Bounds on General Randić Index for F-Sum Graphs
A topological invariant is a numerical parameter associated with molecular graph and plays an imperative role in the study and analysis of quantitative structure activity/property relationships (QSAR/QSPR). The correlation between the entire -electron energy and the structure of a molecular graph was explored and understood by the first Zagreb index. Recently, Liu et al. (2019) calculated the first general Zagreb index of the -sum graphs. In the same paper, they also proposed the open problem to compute the general Randić index of the -sum graphs, where and denote the valency of the vertex in the molecular graph . Aim of this paper is to compute the lower and upper bounds of the general Randić index for the -sum graphs when . We present numerous examples to support and check the reliability as well as validity of our bounds. Furthermore, the results acquired are the generalization of the results offered by Deng et al. (2016), who studied the general Randić index for exactly .
Suppose the ordered pair denotes a finite, simple, and connected molecular graph . The set represented by is the vertex set and the set denoted by , disjoint from , is the edge set. Vertices of correspond to atoms, whereas edges represent bonding between atoms. For any vertex , the number of vertices adjacent with is called the degree of vertex and is denoted by . The smallest and the largest degree of are symbolized by and , respectively. Two primary parameters known as the order (total number of vertices) and the size (total number of edges) of graph are denoted by and . A path is a simple graph having order and size with the property that exactly two vertices have degree 1, and rest of the vertices have degree 2. A cycle is a simple graph with same order and size in such a way that each vertex has degree 2.
Graph theory is playing a remarkable role in various domains of science, especially in mathematical chemistry, computer science, and chemical graph theory since the middle of last century. Let be a collection of simple graphs and be a set of real numbers; then, a topological index (TI) is considered to be a function : that associates a graph to a real number. It is worth noting that all the TIs are invariant for the isomorphic structures. To probe and study the chemical, structural, and physical properties of the molecular graphs within the subject of chemical graph theory, several TIs are proposed and intensely investigated. These TIs helped to study the chemical reactivities and physical features such as heat of evaporation and formation, boiling, melting and freezing point, volume of air and vapor pressure, surface tension and density, and critical temperature of the chemical compounds that are involved in the molecular graphs. Moreover, medical behaviors of the drugs, nanomaterials, and crystalline materials which are very important for the chemical industries including pharmaceutical are studied using TIs. For further reading regarding development and applications of TIs, the readers are referred to [1–10].
2. Preliminaries and Background
Some convinced and significant degree-based TIs closely related to our work are defined below.
Definition 1. Let be a molecular graph; then, the first Zagreb index and second Zagreb index are defined asGutman and Trinajstić  defined the first and second Zagreb indices to establish the relationship between the entire -electron energy and a structure of a molecular graph.
Definition 2. Let be a molecular graph; then, the first general Zagreb index (FGZI) is defined aswhere , , and . Li and Zheng  provided the idea of FGZI. It is evident that we achieve (first Zagreb index) and (forgotten index) by putting , and in FGZI, respectively [11, 13].
Definition 3. Let be a molecular graph; then, the general Randić index (GRI) is defined asBollobás and Erdös originated the concept of GRI . It is clear that gives the classical Randić connectivity index  and provides second Zagreb index .
Definition 4. Let be a molecular graph; then, the general sum-connectivity index (GSCI) is defined asZhou and Trinajstić initiated the idea of GSCI . For , we attain the classical sum-connectivity index (SCI) .
Lučić et al.  studied and observed that there is good correlation between indices and themselves besides their correlation with (-electron energy) of benzenoid hydrocarbons. For further insight and applications related to all above TIs, the readers are referred to [19–23].
The operations on graphs, in the construction of new graphs, also play an important role in graph theory, where the old graphs are called the factors of the new graph. Cartesian product (binary operation) is an elegant technique to construct a broader network from two base graphs and is inevitable for design and analysis of networks . In , Eliasi and Taeri contrived and constructed the -sum graphs () by employing the idea of Cartesian product on graphs and , where are two simple-connected graphs and is obtained after applying on which is elaborated subsequently.
Definition 5. Let be a simple, connected, and finite graph; then, the four significant related graphs can be defined as follows [25, 26]:(1)Subdivision graph is an expansion of graph by introducing an additional vertex on each edge of .(2)Triangle parallel graph is derived from by joining the solid vertices of the original edges of that are incident with hollow vertices.(3)Line superposition graph is obtained from by attaching those pairs of new vertices by edges which have common adjacent (solid) vertex.(4)Total graph is constructed from by applying and , simultaneously.For more details, see Figure 1. For further insight regarding graph operations, see .
Definition 6. Let and be two finite, simple, and connected graphs, be an operation (one of , , , and ), and be a graph (derived from by employing the operation ) with as vertex set and as edge set. Then, the -sum graph is a graph with vertex set such that two vertices and of are adjacent if and only if [ and ] or [ and ].
We observe that the graph has copies of the graph provided that vertices of these copies are labeled with vertices of . In graph , the vertices of are referred as solid vertices, whereas the vertices are referred as hollow vertices. Now join only solid vertices having same label in such that their adjacency in is preserved. For more clarity, see Figure 2.
Following theorems from basic mathematics are of substantial significance in order to obtain core results.
Theorem 1. Binomial and trinomial theorems provide easy and powerful way in expanding expression involving finite higher powers. The algebraic expressions of binomial and trinomial expansions are described beneath, respectively.where .
Although valency as well as spectral based TIs are current topics of increasing interest for researchers and recently Liu et al [28–30] studied weighted edge corona networks viz a viz spectra of various matrices and valency based indices of Eulerian as well as generalized Sierpinski networks. However, among the valency-based TIs, the Randić index and its variations such as general sum-connectivity, general Randić, harmonic, geometric arithmetic, and atom bond connectivity indices have ample applications in pharmacology and medicinal chemistry [31–33]; for detailed study regarding Randić index, see survey . In , Yan et al. computed and analyzed the changes in behavior of Wiener index  and enhanced the results to Hosoya polynomial for graph operations presented in Definition 1. In , Eliasi and Taeri not only introduced the -sum graphs but also computed the Wiener index of these graphs. Li et al. , Shi , and Pan et al.  provided bounds on Randić index for chemical graphs (), bounds on Randić index for triangle-free graph, and sharp bounds on zeroth order general Randić index for unicyclic graphs with fix diameter, respectively. Ali et al. , Jamil et al. , and Elumalai et al.  computed bounds on zeroth order general Randić index for certain type of graphs. Later on, Deng et al. , Imran et al. , Liu et al. , and Ahmad et al. computed the first Zagreb index and second Zagreb index (general Randić index for exactly ) of the -sum graphs, bounds of several indices of the -sum graphs, first generalized Zagreb indices of the -sum graphs, and bounds on general sum-connectivity index for -sum graphs , respectively. Moreover, Liu et al.  proposed the open problem to compute the general Randić index for any . In this paper, we solve this open problem, partially, by computing the lower and upper bounds on general Randić index for the -sum graphs for any .
The rest of the paper is put together as follows. Section 2 covers the materials and methods to determine main results and Section 3 includes some applications of the main results. Section 4 covers the conclusion and further directions of the work.
3. Results and Discussion
In this section, the main results regarding general Randić index on the -sum graphs , , and , are computed, where and are considered to be finite, simple, and connected graphs. Throughout , , , , , , , , , , , , , and .
Theorem 2. Let and be two simple, finite, and connected graphs. For , the general Randić index of -sum graph is , whereand . Equality holds if and only if and are regular graphs with same regularity.
Proof. Suppose that denotes the degree of vertex in the -sum graph . Then, general Randić index for -sum graph is calculated asConsiderUsing trinomial theorem, we getwhere .Using Definitions 1 and 3 and the fact , we haveNow applying binomial theorem, we getUsing Definitions 5 and 6 and property of smallest degree of graph , we haveSubstituting the (12) and (14) in (8), we haveSimilarly,Equality holds if and only if and are regular graphs with same regularity. This completes the proof.
Example 1. Let , , , and . Then, , , , , , , , and . Moreover, Table 1 contains values of some indices related to certain graphs and is crucial to figure out examples throughout.
Now, we compute lower and upper bound of GRI using formulas derived in Theorem 2.Likewise, .
To calculate exact value for , we require edge partition of graph , which is presented in Table 2.
Now, we calculate exact value of GRI of for . .
Evidently,Additionally, we computed actual values along with corresponding bounds of GRI for various cases, and some are presented in Table 3.
Theorem 3. Let and be two simple, finite, and connected graphs. For , the general Randić index of -sum graph is , whereand . Equality holds if and only if and are regular graphs with same regularity.
Proof. Suppose that denotes the degree of vertex in the -sum graph . Then, general Randić index for -sum graph is calculated asApplying trinomial theorem, we getwhere . Now applying summations on the convenient expressions, then replacing with corresponding formulas, and using smallest degree of graph , we getNext sum involves those edges from whose end vertices are in .Now applying trinomial theorem, we haveSubsequent sum includes those edges from whose one end vertex is in while the other is in .Using (22), (25), and (26) in (20), we getEquality holds if and only if and are regular graphs with same regularity. This completes the proof.
Example 2. Let , , , and . Then, , , , , , , , and . Now, we compute lower and upper bound of GRI using formulas derived in Theorem 3.In similar way, we calculate .
To compute exact value for , we need edge partition of graph , which is given in Table 4.
Exact value of GRI of for is given by .
Clearly,Moreover, we computed actual values along with corresponding bounds of GRI for various cases, and some are presented in Table 5.
Theorem 4. Let and be two simple, finite, and connected graphs. For , the general Randić index of -sum graph is , whereand . Equality holds if and only if and are regular graphs with same regularity.
Proof. Suppose that denotes the degree of vertex in the -sum graph . Then general Randić index for -sum graph is calculated asApplying trinomial theorem, we getwhere . involves those edges from whose one end vertex is in and the other is in .Now applying binomial theorem, we have