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

The D-eigenvalues of a graph G are the eigenvalues of its distance matrix D, and the D-energy ED(G) is the sum of the absolute values of its D-eigenvalues. Two graphs are said to be D-equienergetic if they have the same D-energy. In this note we obtain bounds for the distance spectral radius and D-energy of graphs of diameter 2. Pairs of equiregular D-equienergetic graphs of diameter 2, on p = 3t + 1 vertices are also constructed.

https://dyuthi.cusat.ac.in/xmlui/bitstream/handle/purl/626/7)%20ON%20DISTANCE%20ENERGY%20OF%20GRAPHS.pdf?sequence=1

On Distance Energy of Graphs

Sharp bounds on the distance spectral radius and the distance energy of graphs

If G is a graph on n vertices, and di is the degree of its i-th vertex, then the Randic matrix of G is the square matrix of order n whose (i, j)-entry is equal to 1/ � di dj if the i-th and j-th vertex of G are adjacent, and zero otherwise. This matrix in a natural way occurs within Laplacian spectral theory, and provides the non-trivial part of the so-called normalized Laplacian matrix. In spite of its obvious relation to the famous Randic index, the Randic matrix seems to have not been much studied in mathematical chemistry. In this paper we define the Randic energy as the sum of the absolute values of the eigenvalues of the Randic matrix, and establish some of its properties, in particular lower and upper bounds for it.

http://www.ahmetsinancevik.com/pdfler/A13.pdf

Randic Matrix and Randic Energy

https://draganstevanovic.files.wordpress.com/2012/10/equi-d-energy.pdf

The distance spectrum and energy of the compositions of regular graphs

T heD-eigenvalues of a connected graph G are the eigen- values of its distance matrix D ,a nd form theD-spectrum of G.T he D-energy ED(G) of the graph G is the sum of the absolute values of its D-eigenvalues. Two (connected) graphs are said to be D-equienergetic if they have equal D-energies. The D-spectra of some graphs and their D-energies are calculated. A pair of D-equienergetic bipartite graphs on 24 t, t ≥ 3, vertices is constructed.

/pdf/on-the-distance-spectra-of-some-graphs-d7zlouifff.pdf

On the distance spectra of some graphs

Eigenvalue of a graph is the eigenvalue of its adjacency matrix. The energy E(G) of a graph G is the sum of the absolute values of its eigenvalues. In this paper we obtain analytic expressions for the energy of two classes of regular graphs.

https://dyuthi.cusat.ac.in/xmlui/bitstream/handle/purl/627/A_Note_on_energy_of_some_graphs.pdf?sequence=1

G. Indulal

Papers

On Distance Energy of Graphs

Sharp bounds on the distance spectral radius and the distance energy of graphs

The distance spectrum and energy of the compositions of regular graphs

On the distance spectra of some graphs

A note on energy of some graphs