Abstract
For the vertex set of a graph , the sum of reciprocals of the breadth (distances) between the vertex and whole other remaining vertices of is called reciprocal status of . In this study, first of all, we introduced the reciprocal status index and reciprocal status co-index of a graph . Later, we exposed some sharp bounds for these indices. Furthermore, we determined the reciprocal status (co-)index over some standard graphs. Finally, we presented correlations between reciprocal status index and some properties of Butane derivatives via a table and illustrated with a figure.
1. Introduction and Preliminaries
Topological indices are known as special graph invariants, and they are mainly used to calculate (quantitative structure-activity relationship) and (quantitative structure-property relationship) studies, cf. [15, 16]. Although the Wiener index ([17]) is the oldest topological index, later on some distance-based topological indices have defined as hyper Wiener index [13], Harary index [4, 8], and so on. The main idea in the study of topological indices is whether they are useful for identifying chemical properties. In general, octane isomers are the exceptional data for these studies.
If we suppose is connected having order also having size , then it is quite well known that, while the vertex and edge sets are shown by and , respectively, an edge connecting (joining) the vertices is shown by , and the degree of is actually the number of edges incident to . On the contrary, , which notates the distance between any two vertices and , is actually the length of the shortest path connecting and , and the maximum breadth between any pair of vertices of is defined as the diameter . For unmentioned graph theoretic terminologies, we may refer [1]. The following reminders will be needed for the construction of results in this paper:
In [2], for , the index has recently been exposed bywhere and . According to the same reference, the index indicates a nice output in the meaning of polychlorinated biphenyl and octane isomers. Actually, index can also be written as
We will develop some new type of indices, see Definition 1 below.
For the vertex , Harary [3] defined a parameter called status (or, equivalently and transmission), which is the sum of ’s breadths to remaining vertices of , and that is shown by
The reciprocal status [11] of is defined by the sum of reciprocals of ’s breadths (distances) to remaining vertices in . This is shown by
As it mentioned at the beginning, the oldest index, which is namely the Wiener index [17], is given by
In fact, it is also called as total status [3]. In literature knowledge, there are some studies about the different types of transmission topological indices, see, for instance, [5–7, 9, 10, 12, 14]. Additionally, the status and reciprocal status having base (topological) co-indices have recently been introduced in [12], and also, explicit formulae for them via order and size of are obtained.
Based on the construction of different types of reciprocal status-based topological indices and co-indices until now, in here, we present two new topological based indices, namely, reciprocal status index and reciprocal status co-index of a connected graph .
Therefore, the following definition plays a key role for this study:
Definition 1. Let us consider a connected graph . Therefore, the reciprocal status index over is defined bywhereas the reciprocal status co-index is defined by
For an example of and indices, one may see the graph in Figure 1. Clearly, and .
This paper is organised as in the following. After this introductory part, we will present some “nice” bounds for the reciprocal status index, see equation (6) in Section 2. In Section 3, we will state and prove some results about reciprocal status index of standard graphs such as complete , complete bipartite , path , cycle , wheel , windwill , and star graph . In Section 4 and 5, in parallel to Sections 2 and 3, we will compute some nice bounds for the reciprocal status co-index defined in equation (7) and present some results about this new co-index of standard graphs depicted above. In Section 6, the correlation of the reciprocal status index with some properties of Butane derivatives will be presented by means of a table and figure. Section 7 shows the importance of these new indices in terms of heavy atomic count and some future ideas.
2. Bounds for Reciprocal Status Index
In Section 2, we present some lower and upper bounds for defined in equation (6) and then characterize the strictness conditions of these bounds.
The following is needed to obtain our first main result in this paper:
Lemma 1. Assume is connected owning vertices whereas . Thus,where , , , and . Both equalities hold if .
Proof. We first note that, for any , there exist vertices having distance one from and the remaining vertices having distances at least 2.
Upper bound: for ,which implies thatwhere , , , and .
Lower bound: again, for , there isTherefore,where , , , and as required.
For the equality: suppose the diameter of is given as 1 or 2. Thus, equality holds. For the converse part, let , and let us assume that , where , , , and . Thus, there exists one pair of vertices (at least) , having . So,Similarly, for , we get but for remaining whole vertices , it is obtained .
Now, our aim is to reach a contradiction by assuming . So, let us separate the partition into three sets , , and such thatClearly, , , and . Therefore,which gives a contradiction. Further, we get .
By means of direct application of Lemma 1, we can give our first main result after expanding the sums:
Theorem 1. Suppose is connected, having total vertices, and finally, suppose . Thus,and
Above equalities again hold if .
The next two consequences of Theorem 2 can be given.
Corollary 1. Let be as in Theorem 2 having edges. Let and be the min. and max. degrees of the elements in , respectively. After that,where and .
Proof. For the vertex , it is clear that . Therefore, by substituting , in the lower bound, and but in the upper bound of Theorem 2, we get the required result.
Corollary 2. Suppose that is connected -regular on vertices and edges. Also, suppose that . Thus,where and . Moreover, equalities in above hold if .
3. Reciprocal Status Index of Some Standard Graphs
As we mentioned in Section 1, we will determine reciprocal status index defined in equation (6) for some special graphs.
In the following, the first result of this section is about complete graphs.
Proposition 1. For a complete graph on vertices and edges, .
Proof. It is known that for any . Henceforth, by equation (6), we get the result.
The next proposition exposes the result for on complete bipartite graphs.
Proposition 2. For a complete bipartite graph , we have
Proof. One can partition into two different subsets and with the condition every single edge of , and we let and . Therefore, and such that and . Clearly, has vertices and edges, and also, the diameter . Hence, by the equality part of Theorem 2, it is achievedHence, we get the result on as required.
Now, we can investigate the situation for cycle graphs as follows.
Proposition 3. For a cycle graph on vertices,
Proof. Case (i): if is an even number, then for any vertex , we haveTherefore, by equation (6), we obtainwhile is even. Case (ii): if is an odd number, then for , we getand then, by equation (6), we reach toas required.
As an example for Proposition 3.3, we can give and .
Another result in special graphs can be presented for a path as in the following.
Proposition 4. For a path graph on vertices,
Proof. Let be the vertices of Pn where is connected to for . Therefore, for each , we getTherefore,which implies the required result on .
Remaining three propositions will be clarified, indices for wheel, windmill, and star graphs, respectively.
Proposition 5. Let us consider a wheel graph with . Then,
Proof. If we partition the edge set into two sets and , then it is easy to see that such that . Therefore, by the equality part of Theorem 2,
Proposition 6. For a windmill graph with , we have
Proof. In the proof, we will follow the same way as in the proof of Proposition 3. So, let us partition into subsets and which gives and . Also, . After that, by considering the equality part of Theorem 2, we obtain the result on .
Since a star graph is actually a tree on nodes and one node’s degree is and the remaining vertices having degree one, so next result easily follows by Theorem 2:
Proposition 7. For a star graph , .
4. Bounds for the Reciprocal Status Co-Index
Now, we present some bounds for reciprocal status co-index of graphs defined in equation (7) and characterize the equality conditions on them.
Proof of the next result is very similar as in the proof of Theorem 2 and, therefore, will be skipped.
Theorem 2. Let be connected and has vertices and also . Thus,where , , , and . Moreover, equality holds on both sides if .
The following consequences are as important as Theorem 2
Corollary 3. Assume that is connected and has vertices and edges and also . Assume also that and is the min. and max. degrees of vertices in , respectively. Therefore,where and .
Proof. For , we know that which implies . It is also known that the graph under these assuming conditions has pair of nonadjacent vertices. Substituting and in the lower bound and and in the upper bound of Theorem 2, we achieve the result.
Corollary 4. Let be connected -regular and has vertices, and also, let . Then,where and . Equalities hold if .
Proof. , and substituting in Theorem 2, we get the result.
5. Reciprocal Status Co-Index of Some Standard Graphs
With a similar approach as in Section 3, we will determine reciprocal status co-index given in equation (7) of some special graph structures such as complete, complete bipartite, cycle, path, wheel, and windmill graphs.
Since the details of the following proposition are clear by considering the definitions of complete graphs and reciprocal status co-index, the proof will be omitted.
Proposition 8. For , .
Proposition 9. For , there exists
Proof. For and , let us assume and are the partite sets of with every single edge of owning one end in and other end in . If , then , and also, if , then . Therefore, for , , and for , . Hence, by equation (7), we obtainas required.
Proposition 10. For a cycle graph on vertices, we have
Proof. Case (i): as obtained in Proposition 3 in Case (i), if is an even number, for any vertex of , we get . Therefore, Case (ii): similarly, as we obtained in Proposition 3 in Case (ii), if is odd, then for , we have . Therefore, we get
Proposition 11. For a wheel graph , .
Proof. It is known that the nonadjacent pairs of vertices of the wheel graph have degree 3, and there are actually total pairs of nonadjacent vertices. Also, . Therefore, by the equality part of Theorem 2,Hence, the result.
Proposition 12. For a windmill graph with , .
Proof. It is already known that the nonadjacent pairs of vertices of the windmill graph have degrees 2, and there are total such pairs in . Further, . Therefore, again by the equality part of Theorem 2,as required.
6. Correlation of Reciprocal Status Index and Some Properties of Butane Derivatives
QSPR studies have taken attention from both mathematicians and chemists since it opens new ways to new inventions. In particular, obtaining the properties of compounds without any big effort is time and money saving. Here, we see that reciprocal status index has a nice correlation including all three physical properties surface tension, complexity, and heavy atomic count, see Table 1.
Correlation coefficient value of with surface tension, complexity, and heavy atomic count is , and 0.97, respectively, see Figure 2.
Using the data of Table 1, the scatter plot between the surface tension, complexity, heavy atomic count, and index of Butane derivatives is shown in Figure 2:
7. Conclusion
In the meaning of invariants over graphs, graph theoretical indices are used for the computation QSAR and QSPR. A large number indices have been exposed in literature which some of them have placed the application areas such as model physical and molecules in chemistry and pharmacy fields. By introducing the reciprocal status index and reciprocal status co-index of connected graphs as new indices in mathematical part of chemistry, in here, we stated and proved some upper and lower bounds on these new indices. Further, index and co-index of certain standard graphs are obtained. As an important conclusion, it is observed that index has a nice correlation on heavy atomic count 0.98.
Motivating this work, in forthcoming papers, one may sketch the results related to status (co-)index in terms of different parameters and indices. Also, it may be planned to design on the results about status index and co-index of some transmission regular graphs, nanostructures, and certain Archimedean lattice as well as may be planned to create a platform to study different structured indices in provisions of chemical/biological aspects. We are sure this paper will be useful into studies.
Conflicts of Interest
The authors declare that they have no conflicts of interest.