Abstract
Suppose is a simple graph with edge set . The Randić index is defined as , where and denote the vertex degrees of and in , respectively. In this paper, the first and second maximum of Randić index among all vertex cyclic graphs was computed. As a consequence, it is proved that the Randić index attains its maximum and second maximum on two classes of chemical graphs. Finally, we will present new lower and upper bounds for the Randić index of connected chemical graphs.
1. Mathematical Notions and Notations
In this section, we first describe some mathematical notions that will be kept throughout. A pair in which is a finite nonempty set and is a subset of 2-elements subsets of is called a simple graph. Throughout this paper, the term graph means simple graph and the sets and in definition of are called the vertex set and edge set of , respectively.
Suppose is a graph. For simplicity of our argument, an edge in is simply written as . Choose a vertex in . The vertex degree of , , is defined as the number of edges in the form . A chemical graph is a graph in which all vertices have degrees less than or equal to 4 [1]. The reason for this name is from quantum chemistry in which it is convenient to model a molecule in such a way that vertices are used to denote atoms and edges are for chemical bonds.
The set of all vertices adjacent to a vertex is denoted by and notations , , and are used for the maximum degree, the number of vertices of degree , and the number of edges of degree in , respectively. The number of edges connecting a vertex of degree with a vertex of degree in is denoted by . A connected vertex graph is called to be -cyclic if it has edges and the number is said to be the cyclomatic number of .
Suppose is a nonempty subset of vertices in a graph . The subgraph of obtained by deleting the vertices of is denoted by , and similarly, if , then the subgraph obtained by deleting all edges in is denoted by . In the case that or , the subgraphs and will shortly be written as or , respectively. Furthermore, if and are nonadjacent vertices in , then the notation is used for the graph obtained from by adding an edge .
The Randić index of a graph is defined as
This topological index was proposed by Milan Randić [2] under the name “branching index.” The Randić index is suitable for measuring the extent of branching of the carbon-atom skeleton of saturated hydrocarbons. We encourage the interested readers to consult the books [3,4] for more information on this topic.
2. Background Materials
This section aims to briefly review the literature on ordering graphs concerning the Randić index. By referring to Theorems 2.2 and 2.3 in [5], among all -vertex trees, the star has the minimum Randić index and the path attains the maximum Randić index. Caporossi et al. [6] proved that among all 1-cyclic graphs of order , the cycle attains the maximum value, and the unicyclic graphs obtained by attaching a pendant path to a vertex of a cycle attain the second maximum Randić index. These are the starting point of the following problem.
Question 1. Find -vertex -cyclic graphs with maximum and minimum Randić index.
Shiu and Zhang [7] obtained the maximum value of Randić index in the class of all vertex chemical trees with pendants such that . Shi [8] obtained some interesting results for chemical trees with respect to two generalizations of Randić index. Dehghan-Zadeh et al. [9, 10] obtained the first and second maximum of Randić index in the class of all vertex -cyclic graphs when .
Deng et al. [11] considered various degree mean rates of an edge and gave some tight bounds for the variation of the Randić index of a graph in terms of its maximum and minimum degree mean rates over its edges. Gutman et al. in a recent interesting paper [12] investigated the connection between Randić index and the degree-based information content of molecular and also general graphs. This connection is based on the linear correlation between Randić index and the logarithm of the multiplicative version of the Randić index.
The aim of this paper is to proceed with Question 1. We will obtain the first and second maximum of Randić index among all -cyclic graphs. This extends some results in [6, 9, 10].
3. Five Graph Transformations
In this section, five graph transformations will be presented which are useful in computing Randić index of graphs. The transformations I and II were introduced in [13]. (1)Transformation I. Suppose that is a graph with a given vertex . In addition, we assume that and are two paths of lengths and , respectively. Let be the graph obtained from , , and by attaching edges and . Define . The above-referred graphs are illustrated in Figure 1.(2)Transformation II. Suppose that is a graph with given vertices and such that and for all , . In addition, we assume that and are two paths of lengths and , respectively. Define be the graph obtained from , , and by attaching vertices , , and . See Figure 2 for more details.(3)Transformation III. Suppose that is a graph with vertices , and such that . In addition, we assume that is a trivial graph with vertex set . Define and . The above-referred graphs are illustrated in Figure 3.(4)Transformation IV. Suppose that is a graph with given vertex such that , and . In addition, we assume that , , and . Define . See Figure 4 for more details.(5)Transformation V. Suppose that is a graph with vertices , and such that , , , , and or 2. Define . The above-referred graphs are illustrated in Figure 5.
(a)
(b)
(a)
(b)
(a)
(b)
(a)
(b)
(a)
(b)
It is well-known that if the derivative of a continuous function satisfies on an open interval , then is increasing on .
Lemma 1. The following hold:(1)Let and be two graphs satisfying the conditions of Transformation I. If or and or and , then .(2)Let and be two graphs satisfying the conditions of Transformation II and let , , , . If or and or and or and , then .(3)Let and be two graphs as shown in Transformation III.(a)If , and , then (b)If , , and , then (c)If and , then (4)Let and be two graphs satisfying the conditions of Transformation IV. Then, .(5)Let and be two graphs satisfying the conditions of Transformation V. Then, .
Proof. (1)Let , , , , , and . Then, by definition, Now, by Equality (2), , for , , for and , and , for and . The proof of other cases of and is similar, and we omit them.(2)Let , , and . Then, by definition, And, by last equality, for , for and , for and , and for and . The proof of other cases of and is similar, and we omit them.(3)Suppose , , , and . To prove (a), we note that To prove (b), we first calculate the difference between and . Let , for . Then, is increasing on , and hence by equation (5), . For the proof of (c), it is enough to notice that . Thus, , as desired.(4)Suppose that , , and . Then, as desired. (5) Suppose that . Then, by definition,Hence, the result.
For a graph , its first Zagreb index is defined as .
Lemma 2 (see [14]). If is a connected graph with vertices and edges, then
Theorem 1. Let be a connected graph with vertices and edges.(1)If or ( and ), then . The equality holds if and only if .(2)If and or ( and ), then , with equality if and only if .
Proof. By definition,1.The last equality for gives and by Lemma 2, Equality holds if and only if . Let and . Then, by Equality (9), and by Lemma 2 and some simple calculations, with equality if and only if .(2)A similar argument as the case 1, it can be proved that for and ,with equality if and only if . For and ,with equality if and only if . This completes the proof.
Theorem 2 (see [6]). Let be a graph with vertices. Then,
Lemma 3 (see [15]). If is a connected graph with vertices and cyclomatic number , then and .
Corollary 1. Let be a connected graph with vertices and cyclomatic number .(1)If , then and (2)If , then and Define , , , and .
Lemma 4. Let be a connected graph with vertices, edges, and cyclomatic number .(1)Suppose and for some . Then, .(2)Suppose and for some . Then, .
Proof. Since , , for all . Now, the proof follows from this fact that . The part (2) is similar.
Let be a positive integer. DefineIf for , then , , , and .
Theorem 3. The following hold:(1)Let be a connected graph with vertices and cyclomatic number 5. Then, , with equality if and only if .(2)Let be a connected graph with vertices and cyclomatic number 6. Then, , with equality if and only if .
Proof. (1)If , then by Corollary 1 and Theorem 2, , with equality if and only if . For , . Let . Then, again by Corollary 1 and Theorem 2, .(2)Proof is similar to the last case, and we omit it.
Remark 1. (1)Let be a connected graph with vertices and cyclomatic number 5. Then, , with equality if and only if is a 3-regular graph.(2)Let be a connected graph with vertices and cyclomatic number 6. Then, , with equality if and only if is a 3-regular graph.For a positive number , we defineIf and , then and .
Proposition 1. Let be a connected graph with vertices, edges, and cyclomatic number , where is a positive integer. If and , then .
Proof. Those are well-known that and . Therefore, . Thus, , and this implies that . Now, since , .
By Proposition 1 and a similar argument as the proof of Theorem 3, we will have the following general result.
Theorem 4. Let be a connected graph with vertices and cyclomatic number , where is a positive integer.(1)If , then , with equality if and only if .(2)If , then , with equality if and only if is a 3-regular graph.Let be a positive number, and . DefineIf for , then , and .
Theorem 5. The following hold:(1)Let be a connected graph with vertices and cyclomatic number 5. If , then , with equality if and only if .(2)Let be a connected graph with vertices and cyclomatic number 6. If , then , with equality if and only if .
Proof. (1)Let and . For , and Corollary 1 and Theorem 2 give . For , , with equality if and only if . If , then .(2)Proof is similar to Case 1, and we omit it.By a simple calculation, one can easily see that Theorem 5 (1) holds for and Theorem 5 (2) holds for . On the other hand, Theorems 3 and 5 imply the following result.
Corollary 2. The following hold:(1)Suppose . The connected graphs with cyclomatic number 5 in the sets and have the first and second maximum Randić index among all -vertex connected graphs with cyclomatic number 5, respectively.(2)Suppose . The connected graphs with cyclomatic number 6 in the sets and have the first and second maximum Randić index among all -vertex connected graphs with cyclomatic number 6, respectively.Suppose is a positive integer. DefineIf and , then and .
Theorem 6. Let be a connected graph with vertices and cyclomatic number .(1)If and , then , with equality if and only if .(2)If and is not a 3-regular graph, then , with equality if and only if .
Proof. The result follows from Proposition 1 and a similar argument as Theorem 3.
We end this section with the following result that follows from Theorems 4 and 6.
Theorem 7. Let be a connected graph with vertices and cyclomatic number .(1)If , then the connected graphs with cyclomatic number in the sets and have the first and second maximum Randić index among all -vertices connected graphs with cyclomatic number , respectively.(2)If , then the 3-regular connected graphs and the connected graphs in the set with cyclomatic number have the first and second maximum Randić index among all -vertices connected graphs with cyclomatic number , respectively.
4. Connected Chemical Graphs
Let be a connected chemical graph with vertices and edges. By Lemma 3,and by Lemma 2,
Now, by some calculations, we have
Suppose , . Since ,and since and ,
On the other hand,
By equations (25)–(27), we have
Theorem 8. Let be a connected chemical graph with vertices and edges. Then,(1), with equality if and only if or (2)
Proof. It is easy to see that and . Therefore, by equation (28), we havewith equality if and only if or . On the other hand, and . Therefore, by equation (28),This completes the proof.
Data Availability
All data generated or analyzed during this study are included in this published article. There are no experimental data in this article.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this article.