# VL Reciprocal Status Index and Co-Index of Graphs

Authors: Veerebradiah Lokesha; Suvarna; Ahmet Sinan Cevik; Ismail Naci Cangul
Journal: Journal of Mathematics
 (2021)

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## Abstract

For the vertex setVG of a graph G, the sum of reciprocals of the breadth (distances) between the vertex v∈VG 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.


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## Body

## 1. Introduction and Preliminaries

Topological indices are known as special graph invariants, and they are mainly used to calculateQSAR (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. 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 supposeG 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 VG and EG, respectively, an edge connecting (joining) the vertices u1andu2 is shown by u1u2, and the degree du of u∈EG is actually the number of edges incident to u. On the contrary, du1,u2, which notates the distance between any two vertices u1 and u2, is actually the length of the shortest path connecting u1 and u2, and the maximum breadth between any pair of vertices of G is defined as the diameter diamG. 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 G, the VL index has recently been exposed by(1)VLG=12∑uv∈EGde+df+4,where de=du1+du2−2 and df=du1⋅du2−2. According to the same reference, the VL index indicates a nice output in the meaning of polychlorinated biphenyl and octane isomers. Actually, VLG index can also be written as(2)VLG=12∑u1u2∈EGdu1+du2+du1⋅du2.We will develop some new type ofVL indices, see Definition 1 below.For the vertexu1, Harary [3] defined a parameter called status (or, equivalently and transmission), which is the sum of u1’s breadths to remaining vertices of G, and that is shown by(3)σu1=∑u2∈VGdu1,u2.The reciprocal status [11] of u1∈VG is defined by the sum of reciprocals of u1’s breadths (distances) to remaining vertices in G. This is shown by(4)rsu1=∑u2∈VG1du1,u2.As it mentioned at the beginning, the oldest index, which is namely the Wiener index [17], is given by(5)WG=∑u1,u2⊆VGdu1,u2=12∑u1∈VGσu1.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.Therefore, the following definition plays a key role for this study:Definition 1.
Let us consider a connected graphG. Therefore, the VL reciprocal status index over G is defined by(6)VLRSG=12∑u1u2∈EGrsu1+rsu2+rsu1⋅rsu2,whereas the VL reciprocal status co-index is defined by(7)VLRS¯G12∑u1u2∉EGrsu1+rsu2+rsu1⋅rsu2.For an example ofVLRSG and VLRS¯G indices, one may see the graph in Figure 1. Clearly, VLRSG=83.625 and VLRS¯G=17.Figure 1
A graph with five vertices.This paper is organised as in the following. After this introductory part, we will present some “nice” bounds for theVL 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 Kn, complete bipartite Kp,q, path Pn, cycle Cn, wheel Wk+1, windwill Fk, and star graph Sn. 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.


## 2. Bounds forVL Reciprocal Status Index

In Section 2, we present some lower and upper bounds forVLRS 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.
AssumeG is connected owning n vertices whereas diamG=D. Thus,(8)18∑uv∈EG2s+t+n−1s+n+3≤VLRSG≤12∑uv∈EGp2+p+1+pqs+q2t,where s=du+dv, t=du⋅dv, p=n−1/D, and q=1−1/D. Both equalities hold if diamG≤2.Proof.
We first note that, for anyu∈VG, there exist du vertices having distance one from u and the remaining n−1−du vertices having distances at least 2.
Upper bound: foru∈VG,(9)rsu≥du+1Dn−1−du=n−1D+du1−1D,which implies that(10)VLRSG=12∑uv∈EGrsu+rsv+rsu.rsv≤12∑uv∈EGp2+p+1+pqs+q2t,where s=du+dv, t=du⋅dv, p=n−1/D, and q=1−1/D.
Lower bound: again, foru∈VG, there is(11)rsu≤du+12n−1−du=12du+n−1.
Therefore,(12)VLRSG=12∑uv∈EGrsu+rsv+rsu.rsv≥18∑uv∈EG2s+t+n−1s+n+3,where s=du+dv, t=du⋅dv, p=n−1/D, and q=1−1/D as required.
For the equality: suppose the diameter ofG is given as 1 or 2. Thus, equality holds. For the converse part, let diam≥3, and let us assume that Tex translation failed, where s=du+dv, t=du⋅dv, p=n−1/D, and q=1−1/D. Thus, there exists one pair of vertices (at least) u1, u2 having du1,u2≥3. So,(13)rsu1≤du1+13+12n−2−du1=n2−23+du12.
Similarly, foru2, we get rsu2≤n/2−2/3+du2/2 but for remaining whole vertices u∈VG, it is obtained rsu≤n/2−1/2+du/2.
Now, our aim is to reach a contradiction by assumingdiam≥3. So, let us separate the partition EG into three sets E1, E2, and E3 such that(14)E1=u1v|rsu1≤n2−23+du12andrsv≤n2−12+dv2,E2=u1v|rsu2≤n2−23+du22andrsv≤n2−12+dv2,(15)E3=uv|rsu≤n2−12+du2andrsv≤n2−12+dv2.
Clearly,E1=du1, E2=du2, and E3=m−du1−du2. Therefore,(16)VLRSG=12∑uv∈EGrsu+rsv+rsu.rsv=12∑u1v∈E1rsu1+rsv+rsu1.rsv+12∑u2v∈E2rsu2+rsv+rsu2.rsv+12∑uv∈E3rsu+rsv+rsu.rsv>18∑uv∈EG2s+t+n−1s+n+3,which gives a contradiction. Further, we get diamG≤2.By means of direct application of Lemma1, we can give our first main result after expanding the sums:Theorem 1.
SupposeG is connected, having total n vertices, and finally, suppose diamG=D. Thus,(17)18n+1d2+12Ad⋅d+n−1n+3≤VLRSG,and(18)VLRSG≤12mp2+p+1+pqd2+q22Ad⋅d.Above equalities again hold ifdiamG=D.The next two consequences of Theorem2 can be given.Corollary 1.
LetG be as in Theorem 2 having m edges. Let δ and Δ be the min. and max. degrees of the elements in VG, respectively. After that,(19)m8δ+nδ+n+2−3≤VLRSG≤m2p2+p+2qΔ1+p+q2Δ2,where p=n−1/D and q=1−1/D.Proof.
For the vertexu∈VG, it is clear that δ≤du≤Δ. Therefore, by substituting du+dv≥2δ, du.dv≥δ2 in the lower bound, and du+dv≤2Δ but du⋅dv≤Δ2 in the upper bound of Theorem 2, we get the required result.Corollary 2.
Suppose thatG is connected r-regular on n vertices and m edges. Also, suppose that diamG=D. Thus,(20)m8r+nr+n+2−3≤VLRSG≤m2p2+p+2rq1+p+q2r2,where p=n−1/D and q=1−1/D. Moreover, equalities in above hold if D≤2.


## 3.VL Reciprocal Status Index of Some Standard Graphs

As we mentioned in Section 1, we will determineVL 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 graphKn on n vertices and m edges, VLRSKn=mn2−1/2.Proof.
It is known thatrsu=n−1 for any u∈VKn. Henceforth, by equation (6), we get the result.The next proposition exposes the result forVLRS on complete bipartite graphs.Proposition 2.
For a complete bipartite graphKp,q, we have(21)VLRSKp,q=pq82p+q+pq+p+q−12p+2q+3.Proof.
One can partitionVKp,q into two different subsets V1 and V2 with the condition every single edge uv of Kp,q, and we let u∈V1 and v∈V2. Therefore, du=q and dv=p such that V1=p and V2=q. Clearly, Kp,q has n=p+q vertices and m=pq edges, and also, the diameter diamkp,q≤2. Hence, by the equality part of Theorem 2, it is achieved(22)VLRSKp,q=pq82q+p+q.p+p+q−1q+p+p+q+3=pq82p+q+pq+p+q−12p+2q+3.
Hence, we get the result onVLRSKp,q as required.Now, we can investigate the situation for cycle graphs as follows.Proposition 3.
For a cycle graphCn on n≥3 vertices,(23)VLRSCn=n24n+4∑i=1n−2/21i+2n+2∑i=1n−2/21i2,ifnis even,2n∑i=1n−2/21i1+∑i=1n−2/21i,ifnis odd.Proof.
Case (i): ifn is an even number, then for any vertex u∈VCn, we have(24)rsu=21+12+⋯+1n−2/2+1n/2=2n+2∑i=1n−2/21i.
Therefore, by equation (6), we obtain(25)VLRSCn=n24n+4∑i=1n−2/21i+2n+2∑i=1n−2/21i2,while n is even.Case (ii): ifn is an odd number, then for u∈VCn, we get(26)rsu=21+12+⋯+1n−2/2=2∑i=1n−2/21i,and then, by equation (6), we reach to(27)VLRSCn=2n∑i=1n−2/21i1+∑i=1n−2/21i,as required.As an example for Proposition 3.3, we can giveVLRSC3=12 and VLRSC4=22,5.Another result in special graphs can be presented for a path as in the following.Proposition 4.
For a path graphPn on n vertices,(28)VLRSPn==12nn−1+2∑i=1n−21i+∑i=1n−11i+∑i=1n−11i.∑j=1n−21j+12∑i=2n−2nin−i++2∑j=1i−11j+∑j=1n−i−11j+∑j=1i−11j+∑j=1n−i1j∑j=1i1j+∑j=1n−i−11j.Proof.
Letv1,v2,…,vn be the vertices of Pn where vi is connected to vi+1 for i=1,2,…,n−1. Therefore, for each 1≤i≤n, we get(29)rsv1=∑i=1n−11i=rsvnandrsvi=∑j=1i−11j+∑j=1n−i1i,2≤i≤n−1.
Therefore,(30)VLRSPn=12∑uv∈EPnrsu+rsv+rsu.rsv=12rsv1+rsv2+rsv1⋅rsv2+12∑i=2n−2rsvi+rsvi+1+rsvi⋅rsvi+1+rsvn−1+rsvn+rsvn−1⋅rsvn,which implies the required result on VLRSPn.Remaining three propositions will be clarified,VLRS indices for wheel, windmill, and star graphs, respectively.Proposition 5.
Let us consider a wheel graphWk+1 with k≥3. Then,(31)VLRSWk+1=k83k2+22k+27.Proof.
If we partition the edge setEWk+1 into two sets E1=uv|du=k,dv=3 and E2=uv|du=3,dv=3, then it is easy to see that E1=E2=k such that diamWk+1=2. Therefore, by the equality part of Theorem 2,(32)VLRSWk+1=18∑uv∈EWn+12du+dv+du⋅dv+kdu+dv+k+4=18∑uv∈E12du+dv+du⋅dv+kdu+dv+k+4+18∑uv∈E22du+dv+du⋅dv+kdu+dv+k+4=k83k2+22k+27.Proposition 6.
For a windmill graphFk with k≥2, we have(33)VLRSFk=k25k2+14k+5.Proof.
In the proof, we will follow the same way as in the proof of Proposition3. So, let us partition EFk into subsets E1=uv|du=2k,dv=2 and E2=uv|du=2,dv=2 which gives E1=2k and E2=k. Also, diamFk=2. After that, by considering the equality part of Theorem 2, we obtain the result on VLRSFk.Since a star graph is actually a tree onn nodes and one node’s degree is n−1 and the remaining n−1 vertices having degree one, so next result easily follows by Theorem 2:Proposition 7.
For a star graphSn, VLRSSn=m/4n2+2n−2.


## 4. Bounds for theVL Reciprocal Status Co-Index

Now, we present some bounds forVL 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 Theorem2 and, therefore, will be skipped.Theorem 2.
LetG be connected and has n vertices and also diamG=D. Thus,(34)18∑uv∈EG2s+t+n−1s+n+3≤VLRS¯G≤12∑uv∈EGp2+p+1+pqs+q2t,where s=du+dv, t=du⋅dv, p=n−1/D, and q=1−1/D. Moreover, equality holds on both sides if D≤2.The following consequences are as important as Theorem2Corollary 3.
Assume thatG is connected and has n vertices and m edges and also diamG=D. Assume also that δ and Δ is the min. and max. degrees of vertices in VG, respectively. Therefore,(35)116nn−1−2mδ+nδ+n+2−3≤VLRS¯GandVLRS¯G≤14n9−2mp2+p+2qδ1+p+q2Δ2,where p=n−1/D and q=1−1/D.Proof.
Foru∈VG, we know that δ≤du≤Δ which implies 2δ≤du+dv≤2Δ. It is also known that the graph G under these assuming conditions has nn−1/2−m pair of nonadjacent vertices. Substituting du+dv≥2δ and du⋅dv≥δ2 in the lower bound and du+dv≤2Δ and du⋅dv≤Δ2 in the upper bound of Theorem 2, we achieve the result.Corollary 4.
LetG be connected r-regular and has n vertices, and also, let diamG=D. Then,(36)m8r+nr+n+2−3≤VLRS¯G≤m2p2+p+2rq1+p+r2q2,where p=n−1/D and q=1−1/D. Equalities hold if diamG=D.Proof.
∀u∈VG, and substituting du=r in Theorem 2, we get the result.


## 5.VL Reciprocal Status Co-Index of Some Standard Graphs

With a similar approach as in Section3, 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 andVL reciprocal status co-index, the proof will be omitted.Proposition 8.
ForKn, VLRS¯Kn=0.Proposition 9.
ForKp,q, there exists(37)VLRS¯Kp,q=pp−142q+p−1+q+p−122+qq−142p+q−1+p+q−122.Proof.
ForV1=p and V2=q, let us assume V1 and V2 are the partite sets of VKp,q with every single edge of Kp,q owning one end in V1 and other end in V2. If u∈V1, then rsu=q+1/2p−1, and also, if u∈V2, then rsu=p+1/2q−1. Therefore, for u,v∈V1, rsu+rsv=2q+p−1, and for u,v∈V2, rsu+rsv=2p+q−1. Hence, by equation (7), we obtain(38)VLRS¯G=12∑u,v⊆V1rsu+rsv+rsu.rsv+12∑u,v⊆V2rsu+rsv+rsu.rsv=pp−142q+p−1+q+p−122+qq−142p+q−1+p+q−122,as required.Proposition 10.
For a cycle graphCn on n≥3 vertices, we have(39)VLRS¯Cn=nn−1−2n44n+4∑i=1n−2/21i+2n+2∑i=1n−2/21i2,ifnis even,nn−1−2n∑i=1n−2/21i1+∑i=1n−2/21i,ifnis odd.Proof.
Case (i): as obtained in Proposition3 in Case (i), if n is an even number, for any vertex u of Cn, we get rsu=2/n+2∑i=1n−2/21/i. Therefore,(40)VLRS¯Cn=12∑uv∉ECnrsu+rsv+rsu.rsv=nn−1−2n44n+4∑i=1n−2/21i+2n+2∑i=1n−2/21i2.Case (ii): similarly, as we obtained in Proposition3 in Case (ii), if n is odd, then for u∈VCn, we have rsu=2∑i=1n−2/21/i. Therefore, we get(41)VLRS¯Cn=12∑uv∉ECnrsu+rsv+rsu.rsv=nn−1−2n∑i=1n−2/21i+1+∑i=1n−2/21i.Proposition 11.
For a wheel graphWk+1k≥4, VLRS¯Wk+1=1/16k2−3kk2+10k+21.Proof.
It is known that the nonadjacent pairs of vertices of the wheel graphWk+1 have degree 3, and there are actually total kk+1/2−2k pairs of nonadjacent vertices. Also, diamWk+1=2. Therefore, by the equality part of Theorem 2,(42)VLRS¯Wk+1==18∑uv∉EWk+12du+dv+du⋅dv+kdu+dv+k+4=116k2−3kk2+10k+21.
Hence, the result.Proposition 12.
For a windmill graphFk with k≥2, VLRS¯Fk=1/164k2−4k4k2+16k+12.Proof.
It is already known that the nonadjacent pairs of vertices of the windmill graphFk have degrees 2, and there are total 2k2k+1/2−3k such pairs in Fk. Further, diamFk=2. Therefore, again by the equality part of Theorem 2,(43)VLRS¯Fk=18∑uv∉EFk2du+dv+du⋅dv+2kdu+dv+2k+4=1164k2−4k4k2+16k+12.as required.


## 6. Correlation ofVL 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 thatVL reciprocal status index VLRS has a nice correlation including all three physical properties surface tension, complexity, and heavy atomic count, see Table 1.Table 1
Correlation ofVLRS index and some properties of Butane derivatives.
Name of the compoundSurface tensionComplexityHeavy atomic countVLRS1,4-Butanedithiol31.117.5637.99022-Butanone22.938.5528.33331,3-Butanediol34.928.7642.0138Butane dinitrile40.792680.0276Butanediamide5396.6879.8916Butane-1-sulfonamide41.9133885.04021-Butanethiol24.813.1525.15281,4-Diaminobutane35.817.5637.9902Butane-1,4-disulfonic acid77.926612191.8427Butyraldehyde23.124.8525.15282,3-Butanedione27.371.5646.6667Butanedihydrazide5911910137.20341-Butanesulfonyl chloride36.4133885.0402Correlation coefficient value ofVLRS with surface tension, complexity, and heavy atomic count is 0.95,0.98, and 0.97, respectively, see Figure 2.Figure 2
Correlation ofVL reciprocal status index with physical properties of Butane derivatives.Using the data of Table1, the scatter plot between the surface tension, complexity, heavy atomic count, and VLRS 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 theVL 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 toVL 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.



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