COMPLEMENTARY DISTANCE SPECTRA AND COMPLEMENTARY DISTANCE ENERGY OF LINE GRAPHS OF REGULAR GRAPHS

Abstract

The complementary distance (CD) matrix of a graph $G$ is defined as $CD(G) = [c_{ij}]$, where $c_{ij} = 1+D-d_{ij}$ if $i \neq j$ and $c_{ij} = 0$, otherwise, where $D$ is the diameter of $G$ and $d_{ij}$ is the distance between the vertices $v_i$ and $v_j$ in $G$. The $CD$-energy of $G$ is defined as the sum of the absolute values of the eigenvalues of $CD$-matrix. Two graphs are said to be $CD$-equienergetic if they have same $CD$-energy. In this paper we show that the complement of the line graph of certain regular graphs has exactly one positive $CD$-eigenvalue. Further we obtain the $CD$-energy of line graphs of certain regualr graphs and thus constructs pairs of $CD$-equienergetic graphs of same order and having different $CD$-eigenvalues.DOI : http://dx.doi.org/10.22342/jims.22.1.205.27-35