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Eigenvalues of the resistance-distance matrix of complete multipartite graphs
Kinkar Chandra Das1, Yujun Yang2
1Department of Mathematics, Sungkyunkwan University, Suwon, 440-746 Republic of Korea.
Abstract:
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.
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