Combined degree sum energy of graphs

Abstract

The combined degree sum matrix of a graph G is defined by CDS(G)=[c_ij] where c_ij=d_i+d_j+d_id_j whenever i is not equal to j, and zero otherwise, where d_i is the degree of a vertex v_i in G. The combined degree sum polynomial of G is the characteristic polynomial of CDS(G). The combined degree sum energy of G is equal to the sum of the absolute values of the eigen values of CDS(G). The total pi- electron energy of different compounds comprising hetero atoms is connected with their E_CDS(G) values, and we found a good correlation with the correlation coefficient r=0.9823. Combined degree sum polynomial and combined degree sum energy of some graphs are obtained. Further, we establish bounds for the largest CDS eigenvalue and CDS energy of a graphs. Jog et al. gave the expression for degree sum polynomial of semi total line graph T₁(G) of r-regular graph is prone to some error. We give correct expression for it. Finally, we obtain combined degree sum polynomial of graph valued functions and graph operations on regular graphs.