A Study of the Eigenvalues of the Matrix Of Distance Reciprocals in K[r, n-r] And The Cycle

Lee Xu

University of Chinese Academy of Sciences, CAS, Mathematics Department, Beijing, China

Abstract

This paper Deals with the complete bipartite graph K(r, n-r) and the cycle . The matrix of concern is the matrix B which is the (n, n) matrix and whose non zero entries are the reciprocals of the non zero entries of the distance matrix D. A complete characterization of the spectrum of B and a set of n independent eigenvectors of B will be presented. Two special cases will be mentioned, namely the star K(1, n-1) and the graph K(2, n-2). We will also look at the case of infinite graph, i. e if the size n grows big while r stays finite. Finally, some numerical data will be presented. As for the cycle, we present the complete set of eigenvalues of the matrix B.

Keywords: Infinite Graph; Matrix; cycle.