Discrete Dynamics in Nature and Society
Volume 2020 (2020), Article ID 3976274, 4 pages
https://doi.org/10.1155/2020/3976274

Research Article

Minimum Variable Connectivity Index of Trees of a Fixed Order

Shamaila Yousaf,1,2 Akhlaq Ahmad Bhatti,¹ and Akbar Ali3,4

¹Department of Sciences and Humanities, National University of Computer and Emerging Sciences, Lahore Campus, B-Block, Faisal Town, Lahore, Pakistan
²Department of Mathematics, University of Gujrat, Hafiz Hayat Campus, Gujrat, Pakistan
³Department of Mathematics, Faculty of Science, University of Ha’il, Ha’il, Saudi Arabia
⁴Knowledge Unit of Science, University of Management and Technology, Sialkot, Pakistan

Correspondence should be addressed to Akbar Ali; akbarali.maths@gmail.com

Received 13 May 2020; Accepted 29 June 2020; Published 20 July 2020

Academic Editor: Juan L. G. Guirao

Copyright © 2020 Shamaila Yousaf et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The connectivity index, introduced by the chemist Milan Randić in 1975, is one of the topological indices with many applications. In the first quarter of 1990s, Randić proposed the variable connectivity index by extending the definition of the connectivity index. The variable connectivity index for graph is defined as , where is a nonnegative real number, is the edge set of , and denotes the degree of an arbitrary vertex in . Soon after the innovation of the variable connectivity index, its various chemical applications have been reported in different papers. However, to the best of the authors’ knowledge, mathematical properties of the variable connectivity index, for , have not yet been discussed explicitly in any paper. The main purpose of the present paper is to fill this gap by studying this topological index in a mathematical point of view. More precisely, in this paper, we prove that the star graph has the minimum variable connectivity index among all trees of a fixed order , where .

1. Introduction

All the graphs that we discuss in the present study are simple, connected, undirected, and finite. For a graph , the number is called its order, and is the size of . Neighbor of a vertex is a vertex adjacent to . The set of all neighbors of vertex of is denoted by . The number is called the degree of a vertex, , and it is denoted by . If , then is called a pendent vertex or a leaf. A graph of order is called an -vertex graph. Denote by and the -vertex path graph and the -vertex star graph, respectively. The class of all -vertex trees is denoted by . For the (chemical) graph theoretical notation and terminology that are not defined in this paper, refer to [1, 2].

One of the fundamental ideas in CGT (chemical graph theory) is molecular connectivity. Chemical behavior of a compound is dependent upon its structure. QSPR/QSAR (quantitative structure-property/activity relationship) studies are progressive fields of chemical research that focus on the behavior of this dependency. The quantitative relationships are mathematical models that either enable the prediction of a continuous variable (e.g., boiling point and toxicity) or the classification of a discrete variable (e.g., sweet/bitter and toxic/nontoxic) from structural parameters. Actually, CGT has provided many topological indices that have been and are being used in QSPR/QSAR studies for predicting the physicochemical properties of chemical compounds. Topological indices are those graph invariants that found some applications in chemistry [3–6]. For further details about the topological indices and their applications, refer to [3, 7–11] and the references therein.

Molecules can be modeled using graphs in which vertices correspond to atoms of the considered molecules, and the edges correspond to the covalent bonds between atoms [6]. To model the heteroatom molecules, it is better to use the vertex-weighted graphs, which are the graphs whose one or more vertices are distinguished in some way from the rest of the vertices [12]. Let be a vertex-weighted graph with the vertex set , and let be the weight of the vertex for . The augmented vertex-adjacency matrix of is an matrix denoted by and is defined as , where

The variable connectivity index [13, 14], proposed by Randić, for graph is defined as

We associate this index’s name with its inventor Randić by calling it as the variable Randić index. This index was actually introduced within the QSPR/QSAR studies of heteroatom molecules. If is the molecular graph of a homoatomic molecule, then (say), and hence, the variable Randić index becomes

In the rest of this paper, we denote this index by instead of . Clearly, if we take , then the invariant is the classical Randić index [10, 15]. Liu and Zhong [5] showed that the variable Randić index has more flexibility in characterizing polymers, which can lead to simpler correlations with better correlative accuracy. Details about the chemical applications of the variable Randić index can be found in [5, 7, 9, 12, 15–22] and related references listed therein. It needs to be mentioned here that the variable Randić index seems to have more chemical applications than the several well-known variable indices, for example, the indices considered in [23–33]. However, to the best of the authors’ knowledge, mathematical properties of the variable Randić index, for , have not yet been discussed explicitly in any paper. The main purpose of the present paper is to fill this gap by studying this topological index in a mathematical point of view. Since the trees (that are the connected graphs without cycles) form an important class of graphs both in chemical graph theory as well as in general graph theory, in this paper, we study an extremal problem related to the variable Randić index of the class of trees. We prove that the star graph has the minimum variable Randić index among all trees of a fixed order , where .

2. Main Result

To establish the main result, we prove a lemma first.

Lemma 1. If and , then function defined as

is positive-valued.

Proof. We note that the function is strictly increasing in on the interval because

where the last inequality holds because

Also, note that the value of the function at is 0, which implies that the function is positive-valued for , and hence, function is strictly increasing in on the interval . Due to the identity , function is strictly increasing also in on the interval . It holds that , and hence, for all and .

Transformation 1. For , let be a tree of order containing at least two nonpendent vertices. Let be a vertex of maximum degree, and let , where for . Also, take . Note that . We transform into another tree by removing the edges and adding the edges .

Lemma 2. Let and be the trees defined in Transformation 1. For , it holds that

Proof. By using the definition of the variable Randić index, one has

where

Equation (8) gives

Since the vertex has the maximum degree, that is, for every and for , thus, and . Hence, equation (10) yields

By using Lemma 1 in (11), we get , as desired.
Next result is a direct consequence of Lemma 2.

Theorem 1. For and , among all trees of a fixed order , star graph is the unique tree with minimum variable Randić index , which is

For and small values of , we calculate the variable Randić index of the trees of order and find that the value of this index does not exceed from

This suggests the following conjecture.

Conjecture 1. For and , among all trees of a fixed order , path graph is the unique tree with maximum variable Randić index , which is

Data Availability

There are no data concerning the present study except those presented in this manuscript.

Conflicts of Interest

The authors declare that there are no conflicts of interest.

Acknowledgments

This work was supported by the National University of Computer and Emerging Sciences, Lahore, Pakistan.

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