Using Huckel molecular orbital (HMO) theory of total p-electron energy, I. Gutman conceived the idea of energy of a graph with the help of adjacency matrix. Adjacent atoms may or may not have same degree. Hence, we are thinking that there could be some physical/ chemical properties between the adjacent atoms of the molecules having same or different degrees. This is the motivation for defining the new type of energy of a graph called “vertex degree energy”,  of a graph which is the generalization of the ordinary energy of a graph. In this paper, we study only the mathematical properties of vertex degree energy of a graph and its physical/chemical properties are yet to be investigated. Here, we have established upper and lower bounds for  and also computed its value for some standard graphs like star graph, complete graph, crown graph and cocktail party graphs, etc.

vertex degree set, vertex degree matrix, vertex degree energy.