On The Distance Spectrum And Wiener Index Of Graph

As an important branch of graph theory and combination matrix theory, the theory of graph spectra has attracted more and more attention from the researchers. It studies mainly about the relationship between graph structure and graph matrice together with their eigenvalues. It consists of adjacent spectrum, Laplacian spectrum, signless Laplacian spectrum, normalized spectrum, etc. Recently distance spectrum has become popular in spectra theory. In this paper, we obtain the graphs with the least distance spectrum among all the graphs with given independent number, the graphs with the least distance spectrum among all the digraphs with given independent number, the graphs with the least distance spectrum among all the graphs with given chromatic number. And we sum up the general methods to study the distance spectra by comparing the characteristics and conclusions between adjacent spectrum and distance spectrum. Moreover, we discuss a closely-related graph parameter---the Wiener index. The Wiener index is the sum of the distances between all unordered pairs of vertices in the graph, i.e. half the sum of all the elements in the distance matrix. We obtain the graphs with the least Wiener index among all the unicyclic graphs with n vertices and diameter d. Moreover, if4≤d≤n-3, d=0(mod2), then the unicyclic graphs with the second least Wiener index are obtained.