On the complementary equienergetic graphs.

Abstract

Energy of a simple graph G, denoted by E(G), is the sum of the absolute values of the eigenvalues of G. Two graphs with the same order and energy are called equienergetic graphs. A graph G with the property G cong overlineG is called self-complementary graph, where overlineG denotes the complement of G. Two non-self-complementary equienergetic graphs G_1 and G_2 satisfying the property G_1 cong overlineG_2 are called complementary equienergetic graphs. Recently, Ramaneet al.[Graphs equienergetic with their complements, MATCH Commun. Math. Comput. Chem. 82 (2019) 471-480] initiated the study of the complementary equienergetic regular graphs and they asked to study the complementary equienergetic non-regular graphs. In this paper, by developing some computer codes and by making use of some software like Nauty, Maple and GraphTea, all the complementary equienergetic graphs with at most 10 vertices as well as all the members of the graph class Omega = G: E(L(G)) = E( overlineL(G) ), the order of G is at most 10 are determined, where L(G) denotes the line graph of G. In the cases where we could not find the closed forms of the eigenvalues and energies of the obtained graphs, we verify thegraph energies using a high precision computing (2000 decimal places) of Maple. Aresult about a pair of complementary equienergetic graphs is also given at the end of this paper.