Login Register Cart 0
Generic placeholder image

Combinatorial Chemistry & High Throughput Screening

Editor-in-Chief

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

Back
Journal Home About Journal Editorial Board Journal Insight Current Issue Volumes/Issues
Subscribe
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.

« Previous Next »
Graphical Abstract

[1]
Balasubramanian, K.; Gupta, S.P. Quantum molecular dynamics, topological, group theoretical and graph theoretical studies of protein-protein interactions. Curr. Top. Med. Chem., 2019, 19(6), 426-443.
[http://dx.doi.org/10.2174/1568026619666190304152704] [PMID: 30836919]
[2]
Balasubramanian, K. Mathematical and computational techniques for drug discovery: Promises and developments. Curr. Top. Med. Chem., 2018, 18(32), 2774-2799.
[http://dx.doi.org/10.2174/1568026619666190208164005] [PMID: 30747069]
[3]
Balasubramanian, K. Integration of graph theory and quantum chemistry for structure-activity relationships. SAR QSAR Environ. Res., 1994, 2(1-2), 59-77.
[http://dx.doi.org/10.1080/10629369408028840] [PMID: 8790640]
[4]
Balaban, A.T. Topological and stereochemical molecular descriptors for databases useful in QSAR, similarity/dissimilarity and drug design. SAR QSAR Environ. Res., 1998, 8, 1-21.
[http://dx.doi.org/10.1080/10629369808033259]
[5]
Ali, A.; Raza, Z.; Bhatti, A.A. Extremal pentagonal chains with respect to degree-based topological indices. Can. J. Chem., 2016, 94(10), 870-876.
[http://dx.doi.org/10.1139/cjc-2016-0308]
[6]
Ali, A.; Raza, Z.; Bhatti, A.A. Bond incident degree (BID) indices of polyomino chains: A unified approach. Appl. Math. Comput., 2016, 287(9), 28-37.
[http://dx.doi.org/10.1016/j.amc.2016.04.012]
[7]
Akhter, S. Two degree distance based topological indices of trees. IEEE Access, 2019, 7, 95653-95658.
[http://dx.doi.org/10.1109/ACCESS.2019.2927091]
[8]
Akhter, S.; Imran, M.; Raza, Z. Bounds for the general sum-connectivity index of composite graphs. J. Inequal. Appl., 2017, 2017(1), 76.
[http://dx.doi.org/10.1186/s13660-017-1350-y] [PMID: 28469353]
[9]
Akhter, S.; Imran, M.; Gao, W.; Frahani, M.R. On topological indices of honeycomb networks and Graphene networks. Hacet. J. Math. Stat., 2018, 47(1), 1-17.
[10]
Akhter, S.; Farooq, R. The eccentric adjacency index of unicyclic graphs and trees. Asian Eur. J. Math., 2020, 13(1)
[11]
Akhter, S.; Farooq, R. Eccentric adjacency index of graphs with a given number of cut edges. Bull. Malays. Math. Sci. Soc., 2020, 43, 2509-2522.
[http://dx.doi.org/10.1007/s40840-019-00820-x]
[12]
Chen, S.; Liu, W. Extremal modified Schultz index of bicyclic graphs. MATCH Commun. Math. Comput. Chem., 2010, 64(3), 767-782.
[13]
Chen, S.; Liu, W. The modified Schultz index of trees. Int. J. Contemp. Math. Sciences, 2010, 5(24), 1183-1186.
[14]
Imran, M.; Akhter, S.; Iqbal, Z. Edge Mostar index of chemical structures and nanostructures using graph operations. Int. J. Quantum Chem., 2020, 120e26259
[http://dx.doi.org/10.1002/qua.26259]
[15]
Imran, M.; Akhter, S.; Iqbal, Z. On the eccentric connectivity polynomial of F-sum of connected graphs. Complexity, 2020.20205061682
[http://dx.doi.org/10.1155/2020/5061682]
[16]
Liu, J.B.; Wang, C.; Wang, S.; Wei, B. Zagreb indices and multiplicative Zagreb indices of Eulerian graphs. Bull. Malays. Math. Soc., 2019, 42(1), 67-78.
[http://dx.doi.org/10.1007/s40840-017-0463-2]
[17]
Liu, J.B.; Zhao, J.; Zhu, Z.X. On the number of spanning trees and normalized Laplacian of linear octagonal quadrilateral networks. Int. J. Quantum Chem., 2019, 119, 25971.
[http://dx.doi.org/10.1002/qua.25971]
[18]
Liu, J.B.; Zhao, J.; Cai, Z.Q. On the generalized adjacency, Laplacian and signless Laplacian spectra of the weighted edge corona networks. Physica A, 2020.540123073
[http://dx.doi.org/10.1016/j.physa.2019.123073]
[19]
Liu, J.B.; Zhao, J.; He, H.; Shao, Z. Valency-based topological descriptors and structural property of the generalized sierpinski networks. J. Stat. Phys., 2019, 177, 1131-1147.
[http://dx.doi.org/10.1007/s10955-019-02412-2]
[20]
Liu, J.B.; Shi, Z.Y.; Pan, Y.H.; Cao, J.; Abdel-Aty, M.; Al-Juboori, U. Computing the Laplacian spectrum of linear octagonal-quadrilateral networks and its applications. Polycycl. Aromat. Compd., 2020.
[http://dx.doi.org/10.1080/10406638.2020.1748666]
[21]
Raza, Z.; Ali, A. Bounds on the Zagreb Indices for Molecular (n,m)-Graphs. Int. J. Quantum Chem., 2020.
[http://dx.doi.org/10.1002/qua.26333]
[22]
Wiener, H. Structural determination of paraffin boiling points. J. Am. Chem. Soc., 1947, 69(1), 17-20.
[http://dx.doi.org/10.1021/ja01193a005] [PMID: 20291038]
[23]
Ali, P.; Mukwembi, S.; Munirya, S. Degree distance and vertex-connectivity. Discrete Appl. Math., 2013, 161(18), 2802-2811.
[http://dx.doi.org/10.1016/j.dam.2013.06.033]
[24]
Ali, P.; Mukwembi, S.; Munirya, S. Degree distance and edge-conectivity. Australas. J. Combin, 2014, 60(1), 50-68.
[25]
Das, K.C.; Su, G.; Xiong, L. Relation between degree distance and Gutman index of graphs. MATCH Commun. Math. Comput. Chem., 2016, 76, 221-232.
[26]
Feng, L.; Liu, W.; Ilić, A.; Yu, G. Degree distance of unicyclic graphs with given matching number. Graphs Comb., 2013, 29(3), 449-462.
[http://dx.doi.org/10.1007/s00373-012-1143-5]
[27]
Li, S.; Song, Y.; Zhang, H. On the degree distance of unicyclic graphs with given matching number. Graphs Comb., 2015, 31(6), 2261-2274.
[http://dx.doi.org/10.1007/s00373-015-1527-4]
[28]
Mukwembi, S.; Munirya, S. Degree distance and minimum degree. Bull. Aust. Math. Soc., 2013, 87(2), 255-271.
[http://dx.doi.org/10.1017/S0004972712000354]
[29]
Xu, K.; Klavzar, S.; Das, K.C.; Wang, J. Extremal (n,m)-graphs with respect to distance degree based topological indices. MATCH Commun. Math. Comput. Chem., 2014, 72, 865-880.
[30]
Xu, K.; Liu, M.; Das, K.C.; Gutman, I.; Furtula, B. A survey on graphs extremal with respect to distance-based topological indices. MATCH Commun. Math. Comput. Chem, 2014, 71, 461-508.
[31]
Kazemi, R.; Meimondari, L.K. Degree distance and Gutman index of increasing trees. Tran. Combina, 2016, 5(2), 23-31.
[32]
Arockiaraj, M.; Kavitha, S.R.J.; Balasubramanian, K. Vertex cut method for degree and distance-based topological indices and its applications to silicate networks. J. Math. Chem., 2016, 54(8), 1728-1747.
[http://dx.doi.org/10.1007/s10910-016-0646-3]
[33]
Arockiaraj, M.; Shalini, A.J. Extended Cut Method for Edge Wiener, Schultz and Gutman Indices with Applications. MATCH Commun. Math. Comput. Chem., 2016, 76, 233-250.
[34]
Li, S.; Zhang, H. Some extremal properties of the multiplicatively weighted Harary index of a graph. J. Comb. Optim., 2016, 31, 961-978.
[http://dx.doi.org/10.1007/s10878-014-9802-5]
[35]
Hua, H. Wiener and Schultz molecular topological indices of graphs with specified cut edges. MATCH Commun. Math. Comput. Chem., 2009, 61(3), 643-651.
[36]
Li, S.; Meng, X. Four edge-grafting theorems on the reciprocal degree distance of graphs and their applications. J. Comb. Optim., 2015, 30, 468-488.
[http://dx.doi.org/10.1007/s10878-013-9649-1]
[37]
He, C.; Li, S.; Tu, J. Edge-grafting transformations on the average eccentricity of graphs and their applications. Discrete Appl. Math., 2018, 238, 95-105.
[http://dx.doi.org/10.1016/j.dam.2017.11.032]
[38]
Andova, V.; Dimitrov, D.; Fink, J.; Skrekovski, R. Bounds on Gutman index. MATCH Commun. Math.˘. Comput. Chem., 2011, 67(2), 515-524.
[39]
Haynes, T.W.; Hedetniemi, S.; Slater, P. Fundamentals of domination in graphs; Marcel Dekker, Inc.: New York, 1998.

Rights & Permissions Print Cite
Article Metrics
22
1
© 2024 Bentham Science Publishers | Privacy Policy