Reciprocal complementary distance spectra and reciprocal complementary distance energy of line graphs of regular graphs

Abstract

The reciprocal complementary distance (RCD) matrix of a graph $G$ is defined as $RCD(G) = [rc_{ij}]$ where $rc_{ij} = \frac{1}{1+D-d_{ij}}$ if $i \neq j$ and $rc_{ij} = 0$, 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 $RCD$-energy of $G$ is defined as the sum of the absolute values of the eigenvalues of $RCD(G)$. Two graphs are said to be $RCD$-equienergetic if they have same $RCD$-energy. In this paper we show that the line graph of certain regular graphs has exactly one positive $RCD$-eigenvalue. Further we show that $RCD$-energy of line graph of these regular graphs is solely depends on the order and regularity of $G$. This results enables to construct pairs of $RCD$-equienergetic graphs of same order and having different $RCD$-eigenvalues.