A Study of Energy on k-Splitting and k-Shadow Graphs

Let G=(V,E) be a graph. The k-splitting graph of G, denoted by S_k (G), is a graph constructed by adding to each vertex v of G as many as k new vertices such that the k new vertices are adjacent to the vertices that are adjacent to v. The k-shadow graph of G, denoted by D_k (G), is a graph constructed by taking k copies of G and connecting each of these vertices with each of its neighboring vertices. The energy of a graph G is defined as the sum of the absolute values of all eigenvalues in the matrix for the graph. In this article, we study the energy of the k-splitting graph and the k-shadow graph, which are the energy of the adjacency matrix, the energy of the maximum degree matrix, the energy of the minimum degree matrix. We also revise the Sombor energy of the k-splitting graph and the k-shadow graph and we compare this result with the results carried out by previous researchers.