Background: Topological indices have numerous implementations in chemistry, biology and in lot of other areas. It is a real number associated to a graph, which provides information about its physical and chemical properties and their correlations. For a connected graph H, the degree distance defined as DD(H)=∑_(\h_1,h_2⊆V(H))〖(〖deg〗_H (h_1 )+〖deg〗_H (h_2 )) d_H (h_1,h_2 ) 〗, where 〖deg〗_H (h_1 ) is the degree of vertex h_1and d_H (h_1,h_2 ) is the distance between h_1and h_2in 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: By 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 the all trees of diameter d. It is also proved that the matching number and domination numbers are equal for trees having minimum DD index.

In this article, we compute the vertex Padmakar-Ivan (PIv ) index, vertex Szeged (Szv ) index, edge Padmakar-Ivan (PIe ) index, edge Szeged (Sze ) index, weighted vertex Padmakar-Ivan (wPIv ) index, and weighted vertex Szeged (wSzv ) index of a graph product called subdivision vertex-edge join of graphs.

Topological indices are very useful to assume certain physiochemical properties of the chemical compound. A molecular descriptor which changes the molecular structures into certain real numbers is said to be a topological index. In chemical graph theory, to create quantitative structure activity relationships in which properties of molecule may be linked with their chemical structures relies greatly on topological indices. The benzene molecule is a common chemical shape in chemistry, physics, and nanoscience. This molecule could be very beneficial to synthesize fragrant compounds. The circumcoronene collection of benzenoid H m is one family that generates from benzene molecules. The purpose of this study is to calculate the topological indices of the double and strong double graphs of the circumcoronene series of benzenoids H m . In addition, we also present a numerical and graphical comparison of topological indices of the double and strong double graphs of the circumcoronene series of benzenoid H m .