Some energy of a line graph of prime graph

From a mathematical perspective, the energy of a graph is described as the total absolute values of the eigenvalues of the adjacency matrix of the graph. It was commonly related to the theory of spectral graphs. The energy of the graph is initially characterized by Ivan Gutman in 1978. However, an exploration of energy was inspired by Hukel’s work in the 1930s when accustomed to calculating the total amount of π-electron energy of molecules. Since then, several mathematicians have become concern about the energy of the graph and a few modifications have been proposed. Previously, the line graph of prime graph, L(PG(R)) has been introduced. In a prime graph, PG(R) either two different vertices x, y ∈ R are adjacent in the case that yRx = 0 or xRy = 0. Meanwhile, the line graph of prime graph, L(PG(R)) was defined as a graph whose vertex set is made up of prime graph edges, where the two vertices are adjacent if their respective edges in the prime graph share the same vertex. The goal of this paper is to determine the energy of L(PG(Zn)) for n = 3, 4, 5 by using the eigenvalues of the graph. The eigenvalues are obtained by mapping the elements onto their adjacency matrix. Finally, the energy of L(PG(Zn)) for n = 3, 4, 5 is computed. It is found that the energy of L(PG(Zn)) is equal to 2n − 4.