Inverse Problem for Leap Zagreb Indices

Abstract

The structure of a chemical compound is usually modeled as a graph, which is so-called a molecular graph. It has been found that some topological indices of a molecular graph are closely related to many physicochemical properties of its chemical compounds. From this relation, it arises the important inverse topological indices problem, that carry out a thorough search of the existence of a graph having its index value equal to a given integer. In this paper, we are interested in solving this problem for the first, second and third leap Zagreb indices of connected graphs. We are also restricting the solutions to trees and unicyclic graphs. It is shown that for every even non-negative integer $k$ there exists a graph having its first leap Zagreb index value equal to $k$. For every non-negative integer $k$, except $2$, there exists a graph having its second leap Zagreb index value equal to $k$ and for every non-negative integer $k$, except $1,3,5,7,9,11,17$, there exists a graph having its third leap Zagreb index value equal to $k$. The general formulas of leap Zagreb indices values for some certain trees and unicyclic graphs which are useful in this work are presented.