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

S Pirzada ; Saleem Khan

Filomat, Tome 37 (2023) no. 23, p. 7973

Résumé

The reciprocal distance Laplacian matrix of a connected graph G is defined as RD L (G) = RT(G) − RD(G), where RT(G) is the diagonal matrix of reciprocal distance degrees and RD(G) is the Harary matrix. Since RD L (G) is a real symmetric matrix, we denote its eigenvalues as λ 1 (RD L (G)) ≥ λ 2 (RD L (G)) ≥ · · · ≥ λ n (RD L (G)). The largest eigenvalue λ 1 (RD L (G)) of RD L (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 RD L (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 S n 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 K n , K n − e, the star S n, the complete balanced bipartite graph K n 2, n 2 and the complete split graph CS(n, α) are all determined from the RD L-spectrum.

Détail

DOI : 10.2298/FIL2323973P

Classification : 05C50, 05C12, 15A18
Keywords: Distance matrix: Distance Laplacian matrix: Reciprocal Distance Laplacian matrix: Reciprocal Distance Laplacian eigenvalues, Harary matrix