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Combinatorial Chemistry & High Throughput Screening

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ISSN (Print): 1386-2073
ISSN (Online): 1875-5402

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Research Article

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

Author(s): Muhammad Imran*, Shehnaz Akhtar, Uzma Ahmad, Sarfraz Ahmad and Ahsan Bilal

Volume 25, Issue 3, 2022

Published on: 24 December, 2020

Page: [560 - 567] Pages: 8

DOI: 10.2174/1386207323666201224123643

Price: $65

Abstract

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) = Σ{h1,h2}⊆V(H) [degH(h1)+degH (h2)]dh (h1,h2), where degH (h1)is the degree of vertex h1 and dH (h1,h2) is the distance between h1 and h2 in 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 Cn,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.

Keywords: Topological indices, degree distance index, extremal graphs, tree, vertex, edge.

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Graphical Abstract

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