On the distribution of eigenvalues of the reciprocal distance Laplacian matrix of graphs

Abstract

The reciprocal distance Laplacian matrix of a connected graph G is defined as RDL(G) = RT(G) ? RD(G), where RT(G) is the diagonal matrix of reciprocal distance degrees and RD(G) is the Harary matrix. Since RDL(G) is a real symmetric matrix, we denote its eigenvalues as ?1(RDL(G)) ? ?2(RDL(G)) ?...? ?n(RDL(G)). The largest eigenvalue ?1(RDL(G)) of RDL(G) is called the reciprocal distance Laplacian spectral radius. In this article, we prove that the multiplicity of n as a reciprocal distance Laplacian eigenvalue of RDL(G) is exactly one less than the number of components in the complement graph G of G. We show that the class of the complete bipartite graphs maximize the reciprocal distance Laplacian spectral radius among all the bipartite graphs with n vertices. Also, we show that the star graph Sn is the unique graph having the maximum reciprocal distance Laplacian spectral radius in the class of trees with n vertices. We determine the reciprocal distance Laplacian spectrum of several well known graphs. We prove that the complete graph Kn, Kn ? e, the star Sn, the complete balanced bipartite graph Kn/2 ,n/2 and the complete split graph CS(n, ?) are all determined from the RDL-spectrum.