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

Let $$\mathcal {A}(G)$$ A ( G ) and $$\mathfrak {D}(G)$$ D ( G ) be the adjacency and distance matrices of a graph G respectively. The adjacency energy or $$\mathcal {A}$$ A -energy $$\mathcal {E}_{\mathcal {A}}(G)$$ E A ( G ) of a graph G is defined as the sum of the absolute values of the eigenvalues of $$\mathcal {A}(G)$$ A ( G ) . Analogously, the $$\mathfrak {D}$$ D -energy $$\mathcal {E}_{\mathfrak {D}}(G)$$ E D ( G ) is defined to be the sum of the absolute values of the eigenvalues of $$\mathfrak {D}(G)$$ 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 $$\mathcal {A}$$ A -energy and $$\mathfrak {D}$$ 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)