Abstract
The variable connectivity index, introduced by the chemist Milan Randić in the first quarter of 1990s, for a graph is defined as , where is a non-negative real number and is the degree of a vertex in . We call this index as the variable Randić index and denote it by . In this paper, we show that the graph created from the star graph of order by adding an edge has the minimum value among all unicyclic graphs of a fixed order , for every and .
1. Introduction
All 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 called its size. For a vertex , denoted by , the set of all those vertices of are adjacent with . The number is called degree of . If , then is called a pendent vertex or a leaf. A graph of order is known as an -vertex graph. As usual, the -vertex path and star graphs are denoted by and , respectively. An -vertex graph containing exactly one cycle is called a unicyclic graph. The class of all -vertex unicyclic graphs is denoted by . The graph obtained from by adding an edge is denoted by . 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 [1–3].
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 be a vertex-weighted graph with the vertex set and the vertex weight of the vertex is for . The augmented vertex-adjacency matrix of is an matrix denoted by and is defined as , where
The variable connectivity index [5, 6], proposed by Randić, for the 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 (quantitative structure-property/activity relationship) studies of heteroatoms 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 [7, 8]. Details about the chemical applications of the variable Randić index can be found in [4, 7, 9–17] 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 well-known variable indices, see, for example, the variable indices considered in the papers [19–25].
For convenience, we introduce some further notation and terminology. Let be the unicyclic graph constructed from a path by connecting pendant vertices to and a cycle to , respectively. Let be the unicyclic graph constructed from cycle by connecting pendant vertices to one vertex of and be the unicyclic graph created from cycle by attaching , , and to the vertices of . Let be the such class of whose every member has unique 3-cycle, and the vertices apart from 3-cycle are pendent vertices. Let be the such class of 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 vertices that minimize variable Randić index. We further show that, for , has minimum variable Randić index among the collection andwhere equality holds if and only if .
2. Transformations Which Decrease the Variable Randić Index
We introduce some transformations and prove some lemmas to establish main results.
Lemma 1. If and , then the function defined asis positive valued.
Proof. We note that the function is strictly increasing in on the interval becauseAlso, it can be seen that the value of the function at is , which implies that the function is positive valued for , and hence the function is strictly increasing w.r.t. on the interval . In our case , the function is also strictly increasing w.r.t. on the interval . Hence, for all and .
Lemma 2. If and , then the function defined asis positive valued.
The proof of Lemma 2 is analogous to the proof of Lemma 1.
Lemma 3. If , , and , it holds that
Proof. LetSince , so , also . Hence, .
Lemma 4. For , it holds thatThe proof of Lemma 4 is analogous to the proof of Lemma 3.
Lemma 5. For , it holds thatThe proof of Lemma 5 is analogous to the proof of Lemma 3.
Now, we use three transformations, which will reduce the variable Randić index.
2.1. Transformation 1
Let and be an edge of such that . , where are leaves.
Define . and are depicted in Figure 1.
Lemma 6. Let . By applying Transformation 1 and for , we have
Proof. Let :Using Lemma 3, one can see that inequality (13) holds.
2.2. Transformation 2
Let such that and their unique cycles are and , respectively, . For both graphs, the vertices apart from the cycle are leaf vertices joined to exactly one vertex of the cycle.
Lemma 7. Let such that . By applying Transformation 2 and for , we have
Proof. SinceNow, there will be three cases. Case I: if , then Using Lemma 4, one can see that relation (16) is positive valued. Case II: if , then Using Lemma 5, one can see that relation (17) is positive valued. Case III: if , then
2.3. Transformation 3
Let with and are on the unique cycle of . Pendant neighbors of and are , where , and , where , respectively. Assume that the path between and on cycle is such that , if and . Construct from by removing the edges and , reducing the path into one vertex and attaching a star to , by making sure that . and are depicted in Figure 2.
Lemma 8. Let such that . By applying Transformation 3 and for , , we have
Proof. Case I: let and , then Using Lemma 1, one can see that relation (20) is positive valued. Case II: now let , thenSince and keeping in mind the fact thatis an increasing function becauseWe haveUsing Lemma 2, one can see that relation (24) is positive valued.
Let be the such class of that is obtained from cycle by attaching , , and to the vertices of . Let be the such class of that is obtained from cycle by attaching and to the nonadjacent vertices of .
Lemma 9. Let , , and be three unicyclic graphs in as in Figure 3. If , thenfor .
Proof. Let be three unicyclic graphs in as in Figure 3.where . Similarly, we havewhere .
Bearing in mind the fact that , one can see thatHence, .
Lemma 10. Let be three unicyclic graphs in as in Figure 4. If , thenfor .
Proof. Let and keeping in mind Figure 4, we havewhere . Similarly, we havewhere .
Bearing in mind the fact that , for all . Also, , which implies that .
As a consequence of Lemmas 6–10, one has the following.
Theorem 1. Among all the unicyclic graphs of order , has the minimum variable Randić index for and its value is
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by the National University of Computer and Emerging Sciences, Lahore, Pakistan.