Minimum Variable Connectivity Index of Trees of a Fixed Order

The connectivity index, introduced by the chemist Milan Randi´c in 1975, is one of the topological indices with many applications. In the ﬁrst quarter of 1990s, Randi´c proposed the variable connectivity index by extending the deﬁnition of the connectivity index. The variable connectivity index for graph G is deﬁned as 􏽐 vw ∈ E ( G ) (( d ( v ) + c )( d ( w ) + c )) − 1/2 , where c is a nonnegative real number, E ( G ) is the edge set of G , and d ( t ) denotes the degree of an arbitrary vertex t in G . 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 c > 0, 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 n , where n ≥


Introduction
All the graphs that we discuss in the present study are simple, connected, undirected, and finite. For a graph G � (V, E), the number |V(G)| is called its order, and |E(G)| is the size of G. Neighbor of a vertex v ∈ V(G) is a vertex adjacent to v. e set of all neighbors of vertex v of G is denoted by N(v). e number |N(v)| is called the degree of a vertex, v ∈ G, and it is denoted by d (v). If d(v) � 1, then v is called a pendent vertex or a leaf. A graph of order n is called an n-vertex graph. Denote by P n and S n the n-vertex path graph and the n-vertex star graph, respectively. e class of all n-vertex trees is denoted by T n . 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. e quantitative relationships are mathematical models that either enable the prediction of a continuous variable (e.g., boiling point and LC 50 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][4][5][6]. For further details about the topological indices and their applications, refer to [3,[7][8][9][10][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 G be a vertex-weighted graph with the vertex set v 1 , v 2 , . . . , v n , and let w i be the weight of the vertex v i for i � 1, 2, . . . , n. e augmented vertex-adjacency matrix of G is an n × n matrix denoted by av A(G) and is defined as e variable connectivity index [13,14], proposed by Randić, for graph G is defined as (2) We associate this index's name with its inventor Randić by calling it as the variable Randić index. is index was actually introduced within the QSPR/QSAR studies of heteroatom molecules. If G is the molecular graph of a homoatomic molecule, then w 1 � w 2 � · · · � w n � c (say), and hence, the variable Randić In the rest of this paper, we denote this index by [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][16][17][18][19][20][21][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][24][25][26][27][28][29][30][31][32][33]. However, to the best of the authors' knowledge, mathematical properties of the variable Randić index, for c > 0, have not yet been discussed explicitly in any paper. e 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 S n has the minimum variable Randić index among all trees of a fixed order n, where n ≥ 4.

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

Lemma 1.
If c ≥ 0 and s, t ≥ 2, then function f defined as is positive-valued. Proof. We note that the function zf/zs is strictly increasing in t on the interval (1, ∞) because where the last inequality holds because Also, note that the value of the function zf/zs at t � 1 is 0, which implies that the function zf/zs is positive-valued for t > 1, and hence, function f is strictly increasing in s on the interval (1, ∞). Due to the identity f(c, s, t) � f(c, t, s), function f is strictly increasing also in t on the interval Note that s ≥ t ≥ 2. We transform T into another tree T � by removing the edges vv 1 , vv 2 , . . . , vv t− 1 and adding the edges uv 1 , uv 2 , . . . , uv t− 1 .

Lemma 2. Let T and T
� be the trees defined in Transformation Proof. By using the definition of the variable Randić index, one has where Equation (8) gives Since the vertex u ∈ V(T) has the maximum degree, that is, Hence, equation (10) yields (11) By using Lemma 1 in (11) Next result is a direct consequence of Lemma 2.
□ Theorem 1. For n ≥ 4 and c ≥ 0, among all trees of a fixed order n, star graph S n is the unique tree with minimum variable Randić index v R c , which is For c ≥ 0 and small values of n, we calculate the variable Randić index v R c of the trees of order n and find that the value of this index does not exceed from v R c P n � 2 ������������ (1 + c)(2 + c) This suggests the following conjecture.

Conjecture 1.
For n ≥ 4 and c ≥ 0, among all trees of a fixed order n, path graph P n is the unique tree with maximum variable Randić index v R c , which is Data Availability ere are no data concerning the present study except those presented in this manuscript. Discrete Dynamics in Nature and Society 3

Conflicts of Interest
e authors declare that there are no conflicts of interest.