipartite graphs. Moreover, we obtain a lower bound on the second largest eigenvalue of the resistance-distance matrix of complete multipartite graphs.