Abstract
Topological index is a mapping which corresponds underlying graph with a numeric value and invariant up to all the isomorphisms of graph. Our study is based on a partial open question regarding topological indices: for which members of n-vertex graph family, certain index has minimum or maximum value? In this work, we answered the above-mentioned question regarding and for transformed families of graphs and . We investigated the fact of pendent paths and the transformation over these indices and developed the tight upper bounds regarding these families of graphs. Moreover, we characterized transformed graphs associated with maximum values of these indices.
1. Introduction
Nowadays, graph theory has potential applications in different fields of science. It is especially used for theoretical study of chemical compounds in chemistry. This area of study named as chemical graph theory deals with the problems related to the properties in chemistry. In the middle of last century, theoretical study of chemical compounds attracted the researchers due to its effective applications such as prediction of physiochemical properties of substances in cheminformatics, pharmaceutical sciences, materials science, engineering, and so forth [1]. Cheminformatics is comparatively the latest area of information technology which comprises chemistry, mathematics, and other informational sciences that concentrate on gathering, storing, treating, and examining chemical data. There are many theoretical molecular descriptors in literature used to predict properties of chemical compounds. Among these molecular descriptors, topological indices have an impact in chemistry due to the prediction of physiochemical properties of underlying substance. Its role in “quantitative-structure property relationship” (QSPR) or “quantitative-structure activity relationship” (QSAR) investigation models is also remarkable [2, 3].
In 1947, Wiener for the first time introduced the use of topological index during his work on paraffin’s boiling points [4] and provided that it has best correlation with the boiling points of alkanes. The discovery of the Wiener index provided emerging research platform to the research community. In the later years, researchers of different communities proposed many other topological indices and used them for approximation of the chemical properties of their own interest.
In the race for better prediction, Randic [5] in introduced degree-based topological index named Randić connectivity index which was the best predictive invariant in those days. The Randić index was reported as the first degree-based index in QSPR study because Zagreb indices by Gutman and Trinajstic̀ [6] were used for totally different purpose before Randić index. In 1998, parallel to the work of Bollobás and Erdös [7], Estrada et al. [8] defined atom bond connectivity (ABC) index aswhich has a good correlation with the heat of formation of alkanes. Star graph among trees and complete graph in general for fixed number of vertices have maximal value for ABC index [9]. For more details, one can see [10, 11]. Furtula et al. [12] made a generalization of index asby replacing with . The augmented Zagreb index AZI is as,
Its correlation potential reported is even better than that of other indices in [13–15].
Mathematical study of index and [16–35] encouraged us to answer the fundamental question regarding characterization of transformed families of graphs with maximum and minimum values for ABC index and AZI. Gupta et.al. determined bounds for symmetric division degree index in [36]. In this work, we studied index and for transformed graphs and under the fact of transformation , . We characterize extremal graphs of these transformed families of graphs for and established their bounds for and . When a path is attached with the fully connected vertex (vertex with degree greater than one) of the graph, then it has an impact over the increase and decrease of the index under study. Throughout this work, consider graph [37]. It comprises -vertex simple connected graph along with pendent paths of length attached with having degree . Let be the degree sequence of . is shown in Figure 1.
1.1. Graph Transformations
Let , be the new graph generated by removing set edges of , and be the new graph generated by deleting set of vertices . We define following transformations.
Let be the transformation defined over pendent paths attached with the graph [38]. has solid effect over increase and decrease of and .
1.1.1. Transformation A
Let , for and paths pendent at of the form comprise . Then,
The transformation is shown in Figure 2.
1.1.2. Transformation
is the time repetition of transformation .
Let graph with degree of vertex , and be the degree of .
2. Upper Bounds for and
Initially, we proved Proposition 1, which is helpful to prove the main results for .
Proposition 1. LetThen, for and , .
Proof. LetThis implies
Lemma 1 (see [15]). LetThen,(1)is decreasing for.(2)for any real number.(3)For fixed,is increasing andfor.In Theorem 1, we discuss the effect of pendent paths over and determine its upper bound.
Theorem 1. Let graph comprise -vertex graph having edges and pendent vertices. Then,
Equality holds for a complete graph of size with pendent paths of length at each vertex, i.e., .
Proof. Let be the graph formed by number of paths having length pendent at distinct vertices such that . Then,The construction of , implies , and forThe edge set of is partitioned asThe construction of implies that the cardinality of is , , , and . For minimum degree of vertices of and maximum degree , using Proposition 1 and Lemma 1, we haveNow, from equation (12), we getAfter simplification, we getInequality (15) completes the proof.
In Theorem 2, we discussed the effect of successive applications of transformation as shown in Figure 2 over .
Theorem 2. Let graph comprise -vertex simple connected graph . Then,
Equality holds for a complete graph of size with pendent paths of length at each vertex, i.e., .
Proof. Let a simple graph of order , Size having pendent vrtives. The augmented Zagreb index of any graph isThe construction of , implies . After successive applications of transformation as , , the edge set of is partitioned as wherewhich impliesThe construction of implies that the cardinality isand . For minimum degree of vertices of and maximum degree , using Lemma 1, we haveSubstituting these changes in equation (19), we have following inequality.After simplification, we getInequality (23) completes the proof.
3. Upper Bounds for and
Then,(1)is increasing for.(2)for any real number.(3)For fixed,is decreasing for.
Proposition 2 is related to the index.
Proposition 2. LetFor and .
Proof. LetBy Lemma 2, for or and ,Now for or and , let , , and or , ,Since is a symmetric function, one can let or , so the factor along with , , , and . All the factors involved in equation (29) are positive. This impliesHence, for all and ,In Theorem 3, we discuss the effect of pendent paths over index and determine its upper bound.
Theorem 3. Let graph and having order , size , and pendent vertices. Then,
Equality holds for with pendent paths at each vertex, i.e., .
Proof. Let be the graph. isThe construction of , implies , and forWe use edge set partition of defined in Theorem 1:The construction of implies that the cardinality of is , , , and . For minimum degree of vertices of and maximum degree , using Lemma 2 and Proposition 2, we haveNow, from equation (35),After simplification,Inequality (38) completes the proof.
Theorem 4 gives the discussion about the effect of successive applications of transformation as shown in Figure 2 over index.
Theorem 4. Let graph with maximum degree of and minimum . Then,
Equality holds for a complete graph of size with pendent paths of length at each vertex, i.e., .
Proof. Let graph having pendent paths and be the time repetition of transformation . isAfter successive applications of transformation as , , the edge set of is partitioned as whereThe construction of showsThe construction of implies thatand . For minimum degree of vertices of and maximum degree , using Lemma 2 and Proposition 2, we haveSubstituting these changes in equation (42), we have following inequality.After simplification, we getInequality (46) completes the proof.
4. Conclusion
The study of mathematical aspect regarding topological indices is a partially open problem: for which members of graph family, certain index has minimal or maximal value? In this work, we deal with this fundamental question. We considered graphs of family with pendent paths of length and transformed family where transformation is the graph transformation. The concluding key points concerning our study are given as follows. We studied the fact of pendent paths over the increase and decrease of and index in addition to defining upper bounds for these indices and mentioned graphs with maximum values of these indices. We discussed the fact of defined transformation as ; , over and indices. We determined upper bounds and characterized extremal graphs for such bounds.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
All authors contributed equally to this study.
Acknowledgments
The authors deeply acknowledge the Researchers Supporting Program (TUMA-Project-2021-27) Almaarefa University, Riyadh, Saudi Arabia, for supporting steps of this work.