The eigenvalues of the matrix of distance recirocals in the graphs K(r, n-r)‎ and Cn

الملخص EN

-complete bipartite graph K(r, n-r) and the cycle Cn.

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.