Quasi-Laplacian energy of fractal graphs

Graph energy is a measurement of determining the structural information content of graphs. The first Zagreb index can be handled with its connection to graph energy. In this paper, a novel and significant application of the first Zagreb index to composite graphs based on fractal graphs is revealed, and by the relation between quasi-Laplacian energy and the vertex degrees of a graph, we derive closed-form formulas for the quasi-Laplacian energy of fractal graphs or namely R-graphs, R-vertex and edge join, R-vertex and edge corona, R-vertex and edge neighborhood graphs in terms of the corresponding energy, the first Zagreb indices, number of vertices and edges of the underlying graphs.