Eigenvalues of the resistance-distance matrix of complete multipartite graphs

Let [Formula: see text] be a simple graph. The resistance distance between [Formula: see text] , denoted by [Formula: see text] , 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 [Formula: see text] , is a [Formula: see text] matrix whose diagonal entries are 0 and for [Formula: see text] , whose ij-entry is [Formula: see text] . 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.