GOURAVA AND HYPER-GOURAVA INDICES OF SOME CACTUS CHAINS

The physico-chemical characteristics of molecules are theoretically explored using the theory of graphs and mathematical chemistry. A graph’s topological index is a numerical value derived from the graph mathematically. The Gourava and hyper-Gourava indices of various cactus chains are determined in this study.


INTRODUCTION
A molecular graph, also known as a chemical graph, is a graph in which the atoms are represented by the vertices, while the bonds are represented by the edges. Topological indices are numeric quantities obtained from a molecular graph that correlate the molecular graph's physico-chemical characteristics and have been shown to be beneficial in isomer discrimination, QSAR and QSPR analysis.
Only simple, finite, connected graphs with V (G) as vertex set and E(G) as edge set are considered throughout this study. The degree d G (a) of a vertex a is the number of vertices adjacent to a.
A cactus graph is a connected graph in which no edge lies in more than one cycle. Every cactus graph cycle is chordless, and every cactus graph block is either an edge or a cycle. A cactus graph is said to be triangular if all of its blocks are triangular. A triangular cactus graph is described as a chain triangular cactus if all of its triangles have at most two cut-vertices and each cut-vertice is shared by precisely two triangles. A square cactus graph is a type of cactus graph and all of its blocks are square. A square cactus graph is said to be a chain square cactus if all of its squares have at most two cut-vertices and each cut-vertice is shared by precisely two squares. It's worth noting that the internal squares' connections to their neighbours may vary. A chain square cactus is called ortho-chain square cactus if the cut-vertices are nearby. A para-chain square cactus is one in which the cut-vertices are not contiguous in a chain square cactus. The Gourava and hyper-Gourava indices of various generic ortho and para cactus chains are studied in this paper, and particular situations such as the triangular chain cactus T n , ortho-chain square cactus O n , and para-chain square cactus Q n are considered. Latest investigations on several cactus chains can be found in [1,3,13,14] and references cited therein. For undefined terms and notations refer to [5].
The first and second Gourava indices of a molecular graph were introduced by Kulli [6] and are defined as: Kulli proposed the first and second hyper-Gourava indices of a molecular graph G in [7], and they are defined as Several topological indices were investigated. For further information, see [2,4,8,9,10,11,12].

MAIN RESULTS
We look at two types of cactus chains in this section: the para cacti chain and the ortho cacti chain of cycles. We start with a para cacti chain of length n cycles C m , where each block is a cycle C m . Let C n m be the symbol for it. We compute an exact expression of GO 1 , GO 2 , HGO 1 and HGO 2 of C n m in the following theorem.
Theorem 2.1. For a para cacti chain of cycles C n m (m ≥ 4, n ≥ 2), Proof. 1. By utilizing the definition of GO 1 and entries in Table 1, we have 2. By making use the definition of GO 2 and values in Table 1, we have 3. By the usage of the definition of HGO 1 and facts in table 1, we have 4. By using the concept of HGO 2 as well as the data in Table 1, we have The graph Q n is pictured in Figure 1.
For a para-chain square cactus graph Q n (n ≥ 2), .
Proof. Replace m = 4 in Theorem 2.1 to complete the proof.
The graph L n is indicated in Figure 2. Proof. We get the required outcome if we set m = 6 in the Theorem 2.1.
The ortho-chain cacti of cycles with neighbouring cut-vertices is now considered. Let CO n m be an ortho-chain cactus graph, where m is the cycle length and n is the chain length. |V (CO n m )| = mn − n + 1 and |E(CO n m )| = mn are self-evident. GO 1 , GO 2 , HGO 1 and HGO 2 of CO n m are obtained by utilizing the following theorem. Proof. 1. By using the concept of GO 1 as well as the data in Table 2, we have = (mn − 3m + 2)(4 + 4) + 2n(6 + 8) + (n − 1)(8 + 16) = 8mn − 24m + 52n − 8.
2. By making use the definition of GO 2 and values in Table 2, we have 3. By utilizing the description of HGO 1 and entries in Table 2, we have Then, as illustrated in Figure 3, we consider a chain triangular cactus, designated by T n , where n is the length of the T n . For m = 3, T n is a special case of CO n m .  Proof. We get the required outcome if we set m = 4 in the Theorem 2.4.
By identifying every node of K m with a node of one K y , the graph Q(m, y) is formed from K m and m copies of K y . GO 1 , GO 2 , HGO 1 and HGO 2 of Q(m, y) are computed in the following theorem. Figure 5 depicts the graph Q(m, y). Table 3. Partitioning at the edge of Q(m, y).
Proof. 1. By using the concept of GO 1 as well as the data in Table 3, we have 2. By utilizing the description of GO 2 and entries in Table 3, we have 3. By the usage of the definition of HGO 1 and facts in Table 3, we have 4. By making use the definition of HGO 2 and values in Table 3, we have The join of each cycle of length m ≥ 3 and a new vertex in C n m . That is (C m + K 1 ). We term it a wheel chain. W n m is the symbol for it. GO 1 , GO 2 , HGO 1 and HGO 2 of W n m are derived in the following theorem.   (3,3) mn − 4n + 4 (3,6) 4(n − 1) (3, m) mn − 2n + 2 (6, m) 2(n − 1) Proof. 1. By making use the definition of GO 1 and values in Table 4, we have = 4m 2 n + 24mn + 54n − 6m − 54.
3. By using the expression for HGO 1 and data in Table 4, we have

CONCLUDING REMARKS
In this paper, para cactus chain, ortho cactus chain and wheel cactus chain are discussed and explicit expressions of GO 1 , GO 2 , HGO 1 and HGO 2 are derived for them.