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.
Graphical Abstract
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