On families of graphs which are both adjacency equienergetic and distance equienergetic

Let A ( G ) and D ( G ) be the adjacency and distance matrices of a graph G respectively. The adjacency energy or A -energy E A ( G ) of a graph G is defined as the sum of the absolute values of the eigenvalues of A ( G ) . Analogously, the D -energy E D ( G ) is defined to be the sum of the absolute values of the eigenvalues of D ( G ) . One of the interesting problems on graph energy is to characterize those graphs which are equienergetic with respect to both the adjacency and distance matrices. A weaker problem is to construct the families of graphs which are equienergetic with respect to both the adjacency and distance matrices. In this paper, we find the explicit relations between A -energy and D -energy of certain families of graphs. As a consequence, we provide an answer to the above open problem (Indulal in https://icgc2020.wordpress.com/invitedlectures , 2020; http://www.facweb.iitkgp.ac.in/rkannan/gma.html , 2020)