J.
Indones.
Math.
Soc.
Vol.
No.
, pp.
297Ae307.
GOURAVA AND HYPER-GOURAVA INDICES OF SOME
CACTUS CHAINS
BASAVANAGOUD1 AND SHRUTI POLICEPATIL2
Department of Mathematics.
Karnatak University Dharwad - 580 003 Karnataka.
India basavanagoud@gmail.
shrutipatil300@gmail.
Abstract.
The physico-chemical characteristics of molecules are theoretically explored using the theory of graphs and mathematical chemistry.
A graphAos 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.
Key words and Phrases: Gourava indices, hyper-Gourava indices, cactus chains.
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 Topological indices are numeric quantities obtained from a molecular graph that correlate the molecular graphAos 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 dG .
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 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 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.
ItAos worth noting that the internal squaresAo connections 2020 Mathematics Subject Classification: 05C07, 05C09.
Received: 25-02-2021, accepted: 28-08-2021.
Basavanagoud and Shruti Policepatil 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 hyperGourava indices of various generic ortho and para cactus chains are studied in this paper, and particular situations such as the triangular chain cactus Tn , ortho-chain square cactus On , and para-chain square cactus Qn are considered.
Latest investigations on several cactus chains can be found in .
, 3, 13, .
and references cited For undefined terms and notations refer to .
The first and second Gourava indices of a molecular graph were introduced by Kulli .
and are defined as:
GO1 (G) =
G .
dG .
) dG .
dG .
, abOOE(G) GO2 (G) = dG .
dG .
dG .
dG .
abOOE(G) Kulli proposed the first and second hyper-Gourava indices of a molecular graph G in .
, and they are defined as HGO1 (G) = dG .
dG .
dG .
dG .
, abOOE(G) HGO2 (G) = 2 dG .
dG .
dG .
dG .
abOOE(G) Several topological indices were investigated.
For further information, see .
, 4, 8, 9, 10, 11, .
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 Cm , where each block is a cycle Cm .
Let Cm be the symbol for it.
compute an exact expression of GO1 .
GO2 .
HGO1 and HGO2 of Cm in the following Table 1.
Partitioning at the edge of Cm , dC n .
: ab OO E(C Edge count , .
mn Oe 4n 4 , .
Oe .
Theorem 2.
For a para cacti chain of cycles Cm .
Ou 4, n Ou .
GO1 (Cm ) = 8.
Oe .
Gourava and hyper-Gourava indices of some cactus chains GO2 (Cm ) = 16.
Oe .
HGO1 (Cm ) = 16.
Oe .
HGO2 (Cm ) = 256.
Oe .
Proof.
By utilizing the definition of GO1 and entries in Table 1, we have GO1 (Cm n .
dC n .
dC n .
dC n .
abOOE(Cm .
n Oe 4n .
Oe .
Oe .
By making use the definition of GO2 and values in Table 1, we have GO2 (Cm n .
dC n .
dC n .
abOOE(Cm = .
n Oe 4n .
Oe .
= 16.
Oe .
By the usage of the definition of HGO1 and facts in table 1, we have HGO1 (Cm 2 n .
dC n .
dC n .
dC n .
abOOE(Cm .
n Oe 4n .
Oe .
Oe .
By using the concept of HGO2 as well as the data in Table 1, we have HGO2 (Cm 2 n .
dC n .
dC n .
abOOE(Cm .
n Oe 4n .
Oe .
Oe .
The graph Qn is pictured in Figure 1.
Corollary 2.
For a para-chain square cactus graph Qn .
Ou .
GO1 (Qn ) = 8.
n Oe .
Basavanagoud and Shruti Policepatil Figure 1.
The graph Qn .
GO2 (Qn ) = 3n3 9n2 60n.
HGO1 (Qn ) = 16.
n Oe .
HGO2 (Qn ) = 1024.
n Oe .
Proof.
Replace m = 4 in Theorem 2.
1 to complete the proof.
The graph Ln is indicated in Figure 2.
Figure 2.
The graph Ln .
Corollary 2.
For a para-chain hexagonal cactus graph Ln .
Ou .
GO1 (Ln ) = 24.
n Oe .
GO2 (Ln ) = 32.
n Oe .
HGO1 (Ln ) = 16.
n Oe .
HGO2 (Ln ) = 512.
n Oe .
Proof.
We get the required outcome if we set m = 6 in the Theorem 2.
The ortho-chain cacti of cycles with neighbouring cut-vertices is now conn Let COm be an ortho-chain cactus graph, where m is the cycle length and n is the chain length.
|V (COm )| = mn Oe n 1 and |E(COm )| = mn are self-evident.
GO1 .
GO2 .
HGO1 and HGO2 of COm are obtained by utilizing the following theorem.
Table 2.
Partitioning at the edge of COm n .
, dCO n .
: ab OO E(CO dCOm Edge count , .
mn Oe 3m 2 , .
, .
nOe1 Theorem 2.
For a ortho cacti chain of cycles COm .
Ou 3, n Ou .
GO1 (COm ) = 8mn Oe 24m 52n Oe 8.
Gourava and hyper-Gourava indices of some cactus chains GO2 (COm ) = 16mn Oe 48m 224n Oe 96.
HGO1 (COm ) = 64mn Oe 192m 968n Oe 448.
HGO2 (COm ) = 256mn Oe 768m 20992n Oe 15872.
Proof.
By using the concept of GO1 as well as the data in Table 2, we have GO1 (COm n .
dCO n .
dCO n .
dCO n .
dCOm abOOE(COm .
n Oe 3m .
Oe .
8mn Oe 24m 52n Oe 8.
By making use the definition of GO2 and values in Table 2, we have n .
dCO n .
dCO n .
dCO n .
GO2 (COm ) = dCOm abOOE(COm .
n Oe 3m .
Oe .
16mn Oe 48m 224n Oe 96.
By utilizing the description of HGO1 and entries in Table 2, we have 2 n .
dCO n .
dCO n .
dCO n .
HGO1 (COm ) = dCOm abOOE(COm = .
n Oe 3m .
2 2n.
Oe .
2 = 64mn Oe 192m 968n Oe 448.
By the usage of the definition of HGO2 and facts in table 2, we have 2 n .
dCO n .
dCO n .
dCO n .
HGO2 (COm ) = dCOm abOOE(COm .
n Oe 3m .
2 2n.
Oe .
2 256mn Oe 768m 20992n Oe 15872.
Then, as illustrated in Figure 3, we consider a chain triangular cactus, designated by Tn , where n is the length of the Tn .
For m = 3.
Tn is a special case of COm Basavanagoud and Shruti Policepatil Figure 3.
The graph Tn .
Corollary 2.
For a chain triangular cactus Tn .
Ou .
GO1 (Tn ) = 76n Oe 80.
GO2 (Tn ) = 272n Oe 240.
HGO1 (Tn ) = 1160n Oe 1024.
HGO2 (Tn ) = 21760n Oe 18176.
Proof.
Replace m = 3 in Theorem 2.
4 to complete the proof.
Figure 4.
The graph On .
Corollary 2.
For the ortho-chain square cactus On .
Ou .
GO1 (On ) = 84n Oe 104.
GO2 (On ) = 288.
Oe .
HGO1 (On ) = 1224n Oe 1216.
HGO2 (On ) = 22016n Oe 18944.
Proof.
We get the required outcome if we set m = 4 in the Theorem 2.
By identifying every node of Km with a node of one Ky , the graph Q.
, .
is formed from Km and m copies of Ky .
GO1 .
GO2 .
HGO1 and HGO2 of Q.
, .
are computed in the following theorem.
Figure 5 depicts the graph Q.
, .
Table 3.
Partitioning at the edge of Q.
, .
, dQ.
: ab OO E(Q.
, .
) .
Oe 1, y Oe .
Oe 1, m y Oe .
y Oe 2, m y Oe .
Theorem 2.
For a ortho-chain Q.
, .
, y Ou .
Edge count m.
Oe.
Oe.
Oe .
Oe.
Gourava and hyper-Gourava indices of some cactus chains Figure 5.
The graph Q.
, .
Oe.
GO1 (Q.
, .
) = m.
Oe.
Oe.
Oe .
2 ym Oe y Oe .
Oe.
yOe.
GO2 (Q.
, .
) = m.
Oe .
Oe .
Oe .
y 2 Oe 9y .
Oe .
y m Oe .
Oe .
Oe .
y Oe .
3 ].
Oe.
2 m.
Oe .
2 ym Oe y Oe .
2 HGO1 (Q.
, .
) = m.
Oe.
Oe.
Oe.
yOe.
HGO2 (Q.
, .
) = 2m.
Oe .
Oe .
Oe .
y Oe .
6 ] m.
Oe .
2y Oe .
Oe .
y Oe .
]2 .
Proof.
By using the concept of GO1 as well as the data in Table 3, we have GO1 (Q.
, .
) dQ.
abOOE(Q.
) m.
Oe .
Oe .
2 Oe .
Oe .
2 ym Oe y Oe .
Oe .
y Oe .
By utilizing the description of GO2 and entries in Table 3, we have GO2 (Q.
, .
) dQ.
abOOE(Q.
) = m.
Oe .
Oe .
Oe .
y 2 Oe 9y .
Oe .
y m Oe .
Oe .
Oe .
y Oe .
3 ].
By the usage of the definition of HGO1 and facts in Table 3, we have Basavanagoud and Shruti Policepatil HGO1 (Q.
, .
) 2 dQ.
abOOE(Q.
) m.
Oe .
Oe .
2 Oe .
2 m.
Oe .
2 ym Oe n Oe .
2 m.
Oe .
y Oe .
2 By making use the definition of HGO2 and values in Table 3, we have HGO2 (Q.
, .
) 2 dQ.
abOOE(Q.
) 2m.
Oe .
Oe .
Oe .
2y Oe .
Oe .
y Oe .
]2 2m.
Oe .
y Oe .
That is The join of each cycle of length m Ou 3 and a new vertex in Cm (Cm K1 ).
We term it a wheel chain.
Wm is the symbol for it.
GO1 .
GO2 .
HGO1
and HGO2 of Wm are derived in the following theorem.
Figure 6.
The graph W4n .
Table 4.
Partitioning at the edge of Wm n .
, dW n .
: ab OO E(W .
, .
, .
, .
, .
Edge count mn Oe 4n 4 4.
Oe .
mn Oe 2n 2 2.
Oe .
Theorem 2.
For wheel chain Wm .
Ou 3, n Ou .
GO1 (Wm ) = 4m2 n 24mn 54n Oe 6m Oe 54.
GO2 (Wm ) = 3m3 n 15m2 n Oe 6m2 108mn 432n Oe 54m Oe 432.
HGO1 (Wm ) = 16m3 n 354nm 2070n 90m2 n Oe 66m2 Oe 120m Oe 2070.
HGO2 (Wm ) = 9nm5 108nm4 837nm3 2430nm2 Oe 72m4 Oe 864m3 Oe 2592m2 2916nm 93312n Oe 93312.
Gourava and hyper-Gourava indices of some cactus chains Proof.
By making use the definition of GO1 and values in Table 4, we have GO1 (Wm n .
dW n .
dW n .
dW n .
abOOE(Wm .
n Oe 4n .
Oe .
n Oe 2n .
Oe .
4m2 n 24mn 54n Oe 6m Oe 54.
By the usage of the definition of GO2 and facts in Table 4, we have GO2 (Wm n .
dW n .
dW n .
d Wm abOOE(Wm .
n Oe 4n .
Oe .
n Oe 2n .
Oe .
3m3 n 15m2 n Oe 6m2 108mn 432n Oe 54m Oe 432.
By using the expression for HGO1 and data in Table 4, we have X dW n .
dW n .
dW n .
HGO1 (Wm ) = abOOE(Wm .
n Oe 4n .
Oe .
n Oe 2n .
Oe .
2 16m3 n 354nm 2070n 90m2 n Oe 66m2 Oe 120m Oe 2070.
By utilizing the description of HGO2 and entries in Table 4, we have HGO2 (Wm n .
dW n .
dW n .
abOOE(Wm .
n Oe 4n .
Oe .
n Oe 2n .
Oe .
2 9nm5 108nm4 837nm3 2430nm2 Oe 72m4 Oe 864m3 Oe 2592m2 2916nm 93312n Oe 93312.
COMPARATIVE ANALYSIS
The plotting of GO1 .
GO2 .
HGO1 and HGO2 for the cactus graphs are shown in Figures 7 and 8.
We have built the figures using Origin software takn ing m=4.
GO1 (Cm ).
GO1 (Cm ).
GO1 (COm ).
GO1 (Wm ).
GO2 (Cm ).
GO2 (COm GO2 (Wm ).
HGO1 (Cm ).
HGO1 (COm ).
HGO1 (Wm ).
HGO2 (Cm ).
HGO2 (COm Basavanagoud and Shruti Policepatil Figure 7.
Plot of GO1 .
and GO2 .
for cactus chains.
Figure 8.
Plot of HGO1 .
and HGO2 .
for cactus chains.
and HGO2 (Wm ) are linearly increasing and GO1 (Q.
, .
GO2 (Q.
, .
HGO1 (Q.
, .
) and HGO2 (Q.
, .
) are exponentially increasing.
CONCLUDING REMARKS
In this paper, para cactus chain, ortho cactus chain and wheel cactus chain are discussed and explicit expressions of GO1 .
GO2 .
HGO1 and HGO2 are derived for them.
REFERENCES