TY - JOUR
T1 - On Extremal Graphs of Degree Distance Index by Using Edge-Grafting Transformations Method
AU - Imran, Muhammad
AU - Akhtar, Shehnaz
AU - Ahmad, Uzma
AU - Ahmad, Sarfraz
AU - Bilal, Ahsan
N1 - Funding Information:
This research is supported by the UPAR Grants of United Arab Emirates University, Al Ain, UAE via Grant Nos. G00002590 and G00003271.
Publisher Copyright:
© 2022 Bentham Science Publishers.
PY - 2022/3
Y1 - 2022/3
N2 - 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) degH h1 + degH h2 dH h1,h2, where degH h1 is the degree of vertex h1and dH 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 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.
AB - 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) degH h1 + degH h2 dH h1,h2, where degH h1 is the degree of vertex h1and dH 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 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.
KW - Degree distance index
KW - Edge
KW - Extremal graphs
KW - Topological indices
KW - Tree
KW - Vertex
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U2 - 10.2174/1386207323666201224123643
DO - 10.2174/1386207323666201224123643
M3 - Article
C2 - 33357180
AN - SCOPUS:85123647434
SN - 1386-2073
VL - 25
SP - 560
EP - 567
JO - Combinatorial Chemistry and High Throughput Screening
JF - Combinatorial Chemistry and High Throughput Screening
IS - 3
ER -