An-Najah Staff

This work deals with the spectra and eigenspaces of some matrices related to the distance matrix of some connected graphs. In particular, we investigate the (n x n) matrix Bn whose nonzero entries are the reciprocals of the corresponding nonzero entries in the distance matrix. We derive formulas for the eigenvalues and eigenvectors related to the complete bipartite graph K(r, n-r), and the cycle graphs Cn, for any positive integer n. In the beginning, we state some needed facts in graph and matrix theory. In chapter three, we present some known results about the distance matrix and related topics. In chapter four, the discussion was focused on the matrix Bn related to the graphs K(r, n-r), and K(2, n-2). In chapter tive we deal with the matrix Bn related to the cycles Cn for any positive integer n. New Accomplishments In chapter two, we state and prove a theorem for computing the eigenvalues and eigenvectors of circulant matrices, and relate them to the permutation matrices. ln chapter four: We state and prove a theorem for computing the eigenvalues and eigenvectors of a matrix, that appears in our main study. Also, we state for the first time, a theorem for computing the eigenvalues and eigenvectors for the matrix Bn,whose nonzero entries are the reciprocals of the corresponding nonzero entries of the distance matrix of the graph K(r, n-r), and as a special case the graph K(2. n—2). We will construct a table, which contains numerical values of the spectral radius ( 1, and those of some complete bipartite graphs. obtained by direct calculation and by the resulting formulas. Also, we will construct a table which contains numerical values of for K(2,n—2), K(3,n-3), K(4,n-4) as n goes bigger and bigger, then we state and prove a lemma for calculating the limit of as n approaches . In chapter five, we will find the eigenvalues and eigenvectors for Bn related to the cycle graphs Cn, and we state and prove a theorem which shows that the eigenvalues of Bn are real for any positive integer n. We will also calculate the eigenvalues and eigenvectors of` Bn related to the cycle graph Cn, and those of Cn. We will present some graphs, then we will compute the eigenvector that corresponds to the spectral radius, and will note that vertices with greater eigenvector entries are with smaller eccentricities, and tend to be in the center of the graph.