On the Minimum Variable Connectivity Index of Unicyclic Graphs with a Given Order

<jats:p>The variable connectivity index, introduced by the chemist Milan Randić in the first quarter of 1990s, for a graph <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mi>G</mml:mi></mml:math> is defined as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M2"><mml:mstyle displaystyle="true"><mml:msub><mml:mrow><mml:mo stretchy="false">∑</mml:mo></mml:mrow><mml:mrow><mml:mi>v</mml:mi><mml:mi>w</mml:mi><mml:mo>∈</mml:mo><mml:mi>E</mml:mi><mml:mfenced open="(" close=")" separators="|"><mml:mrow><mml:mi>G</mml:mi></mml:mrow></mml:mfenced></mml:mrow></mml:msub><mml:mrow><mml:msup><mml:mrow><mml:mfenced open="(" close=")" separators="|"><mml:mrow><mml:mfenced open="(" close=")" separators="|"><mml:mrow><mml:msub><mml:mrow><mml:mi>d</mml:mi></mml:mrow><mml:mrow><mml:mi>v</mml:mi></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:mi>γ</mml:mi></mml:mrow></mml:mfenced><mml:mfenced open="(" close=")" separators="|"><mml:mrow><mml:msub><mml:mrow><mml:mi>d</mml:mi></mml:mrow><mml:mrow><mml:mi>w</mml:mi></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:mi>γ</mml:mi></mml:mrow></mml:mfenced></mml:mrow></mml:mfenced></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mrow><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:mrow></mml:msup></mml:mrow></mml:mstyle></mml:math>, where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M3"><mml:mi>γ</mml:mi></mml:math> is a non-negative real number and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M4"><mml:msub><mml:mrow><mml:mi>d</mml:mi></mml:mrow><mml:mrow><mml:mi>w</mml:mi></mml:mrow></mml:msub></mml:math> is the degree of a vertex <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M5"><mml:mi>w</mml:mi></mml:math> in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M6"><mml:mi>G</mml:mi></mml:math>. We call this index as the variable Randić index and denote it by <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M7"><mml:msub><mml:mrow><mml:mstyle displaystyle="true"><mml:mrow><mml:mmultiscripts><mml:mi>R</mml:mi><mml:mprescripts /><mml:none /><mml:mi>v</mml:mi></mml:mmultiscripts></mml:mrow></mml:mstyle></mml:mrow><mml:mrow><mml:mi>γ</mml:mi></mml:mrow></mml:msub></mml:math>. In this paper, we show that the graph created from the star graph of order <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M8"><mml:mi>n</mml:mi></mml:math> by adding an edge has the minimum <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M9"><mml:msub><mml:mrow><mml:mstyle displaystyle="true"><mml:mrow><mml:mmultiscripts><mml:mi>R</mml:mi><mml:mprescripts /><mml:none /><mml:mi>v</mml:mi></mml:mmultiscripts></mml:mrow></mml:mstyle></mml:mrow><mml:mrow><mml:mi>γ</mml:mi></mml:mrow></mml:msub></mml:math> value among all unicyclic graphs of a fixed order <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M10"><mml:mi>n</mml:mi></mml:math>, for every <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M11"><mml:mi>n</mml:mi><mml:mo>≥</mml:mo><mml:mn>4</mml:mn></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M12"><mml:mi>γ</mml:mi><mml:mo>≥</mml:mo><mml:mn>0</mml:mn></mml:math>.</jats:p>


Introduction
All 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 called its size. For a vertex v 1 ∈ V(G), denoted by N(v 1 ), the set of all those vertices of G are adjacent with v 1 . e number d v 1 � |N(v 1 )| is called degree of v 1 . If d v 1 � 1, then v 1 is called a pendent vertex or a leaf. A graph of order n is known as an n-vertex graph. As usual, the n-vertex path and star graphs are denoted by P n and S n , respectively. An n-vertex graph containing exactly one cycle is called a unicyclic graph. e class of all n-vertex unicyclic graphs is denoted by U n . e graph obtained from S n by adding an edge is denoted by S 1 n . For the (chemical) graph theoretical notation and terminology that are not defined in this paper, we refer the reader to some standard books, such as [1][2][3].
To model the heteroatoms 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 [4]. Let G be a vertex-weighted graph with the vertex set v 1 , v 2 , . . . , v n and the vertex weight w i of the vertex v i is 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 av A(G) � [a i,j ] n×n , where e variable connectivity index [5,6], proposed by Randić, for the graph G is defined as 1 (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 (quantitative structure-property/activity relationship) studies of heteroatoms 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 index 1 (3) In the rest of this paper, we denote this index by v R c instead of 1 χ f . Clearly, if we take c � 0 then the invariant v R c (G) is the classical Randić index [7,8]. Details about the chemical applications of the variable Randić index can be found in [4,7,[9][10][11][12][13][14][15][16][17] and related references listed therein. In [18], a mathematical study of the variable Randić index was initiated and it was proved that the star graph has the minimum variable Randić index among all trees of a fixed order n, where n ≥ 4. It needs to be mentioned here that the variable Randić index seems to have more chemical applications than the several wellknown variable indices, see, for example, the variable indices considered in the papers [19][20][21][22][23][24][25].
For convenience, we introduce some further notation and terminology. Let A k n,l,j be the unicyclic graph constructed from a path P � u 0 u 1 . . . u j (j ≥ 1) by connecting k pendant vertices to u 0 and a cycle C l , l � n − j − k to u j , respectively. Let A k n,l be the unicyclic graph constructed from cycle C l , l � n − k by connecting k pendant vertices to one vertex of C l and A i,j,k n,l be the unicyclic graph created from cycle C l , n � i + j + k + l by attaching K 1,i , K 1,j , and K 1,k to the vertices of C l . Let D be the such class of U n whose every member has unique 3-cycle, and the vertices apart from 3-cycle are pendent vertices. Let F be the such class of U n whose every member has unique 4-cycle, and the vertices apart from 4-cycle are pendent vertices and are joined to nonadjacent vertices of the unique 4-cycle.
In this paper, we characterize the collection of unicyclic graphs on n vertices that minimize variable Randić index. We further show that, for c ≥ 0, S 1 n has minimum variable Randić index among the collection U n and where equality holds if and only if G � S 1 n .

Transformations Which Decrease the Variable Randić Index
We introduce some transformations and prove some lemmas to establish main results.

Lemma 1.
If c ≥ 0 and s, t ≥ 1, then the function Ψ defined as is positive valued.
Proof. We note that the function zΨ/zs is strictly increasing in t on the interval (1, ∞) because Also, it can be seen that the value of the function zΨ/zs at t � 1 is > 0, which implies that the function zΨ/zs is positive valued for t > 1, and hence the function Ψ is strictly increasing w.r.t. s on the interval (1, ∞). In our case Ψ(c, s, t) � Ψ(c, s, t), the function Ψ is also strictly increasing w.r.t. t on the interval (1, ∞). Hence, Ψ(c, s, t) > 0 for all s, t > 1 and c ≥ 0. □ Lemma 2. If c ≥ 0 and s, t ≥ 1, then the function Φ defined as is positive valued. The proof of Lemma 2 is analogous to the proof of Lemma 1.

Lemma 3.
If c ≥ 0, t > 2, and r > 1, it holds that 2 Discrete Dynamics in Nature and Society Proof. Let Since The proof of Lemma 4 is analogous to the proof of Lemma 3.
The proof of Lemma 5 is analogous to the proof of Lemma 3. Now, we use three transformations, which will reduce the variable Randić index.
Using Lemma 3, one can see that inequality (13) holds. Proof.
Since Now, there will be three cases.
Using Lemma 4, one can see that relation (16) is Using Lemma 5, one can see that relation (17) is positive valued. Since respectively. Assume that the path between u and v on cycle is P uv such that |E(P uv )| � r, d(V(P uv )) � 2 if r ≥ 2 and |E(P vu )| ≥ 3. Construct G � from G by removing the edges uu i and vv j , reducing the path P uv into one vertex u(v) and attaching a star K 1,r+s+t+1 to u(v), by making sure that | � G| � |G|. G and G � are depicted in Figure 2. Proof Using Lemma 1, one can see that relation (20) is positive valued.
Since r ≥ 2 and keeping in mind the fact that Discrete Dynamics in Nature and Society is an increasing function because zΨ zr We Using Lemma 2, one can see that relation (24) is positive valued.
Let D i,j,k n,3 be the such class of D that is obtained from cycle C l , l � 3 by attaching K 1,i , K 1,j , and K 1,k to the vertices of C l , l � 3. Let F r,s n,4 be the such class of F that is obtained from cycle C l , l � 4 by attaching K 1,r and K 1,s to the nonadjacent vertices of C l , l � 4. be three unicyclic graphs in D as in Figure 3.
where ξ 3 , ξ 4 ∈ (i + 2, i + 3). Bearing in mind the fact that i ≥ j, one can see that Discrete Dynamics in Nature and Society 7 be three unicyclic graphs in F as in Figure 4.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.