Labeled Graph Energy and Their Properties

Abstract

The energy of graph was first introduced and defined by Ivan Gutman in 1978 which was a result of his motivation by Erich Huckel’s Huckel Molecular Orbital Theory. The energy of graph Γ is defined as the sum of the absolute values of the eigenvalue of the adjacency matrix of the graph Γ. Throughout the work simple connected undirected graphs are considered. In this work vertex labeled graph energy is introduced. Here a detailed study on the energy, bounds and properties of different classes of graphs are carried out and also it gives the emphasis to find bounds for energy of a class of graphs to another class. The vertex labeled graph energy is defined as the sum of absolute value of eigenvalue of the matrix of the vertex labeled graph VL(Γ). Energy of different class of graphs along with spectra is obtained. The spectrum and energy of vertex labeled complement of distinct graphs VL(Γ) along with their bounds is found and extended it to applications for the complements of some alkanes...