On Extremal Graphs of Degree Distance Index by Using Edge-Grafting Transformations Method

Background: Topological indices have numerous implementations in chemistry, biology and a lot of other areas. It is a real number associated with a graph, which provides information about its physical and chemical properties and their correlations. For a connected graph H, the degree distance DD index is defined as DD H = {h₁,h₂}v(H) deg_H h₁ + deg_H h₂ d_H h₁,h₂, where deg_H h₁ is the degree of vertex h₁and d_H h₁,h₂ is the distance between h₁and h₂in the graph H. Aim and Objective: In this article, we characterize some extremal trees with respect to degree distance index which has a lot of applications in theoretical and computational chemistry. Materials and Methods: A novel method of edge-grafting transformations is used. We discuss the behavior of DD index under four edge-grafting transformations Results: With the help of those transformations, we derive some extremal trees under certain parameters, including pendant vertices, diameter, matching and domination numbers. Some extremal trees for this graph invariant are also characterized. Conclusion: It is shown that balanced spider approaches to the smallest DD index among trees having given fixed leaves. The tree C_n,d has the smallest DD index, among all trees of diameter d. It is also proved that the matching number and domination numbers are equal for trees having a minimum DD index.