Distance spectra of graphs: A survey

In 1971, Graham and Pollack established a relationship between the number of negative eigenvalues of the distance matrix and the addressing problem in data communication systems. They also proved that the determinant of the distance matrix of a tree is a function of the number of vertices only. Since then several mathematicians were interested in studying the spectral properties of the distance matrix of a connected graph. Computing the distance characteristic polynomial and its coefficients was the first research subject of interest. Thereafter, the eigenvalues attracted much more attention. In the present paper, we report on the results related to the distance matrix of a graph and its spectral properties.

Distance Spectra of Graphs: A Survey

On distance spectral radius of graphs

In this article, we prove that among trees on n vertices and matching number m, the dumbbell is the unique tree that maximizes the distance spectral radius, which was conjectured by Aleksandar Ilic [Distance spectral radius of trees with given matching number, Discr. Appl. Math. 158 (2010), pp. 1799–1806]. In addition, we find the unique tree that maximizes the distance spectral radius in the class of trees with a given number of pendent vertices.

Guanglong Yu

Papers

Some graft transformations and its applications on the distance spectral radius of a graph