VL Reciprocal Status Index and Co-Index of Graphs

For the vertex set V(G) of a graph G, the sum of reciprocals of the breadth (distances) between the vertex v ∈ V(G) and whole other remaining vertices of G is called reciprocal status of v. In this study, first of all, we introduced the VL reciprocal status index and VL reciprocal status co-index of a graph G. Later, we exposed some sharp bounds for these indices. Furthermore, we determined the VL reciprocal status (co-)index over some standard graphs. Finally, we presented correlations between VL reciprocal status index and some properties of Butane derivatives via a table and illustrated with a figure.


Introduction and Preliminaries
Topological indices are known as special graph invariants, and they are mainly used to calculate QSAR (quantitative structure-activity relationship) and QSPR (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. e 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 G is connected having order n also having size m, then it is quite well known that, while the vertex and edge sets are shown by V(G) and E(G), respectively, an edge connecting (joining) the vertices u 1 and u 2 is shown by u 1 u 2 , and the degree d u of u ∈ E(G) is actually the number of edges incident to u. On the contrary, d(u 1 , u 2 ), which notates the distance between any two vertices u 1 and u 2 , is actually the length of the shortest path connecting u 1 and u 2 , and the maximum breadth between any pair of vertices of G is defined as the diameter di am (G). For unmentioned graph theoretic terminologies, we may refer [1]. e following reminders will be needed for the construction of results in this paper: In [2], for G, the VL index has recently been exposed by where d e � d(u 1 ) + d(u 2 ) − 2 and d f � d(u 1 ) · d(u 2 ) − 2. According to the same reference, the VL index indicates a nice output in the meaning of polychlorinated biphenyl and octane isomers. Actually, VL(G) index can also be written as (2) We will develop some new type of VL indices, see Definition 1 below.
For the vertex u 1 , Harary [3] defined a parameter called status (or, equivalently and transmission), which is the sum of u 1 's breadths to remaining vertices of G, and that is shown by e reciprocal status [11] of u 1 ∈ V(G) is defined by the sum of reciprocals of u 1 's breadths (distances) to remaining vertices in G. is 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 G 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, VL reciprocal status index and VL reciprocal status co-index of a connected graph G. erefore, the following definition plays a key role for this study: Definition 1. Let us consider a connected graph G. erefore, the VL reciprocal status index over G is defined by whereas the VL reciprocal status co-index is defined by For an example of VLRS(G) and VLRS(G) indices, one may see the graph in Figure 1. Clearly, VLRS(G) � 83.625 and VLRS(G) � 17.
is paper is organised as in the following. After this introductory part, we will present some "nice" bounds for the VL reciprocal status index, see equation (6) in Section 2. In Section 3, we will state and prove some results about VL reciprocal status index of standard graphs such as complete K n , complete bipartite K p,q , path P n , cycle C n , wheel W k+1 , windwill F k , and star graph S n . In Section 4 and 5, in parallel to Sections 2 and 3, we will compute some nice bounds for the VL 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 VL 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.

Bounds for VL Reciprocal Status Index
In Section 2, we present some lower and upper bounds for VLRS defined in equation (6) and then characterize the strictness conditions of these bounds. e following is needed to obtain our first main result in this paper: Proof. We first note that, for any u ∈ V(G), there exist d u vertices having distance one from u and the remaining n − 1 − d u vertices having distances at least 2. Upper bound: for u ∈ V(G), which implies that erefore,  Journal of Mathematics For the equality: suppose the diameter of G is given as 1 or 2.
us, equality holds. For the converse part, let di am ≥ 3, and let us assume that (Tex translation failed), Similarly, for u 2 , we get rs( Now, our aim is to reach a contradiction by assuming di am ≥ 3. So, let us separate the partition E(G) into three sets E 1 , E 2 , and E 3 such that Clearly, which gives a contradiction. Further, we get di am(G) ≤ 2. □ By means of direct application of Lemma 1, we can give our first main result after expanding the sums: Theorem 1. Suppose G is connected, having total n vertices, and finally, suppose di am(G) � D. us, and Above equalities again hold if di am(G) � D. e next two consequences of eorem 2 can be given.

Corollary 1.
Let G be as in eorem 2 having m edges. Let δ and Δ be the min. and max. degrees of the elements in V(G), respectively. After that, where p � (n − 1/D) and q � 1 − (1/D).

VL Reciprocal Status Index of Some Standard Graphs
As we mentioned in Section 1, we will determine VL 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 K n on n vertices and m edges, Proof. It is known that rs(u) � n − 1 for any u ∈ V(K n ). Henceforth, by equation (6), we get the result. □ e next proposition exposes the result for VLRS on complete bipartite graphs.

Proposition 2. For a complete bipartite graph K p,q , we have
Proof. One can partition V(K p,q ) into two different subsets V 1 and V 2 with the condition every single edge uv of K p,q , and we let u ∈ V 1 and v ∈ V 2 . erefore, d u � q and d v � p such that |V 1 | � p and |V 2 | � q. Clearly, K p,q has n � p + q vertices and m � pq edges, and also, the diameter di am(k p,q ) ≤ 2. Hence, by the equality part of eorem 2, it is achieved Hence, we get the result on VLRS(K p,q ) as required. □ Now, we can investigate the situation for cycle graphs as follows.
Proposition 3. For a cycle graph C n on n ≥ 3 vertices, Proof Case (i): if n is an even number, then for any vertex u ∈ V(C n ), we have erefore, by equation (6), we obtain while n is even.
Case (ii): if n is an odd number, then for u ∈ V(C n ), we get and then, by equation (6), we reach to as required.

□
As an example for Proposition 3.3, we can give VLRS(C 3 ) � 12 and VLRS(C 4 ) � 22, 5. Journal of Mathematics Another result in special graphs can be presented for a path as in the following.

Proposition 4. For a path graph P n on n vertices,
Proof. Let v 1 , v 2 , . . . , v n be the vertices of P n where v i is connected to v i+1 for i � 1, 2, . . . , n − 1. erefore, for each 1 ≤ i ≤ n, we get erefore,

□
Remaining three propositions will be clarified, VLRS indices for wheel, windmill, and star graphs, respectively.

Proposition 5. Let us consider a wheel graph W k+1 with (k ≥ 3). en,
Proof. If we partition the edge set E(W k+1 ) into two sets E 1 � uv|d u � k, d v � 3 and E 2 � uv|d u � 3, d v � 3 , then it is easy to see that |E 1 | � |E 2 | � k such that di am(W k+1 ) � 2. erefore, by the equality part of eorem 2, □ Journal of Mathematics Proposition 6. For a windmill graph F k with k ≥ 2, we have Proof. In the proof, we will follow the same way as in the proof of Proposition 3. So, let us partition E(F k ) into subsets After that, by considering the equality part of eorem 2, we obtain the result on VLRS(F k ).

□
Since a star graph is actually a tree on n nodes and one node's degree is n − 1 and the remaining n − 1 vertices having degree one, so next result easily follows by eorem 2: Proposition 7. For a star graph S n , VLRS(S n ) � (m/4)(n 2 + 2n − 2).

Bounds for the VL Reciprocal Status Co-Index
Now, we present some bounds for VL reciprocal status coindex 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 eorem 2 and, therefore, will be skipped.

Theorem 2. Let G be connected and has n vertices and also
di am(G) � D. us, 1 8 uv∈E(G) 2s + t +(n − 1)(s + n + 3) ≤ VLRS(G) e following consequences are as important as eorem 2 Corollary 3. Assume that G is connected and has n vertices and m edges and also di am(G) � D. Assume also that δ and Δ is the min. and max. degrees of vertices in V(G), respectively. erefore, where p � (n − 1/D) and q � 1 − (1/D).
Proof. For u ∈ V(G), we know that δ ≤ d u ≤ Δ which implies 2δ ≤ d u + d v ≤ 2Δ. It is also known that the graph G under these assuming conditions has n(n − 1)/2 − m pair of nonadjacent vertices. Substituting d u + d v ≥ 2δ and d u · d v ≥ δ 2 in the lower bound and d u + d v ≤ 2Δ and d(u) · d(v) ≤ Δ 2 in the upper bound of eorem 2, we achieve the result.
Proof. ∀u ∈ V(G), and substituting d u � r in eorem 2, we get the result.

VL Reciprocal Status Co-Index of Some Standard Graphs
With a similar approach as in Section 3, we will determine VL 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 VL reciprocal status co-index, the proof will be omitted.
□ Proposition 10. For a cycle graph C n on n ≥ 3 vertices, we have Proof Case (i): as obtained in Proposition 3 in Case (i), if n is an even number, for any vertex u of C n , we get rs(u) � (2/n) + 2 ((n− 2)/2)

Correlation of VL 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 VL reciprocal status index VLRS has a nice correlation including all three physical properties surface tension, complexity, and heavy atomic count, see Table 1.
Correlation coefficient value of VLRS with surface tension, complexity, and heavy atomic count is 0.95, 0.98, 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 VLRS index of Butane derivatives is shown in Figure 2:

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 VL reciprocal status index and VL 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, VLRS index and VLRS co-index of certain standard graphs are obtained. As an important conclusion, it is observed that VLRS index has a nice correlation on heavy atomic count 0.98.
Motivating this work, in forthcoming papers, one may sketch the results related to VL status (co-)index in terms of different parameters and indices. Also, it may be planned to design on the results about VL 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 VL indices in provisions of chemical/biological aspects. We are sure this paper will be useful into QSPR/QSAR studies.

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