# On Second Gourava Invariant forq-Apex Trees

Authors: Ying Wang; Salma Kanwal; Maria Liaqat; Adnan Aslam; Uzma Bashir
Journal: Journal of Chemistry
 (2022)

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## Abstract

LetG be a simple connected graph. The second Gourava index of graph G is defined as GO2G=∑θϑ∈EGdθ+dϑdθdϑ where dθ denotes the degree of vertex θ. If removal of a vertex of G forms a tree, then G is called an apex tree. Let L⊂VG with ∣L∣=q. If removal of L from VG forms a tree and any other subset of VG whose cardinality is less than ∣L∣ does not form a tree, then G is known as q-apex tree. In this paper, we have calculated upper bound for 2nd Gourava index with respect to q-apex trees.


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## Body

## 1. Introduction

Topological index is basic tool in chemical modeling. In molecular graph, atoms are considered as vertices and chemical bonds as edges. Short graph is a combination of vertices and edges. First topological index was Wiener index introduced by Wiener in1947 to compare the boiling points of few alkane isomers. He observed that this index is highly correlated with the boiling point of alkanes. Later study on QSAR manifested that this index is also helpful to correlate with other quantities like density, critical point, and surface tension. The mathematical formula of this index is
(1)WG=∑θ,ϑdGθ,ϑ,where dGθ,ϑ denotes the distance between the vertices θ and ϑ in G. The detailed study about this invariant is given in [1]. Among degree-based topological indices, the most studied indices are first Zagreb index and second Zagreb index [2]. The first Zagreb index is defined as the sum of square of degrees of all the vertices of a graph, where in second Zagreb index, we take the sum of product of degrees of all those vertices of graph which are linked by an edge. For more information about these chemical invariants, see [3]. The first and second Gourava indices were presented by Kulli in 2017 [4]. These indices are defined as
(2)GO1G=∑θϑ∈EGdθ+dϑ+dθdϑ,(3)GO2G=∑θϑ∈EGdθ+dϑdθdϑ.A topological index is a numerical number associated with a molecular graph which has significant applications in chemical graph theory, because it is used as a molecular descriptor to investigate physical as well as chemical properties of chemical structure. Therefore, it is a powerful technique in avoiding high cost and long-term laboratory experiments. There are more than3,000 topological invariants registered till now. Most of these indices have their applications in chemical graph theory. In these molecular descriptors, Gourava and hyper-Gourava invariants are used to find out the physical and chemical properties (such as entropy, acentric factor, and DHAVP) of octane isomers. The first and second Gourava invariants are highly correlated with entropy and acentric factors, respectively.All graphs considered in our study are simple and connected. LetG be a simple graph with vertex set V and edge set E. For a graph G, the degree of a vertex θ is defined as the number of edges attached to it. The smallest degree of a vertex in G is denoted by δG. The vertex in a graph whose degree is one is known as pendent vertex. The neighborhood of a vertex θ is the set containing all nodes attached with θ, denoted by Nθ. There are two types of neighborhood, open neighborhood and closed neighborhood. If Nθ includes all the other nodes except θ, then it is called open neighborhood but if it includes the node θ, then it is called closed neighborhood. Closed neighborhood of θ is defined as Nθ=Nθ∪θ. If we remove one vertex (say) θ from G, then the resulting graph is denoted by G−θ. If we remove subset (say) X′⊂V of vertex set of graph G, then the obtained graph is denoted by G−X′. In molecular graph theory, an acyclic connected graph having order λ is known as tree and is denoted by Tλ∗. A tree with order λ is said to be star if central node has degree λ−1 while all other nodes are pendent. In other way, a complete bipartite graph κ1,λ−1 is called a star of order λ. Let G1 and G2 be two vertex disjoint graphs then their join is denoted by G1+G2 having vertex set as a union of their vertex sets, and the edge set contains all the edges of G1 and G2 and all those edges obtained by linking each node of G1 with each node of G2.Ali et al. [5] investigated second and third modified Zagreb invariants for T-sum graph operation. Using the obtained results, they computed the second and third modified Zagreb of certain well-known chemical structures like alkanes. Cao et al. [6] give an exact formulas for the upper bounds of modified first Zagreb connection index and second Zagreb connection index for several binary graph operations like corona, Cartesian, and lexicographic product. Using the obtained closed formulas, they computed these invariants for several well-known graphs. In [7], the Gourava index for several graph operations is presented.In molecular graph theory, apex graphs play a significant role, which can be elucidated as if removal of single vertex or subset of vertex set from a graphG yields a planar graph, then the graph G is known as apex graph [8]. Embedding of apex graphs having face width three are characterized in [9]. By using the same idea, one can explain the apex and q-apex trees. If we remove a vertex from G and the resulting graph is a tree, then that G is called an apex tree [10]. Similarly, if we remove a subset (say) X′ of vertex set of cardinality q from G and it results in a tree, then G is known as q-apex tree, provided that the removal of any other subset of vertex set whose cardinality is less than q does not form a tree [10]. A tree is known as trivial apex tree or 0-apex tree.Therefore, a non-trivial apex tree is one which is not a tree itself but it can be converted into a tree by removing a single or number of vertices. In short,1-apex tree is a graph which can be made a tree by removing single vertex as shown in Figure 1 (2-apex tree can be converted into a tree by removing 2 vertices as shown in Figure 2) and so on. In fact, apex trees are quasitrees which were introduced by Xu et al. [11]. Xu et al. [12] determined bounds on harary indices in case of apex trees. In 2018, Akhter et al. determined k-apex trees with extremal first reformulated Zagreb index [13].Figure 1
1-apex tree.Figure 2
2-apex tree.


## 2. Upper Bound ofGO2 for q-Apex Trees

Letλ≥3, q≥1, and T∗λ, and Tq∗λ denotes the set of all no-trivial apex trees and q-apex trees on λ vertices, respectively.Lemma 1.
LetT∗ be a tree of order λ, and then
(4)GO2T∗≤λ−12λ
with equality holds if and only ifT∗=Sλ.Proof.
SinceT∗ is a tree, it follows that for any edge θϑ∈ET∗, we have Nθ∩Nϑ=∅. Hence, dT∗θ+dT∗ϑ=∣Nθ∣+∣Nϑ∣≤λ. Now, using the value of dT∗θ+dT∗ϑ in the definition of second Gourava index, we get
(5)GO2T∗=∑θϑ∈ET∗dθ+dϑdθdϑ≤λ∑θϑ∈ET∗dθdϑ≤λλ−12.Example 2.
LetBSa,b be bistar graph obtained by joining the apex vertices of two star graph K1,a and K1,b on different vertices with a,b≥1, and Sλ,a be the graph obtained by joining a−1 pendent edges to the end vertex of path Pλ−a+1. Figure 3 depicts the graphs of BS4,4 and S10∗, respectively. By definition of second Gourava index, we have GO2S10∗=810>GO2BS4,4=490. In Table 1, we have computed the second Gourava index of some classes of trees on 8 vertices. Observe that GO2S8∗ has the maximum value among all other trees.Lemma 3.
Letu′,v′∈VG be two nonadjacent vertices of G, and then
(6)GO2G+u′v′>GO2G.Proof.
By definition ofGO2G, we have
(7)GO2G=∑θϑ∈EGdθ+dϑdθdϑ,GO2G+u′v′=∑θϑ∈EGdθ+dϑdθdϑ+du′+dv′du′dv′.
Sincedu′,dv′>0, it follows that GO2G+u′v′>GO2G.Lemma 4.
LetG∈T∗λ with maximum GO2 value, and let x′∈VG be an apex vertex, and then
(8)δG=2,(9)dx′=λ−1.Proof.
(1)
SupposeδG=1 and y′ is a leaf node in G, then x′y′ is not an edge in G and G+x′y′∈T∗λ. Then, by Lemma 3, GO2G+x′y′>GO2G is a contradiction. Next, we prove that δG>2 is not possible. Suppose dθ≥3 for all θ∈VG, then for any θ∈VG, the degree of all vertices in G−θ is greater or equal to two. This implies that G−θ is not a tree. Hence, δG=2.(2)
Supposedx′<λ−1, then there exists a vertex y′∈VG with x′y′∉EG. Now, G+x′y′∈T∗λ and GO2G+x′y′>GO2G are a contradiction. Hence, dx′=λ−1.Lemma 5.
LetG1 and G2 be two graphs on disjoint vertex sets with ∣VG1∣=π1, ∣VG2∣=π2, ∣EG1∣=ϖ1, and EG2=ϖ2. Then,
(10)GO2G1+G2=GO2G1+GO2G2+4π2M2G1+4π1M2G2+3π22M1G1+3π12M1G2+2π23ϖ1+2π13ϖ2+FG1π2+FG2π1+2M1G1ϖ2+2M1G2ϖ1+π1π2M1G1+M1G2+8ϖ1ϖ2π1+π2+4π1π2ϖ1π2+ϖ2π1+2π1π2ϖ1π1+ϖ2π2+π12π22π1+π2.Proof.
By definition of second Gourava index,(11)GO2G1+G2=∑θϑ∈EG1+G2dG1+G2θ+dG1+G2ϑdG1+G2θdG1+G2ϑ.
Now, using the definition of joint of two graphs having disjoint vertex sets, we get(12)GO2G1+G2=∑θϑ∈EG1dG1+G2θ+dG1+G2ϑdG1+G2θdG1+G2ϑ+∑θϑ∈EG2dG1+G2θ+dG1+G2ϑdG1+G2θdG1+G2ϑ+∑θϑ;θ∈VG1,ϑ∈VG2dG1+G2θ+dG1+G2ϑdG1+G2θdG1+G2ϑ.
Let(13)A1=∑θϑ∈EG1dG1+G2θ+dG1+G2ϑdG1+G2θdG1+G2ϑ,B1=∑θϑ∈EG2dG1+G2θ+dG1+G2ϑdG1+G2θdG1+G2ϑ,C1=∑θϑ∈θϑ;θ∈VG1,ϑ∈VG2dG1+G2θ+dG1+G2ϑdG1+G2θdG1+G2ϑ.
Now, using the factdG1+G2θ=dG1θ+π2;θ∈VG1dG2θ+π1;ϑ∈VG2 the value of A1 can be computed as
(14)A1=∑θϑ∈EG1dG1+G2θ+dG1+G2ϑdG1+G2θdG1+G2ϑ=∑θϑ∈EG1dG1θ+dG1ϑ+2π2dG1θ+π2dG1ϑ+π2=∑θϑ∈EG1dG12θdG1ϑ+dG1θ2π2+dG1θdG1ϑπ2+dG1θπ22+dG12ϑdG1θ+dG1θdG1ϑπ2+dG12ϑπ2+dG1θπ22+2π2dG1θdG1ϑ+2π22dG1θ+2π22dG1ϑ+2π2=∑θϑ∈EG1dG12θdG1ϑ+dG12ϑdG1θ+π2∑θϑ∈EG1dG12θ+dG12ϑ+4π2∑θϑ∈EG1dG1θdG1ϑ+3π22dG1θ+dG1ϑ+2π23∑θϑ∈EG1.
Similarly,(15)B1=GO2G2+π1FG2+4π1M2G2+3π12M1G2+2π13ϖ2.
The value ofC1 can be calculated as
(16)C1=∑θϑ∈θϑ;θ∈VG1,ϑ∈VG2dG1+G2θ+dG1+G2ϑdG1+G2θdG1+G2ϑ=∑θ∈VG1,ϑ∈VG2dG1θ+π2+dG2ϑ+π1dG1θ+π2dG2ϑ+π1=∑θ∈VG1,ϑ∈VG2dG12θdG2ϑ+dG12θπ1+dG2ϑdG1θπ2+dG1θπ2π1+dG1θdG22ϑ+dG1θdG2ϑπ1+dG22ϑπ2+dG2ϑπ1π2+dG1θdG2ϑπ1+dG1θπ12+dG2ϑπ1π2+π12π2+dG1θdG2ϑπ2+dG1θπ1π2+dG2ϑπ22+π1π22=2M1G1ϖ2+2M1G2ϖ1+π1π2M1G1+M1G2+8ϖ1ϖ2π1+π2+4π1π2ϖ1π2+ϖ2π1+2π1π2ϖ1π1+ϖ2π2+π12π22π1+π2.
Now, using the values ofA1,B1, and C1 in Equation (2), we get
(17)GO2G1+G2=GO2G1+GO2G2+4π2M2G1+4π1M2G2+3π22M1G1+3π12M1G2+2π23ϖ1+2π13ϖ2+FG1π2+FG2π1+2M1G1ϖ2+2M1G2ϖ1+π1π2M1G1+M1G2+8ϖ1ϖ2π1+π2+4π1π2ϖ1π2+ϖ2π1+2π1π2ϖ1π1+ϖ2π2+π12π22π1+π2.Theorem 6.
LetG∈T∗λ and λ≥5, and then
(18)GO2G≤6λ3−14λ2+2λ+6,with equality holds if G=κ1+Sλ−1.Proof.
LetG∈T∗λ with maximum GO2 value. By Lemma 4, we have G=κ1+Tλ−1∗. Let the cardinality of vertex sets of κ1 and Tλ−1∗ be ρ1 and ρ2 where cardinality of their edge sets is ς1 and ς2, respectively. Now by, using Lemma 1., we have
(19)GO2κ1+Tλ−1∗=GO2κ1+GO2Tλ−1∗+4ρ2M2κ1+4ρ1M2Tλ−1∗+3ρ22M1κ1+3ρ12M1Tλ−1∗+2ρ23ς1+2ρ13ς2+Fκ1ρ2+FTλ−1∗ρ1+2M1κ1ς2+2M1Tλ−1∗ς1+ρ1ρ2M1κ1+M1Tλ−1∗+8ς1ς2ρ1+ρ2+4ρ1ρ2ς1ρ2+ς2ρ1+2ρ1ρ2ς1ρ1+ς2ρ2+ρ12ρ22ρ1+ρ2≤λ−12λ+3+7λ2−3λ+2+3λ3−12λ2+15λ−6+λ3−6λ2+15λ−14+λ3−2λ2+λ=6λ3−14λ2+2λ+6.Example 7.
LetG1=κ1+S6−1, G2=κ1+T5∗ as depicted in Figure 4 and G3 be 1-apex tree of order 6 as shown in Figure 5. Then, the values of second Gourava index of these 1-apex trees are GO2G1=810, GO2G1=720, and GO2G1=214. Table 2 depicts the values second Gourava index of some graphs in the class of 1-apex trees of order 8. Observe that GO2ofκ1+S7 is maximum among all other graphs.Theorem 8.
Letq≥2, λ≥5, and G∈Tq∗λ, and then
(20)GO2G≤q+q2λ−13+λ+qλ−1λq+12−q+13with equality holds if G=κq+Sλ−qProof.
We prove it by induction the value ofq. For q=1, the result follows from Theorem 6. Suppose the result is true for q−1 apex trees, let G∈Tq∗λ with maximum GO2 and Vq⊂VG be the set of q-apex vertices. Since GO2G+u′v′>GO2G for any nonadjacent edges u′,v′∈VG, it follows that Vq is a clique in G. Hence, du′=λ−1 for all u′∈Vq, and the number of edges m′ of the graph G is
(21)m′=12q2+q+q+1λ−q+12.
Letx′ be an apex vertex and Vq−1=Vq−x′. Observe that dx′=λ−1 and G−x′ are an q−1-apex tree. Then,
(22)GO2G−x′=∑θϑ∈EG−x′dGθ−1+dGϑ−1dGθ−1dGϑ−1=∑θϑ∈EG−x′dGθ+dGϑ−2dGθdGϑ−dGθ−dGϑ+1=∑θϑ∈EG−x′dG2θdGϑ−dG2θ−dGθdGϑ+dGθ+dGθdG2ϑ−dGθdGϑ−dG2ϑ+dGϑ−2dGθdGϑ+2dGθ+2dGϑ−2=∑θϑ∈EG−x′dG2θdGϑ+dGθdG2ϑ−∑θϑ∈EG−x′dG2θ+dG2ϑ−4∑θϑ∈EG−x′dGθdGϑ+3∑θϑ∈EG−x′dGθ+dGϑ−∑θϑ∈EG−x′=∑θϑ∈EG−x′dG2θdGϑ+dGθdG2ϑ−∑θϑ∈EG−x′dG2θ+dG2ϑ+∑x′θ∈EGλ−12dGθ+λ−1dG2θ−∑x′θ∈EGλ−12+dG2θ−∑x′θ∈EGλ−12dGθ+λ−1dG2θ+∑x′θ∈EGλ−12+dG2θ−4∑θϑ∈EG−x′dGθdGϑ+3∑θϑ∈EG−x′dGθ+dGϑ−4∑x′θ∈EGλ−1dGθ+3∑x′θ∈EGλ−1+dGθ+4∑x′θ∈EGλ−1dGθ−3∑x′θ∈EGλ−1+dGθ−2m′−λ+1=∑θϑ∈EGdG2θdGϑ+dGθdG2ϑ−∑θϑ∈EGdG2θ+dG2ϑ−4∑θϑ∈EGdGθdGϑ+3∑θϑ∈EGdGθ+dGϑ−∑x′θ∈EGλ−12dGθ+λ−1dG2θ+∑x′θ∈EGλ−12+dG2θ+4∑x′θ∈EGλ−1dGθ−3∑x′θ∈EGλ−1+dGθ−2m′−λ+1.(23)GO2G−x′=GO2G−FG−4M2G+3M1G−λ−12∑θ∈VG−x′dGθ+λ−1−λ−1−λ−1∑θ∈VG−x′dG2θ+λ−12−λ−12+λ−13+∑θ∈VG−x′dG2θ+λ−12−λ−12+4λ−1∑θ∈VG−x′dGθ+λ−1−λ−1−3λ−12−3∑θ∈VG−x′dGθ+λ−1−λ−1−2m′−λ+1=GO2G−FG−4M2G+3M1G−λ−12∑θ∈VGdGθ−λ−1−λ−1∑θ∈VGdG2θ−λ−12+λ−13∑θ∈VGdG2θ−λ−12+4λ−1∑θ∈VGdGθλ−1−3λ−12−3∑θ∈VG−x′dGθ−λ−1−2m′−λ+1=GO2G−FG−4M2G+3M1G−λ−122m′−λ+1−λ−1M1G−λ−12+λ−13+M1G−λ−12+4λ−12m′−λ+1−3λ−12−32m′−λ+1−2m′−λ+1=GO2G−FG−4M2G+M1G5−λ+3λ−13−8λ−12+8m′λ−16m′+5λ−5−2λ−12m′.
Hence, we get(24)GO2G=GO2G−x′+FG+4M2Gλ−5M1G−3λ−13+8λ−12−8m′λ+16m′−5λ+5+2λ−12m′.
SinceGO2G−x′=q2−qλ−13+λ−q−1λ+q−2q2λ−2q and for q≥2,λ≥5, we have
(25)M1G≤q+1λ2−2λ+1+λq+12−q+13,(26)M2G≤12q2+qλ2−2λ+1+λ−1λq+12−q+13,(27)FG≤q+1λ3−3λ2+3λ−1+λq+13−q+14m′=12q2+q+q+1λ−q+12.
Using all the values in the above equation, we get(28)GO2G≤q2−qλ−13+λ−q−1λ+q−2q2λ−2q+q+1λ3−3λ2+3λ−1+λq+13−q+14+412q2+qλ2−2λ+1+λ−1λq+12−q+13+λ−5q+1λ2−2λ+1+λq+12−q+13−3λ−13+8λ−12−8λ12q2+q+q+1λ−q+12+1612q2+q+q+1λ−q+12−5λ+5+2λ−1212q2+q+q+1λ−q+12.
After simplification, we have(29)GO2G≤q+q2λ−13+λ+qλ−1λq+12−q+13Example 9.
LetG1 and G2=κ2+S5−2 be 2-apex trees of order 5 as shown in Figure 6. Then, GO2G1=224 and GO2G2=888. Table 3 depicts the values of some 2 apex trees on 9 vertices. Observe that GO2 value of κ2+S7 is the maximum among all others.Figure 3
Trees of order10.Table 1
GO2T∗ of certain trees T∗ of order 8.
T∗GO2T∗S8392BS5,1312BS4,2264BS3,3248S8,5212S8,4146S8,3108Figure 4
1-apex tree.Figure 5
1-apex tree.Table 2
GO2 value of some 1-apex trees of order 8.
κ1+Tλ−1∗GO2κ1+Tλ−1∗κ1+S72198κ1+BS4,11962κ1+S7,41758κ1+S7,31634κ1+BS3,21277Figure 6
1-apex tree.Table 3
GO2 values of some 2-apex trees on 9 vertices.
κ2+Tλ−2∗GO2κ2+Tλ−2∗κ2+S77824κ2+BS4,17344κ2+S7,47108κ2+S7,36712κ2+BS3,26684


## 3. Conclusion

In our present discussion, we have determined the maximum value of second Gourava invariant forq-apex trees. In future, it would be interesting to find the same results for those chemical invariants which are not investigated till now.



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