Eigenvalues of the resistance-distance matrix of complete multipartite graphs - Kinkar Chandra Das; Yujun Yang (Journal of Inequalities and Applications, 2017)
Abstract Let G = ( V , E ) $G=(V, E)$ be a simple graph. The resistance distance between i , j ∈ V $i,j\in V$ , denoted by r i j $r_{ij}$ , 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 ) $R(G)$ , is a | V | × | V | $\vert V \vert × \vert V \vert $ matrix whose diagonal entries are 0 and for i ≠ j $i≠ j$ , whose ij-entry is r i j $r_{ij}$ . 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.