Abstract
Graph theory is a dynamic tool for designing and modeling of an interconnection system by a graph. The vertices of such graph are processor nodes and edges are the connections between these processors nodes. The topology of a system decides its best use. Geometric-arithmetic index is one of the most studied graph invariant to characterize the topological aspects of underlying interconnection networks or graphs. Transformation over graph is also an important tool to define new network of their own choice in computer science. In this work, we discuss transformed family of graphs. Let be the connected graph comprises by number of pendent path attached with fully connected vertices of the n-vertex connected graph . Let and be the transformed graphs under the fact of transformations and , , , respectively. In this work, we obtained new inequalities for the graph family and transformed graphs and which involve . The presence of makes the inequalities more general than all those which were previously defined for the index. Furthermore, we characterize extremal graphs which make the inequalities tight.
1. Introduction
The advancement in technology mainly networking, computer, biological, and electrical networks made practicable the accurate data transfer within very small duration. The Internet, social media, biological, ecological, and neural networks are few examples of such networks. Telecommunication is based on interconnection networks which used to share data files. Similarly, data exchange using computing devices is also based on computer network through data linkage, optical fiber cable (OFC), and wireless media such as Wi-Fi. Different algorithms are used for directing, arranging/determining numerical calculations, and image processing. Multiprocessor interconnection networks (MINs) are used to design powerful microprocessors and memory chips [1, 2].
Graph theory provides a fundamental tool for designing and analyzing such networks. Naturally, the interconnection system is modeled by the graph with processor nodes as vertices and links between these nodes as edges of such graph. Graph theory and interconnection networks provide a thorough understanding of these interrelated topics through their topology. The topology of a graph provides information about the manner in which vertices joined in a graph. The topological indices are graph invariants used to study the topology of graphs. Other than computer networks, graph theory is considered as a powerful tool in different areas of research, such as in coding theory, database management system, circuit design, secret sharing schemes, and theoretical chemistry [3]. The topological descriptors of several interconnection networks are already been computed in [4–6]. Along with interconnection networks, these invariants are equally important in chemical graph theory which deals with problems in chemistry using associated graph of chemical compounds [7].
The study of underlying substance using their graph with the help of graph invariants plays an important role in chem-informatics, pharmaceutical sciences, materials science, engineering, and so forth [8, 9]. Among theoretical molecular descriptors, topological indices have an impact in chemistry due to the prediction of physio-chemical properties of the underlying substance. Its role in the QSPR/QSAR analysis to model physical and chemical properties of molecules is also remarkable [10–12]. Actually, topological indices are designed on the ground of transformation which associates a numeric value with the graph which characterizes its topology [13]. The first topological index, named Winner index, was proposed in 1947 by Winner [14]. It provides best correlation with the boiling points of alkanes. The discovery of the Winner index provides emerging research platform to the research community. The interest in accurate prediction of physio-chemical properties encouraged the researchers to propose a large class of topological indices. For the first time, an index is defined on the base of end vertices’ degrees of the edges by Milan Randić named Randić connectivity index [15]:
Due to this reason, it has attained a great attraction of the researchers till now. In 2009, Vukičević and Furtula [16] introduced the geometric-arithmetic index:
has correlation coefficient of 0.972 with heat of formation of benzene hydrocarbons. Also, in case of “standard enthalpy of vaporization,” its accuracy is more than the Randić index. Due to this reason, was studied more than all other indices in the last decade. The bonds and extremal characterization of graphs regarding the index were studied at some extent in [17–24]. It encouraged us to study the index for and transformed graphs and under the fact of transformations and , , respectively. We characterize extremal graphs for all of these families of graphs.
2. Results and Discussion
Throughout this work, let graph comprise with -vertex simple connected graph along with pendent paths of length attached with , having degree . The order of is , size is , and is its degree sequence.
Let graph be with the degree of vertex and and be the degrees of . For validity of our proved results, we defined the following list of useful graphs.
Type I: let , where . of type I is obtained by attaching pendent paths of length with vertices of degree in such a way that the vertices with pendent path are adjacent to the vertices without pendent paths.
The graph of type I is shown in Figure 1(a).
(a)
(b)
Type II: of type II is the graph of type I with .
The graph of type II is shown in Figure 1(b).
Before attempting the major problem, we prove the following preposition.
Proposition 1. Let ; then,
Proof. Let :The above calculations implies
Theorem 1. Let graph comprise of -vertex simple connected graph along with pendent paths of length attached with of degree , maximum degree , and minimum . Then,
Equality holds for graphs of type II. And,
Equality holds for graph of type II.
Proof. Let a simple graph be of order , size , maximum degree , and minimum . be the graph formed by number of paths having length pendent at distinct vertices such that . The geometric-arithmetic index of any graph isThe construction of implies and for .
The edge set of partitioned asand ,The construction of implies that the cardinality of is , , , and . The function is decreasing, where is a constant. So, for minimum degree of vertices of and maximum degree , we haveFrom equation (10), we haveAfter simplification, we obtainNow, again setwhich implies from Proposition 1 and the characteristics of in equation (10). We get the following inequality:After simplification, we obtainInequalities (13) and (16) complete the proof.
Corollary 1 shows generalization of the above defined inequalities. One can get more inequalities of their desire by replacing with already defined bonds of the index.
Corollary 1. Let graph comprise of -vertex simple connected graph along with pendent paths of length attached with of degree , maximum degree , and minimum . Then,
Equality holds for regular graph of the type II.
Proof. Using results of Theorem 1 and inequality regarding the geometric index proved in [25, 26], aswe get desired results.
2.1. 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 use the following transformations as used in [27]. These transformations have solid effect over of .
Transformation A: let , for , and paths pendent at of the form comprise . Then,
The transformation is shown in Figure 2.
In Theorem 2, we discuss the effect of transformation over the index.
Theorem 2. Let graph comprise of -vertex simple connected graph along with pendent paths of length attached with of degree , maximum degree of is , and minimum . Then,
Equality holds for all graphs of type II:
Equality holds for all graphs of the type II and .
Proof. Let a simple graph be of order , size , minimum degree , and maximum . Let be the graph formed by number of paths of length pendent at distinct fully connected vertices of . The geometric-arithmetic index of any graph isThe construction of , implies . After successive applications of transformation as , , the edge set of is partitioned as , :The cardinality of is , , and . The function is decreasing, where is a constant. So, for minimum degree of and maximum, for any graph,Substituting these changes in equation (24), we have the following inequality:After simplification, we get the required resultNow, again, from equation (24) and inequalities,After simplification, we obtainInequalities (27) and (30) complete the proof.
Transformation B: let , for , and paths pendent at of the form which comprises . Then, for fixed vertex ,The transformation is shown in Figure 3 and shown in Figure 4.
Transformation : let and . The transformation is the composition of successive applications of transformation and as and , respectively [27].
In Theorem 3, we discuss the effect of transformation over the index.
Theorem 3. Let graph comprise of -vertex simple connected graph along with pendent paths of length attached with of degree , maximum degree of is , and minimum . Then,
Equality holds for graph of the type II with and .
Proof. Let a simple graph of order , size , minimum degree , and maximum . Let be the graph formed by number of paths of length pendent at distinct fully connected vertices of . The geometric-arithmetic index of any graph isThe construction of , implies . Let be the fixed vertex. Applications of transformation has an effect over the edge set partition as , :The cardinality of is , , , , andThe function is decreasing, where is a constant. So, for minimum degree of and maximum, we haveSubstituting these changes in equation (35), we obtained the following inequality:After simplification, we get the required result:Now, again substituting the following inequalities in equation (35),We obtainAfter simplification, we get the required result:Inequalities (40) and (43) complete the proof.
3. Conclusion
The study of mathematical aspect regarding topological indices is a partially open problem [20, 28, 29]. For which member family of graphs the certain index has a minimal or maximal value? In this work, we discussed, for this fundamental question, general graphs with pendent paths for the most studied index named geometric-arithmetic index and developed tight bounds by characterizing graphs. In Theorems 2 and 3, for the first time, we defined tight bonds for the transformed graphs under the effect of transformations defined in [27].
Data Availability
No data were used to support this study.
Disclosure
This paper has not been published elsewhere and it will not be submitted anywhere else for publication.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
All authors have equally contributed to the study.
Acknowledgments
The authors extend their appreciation to the deputyship for Research and Innovation, Ministry of Education in Saudi Arabia, for funding this research work through the project no. IFP-2020-17.