J.
Indones.
Math.
Soc.
Vol.
No.
, pp.
135Ae149.
CHEMICAL APPLICABILITY OF SECOND ORDER
SOMBOR INDEX
Basavanagouda and Goutam Veerapurb Department of Mathematics.
Karnatak University.
Dharwad - 580 003.
Karnataka.
India, a bbasavanagoud@kud.
in, b samarasajeevana@gmail.
Abstract.
In this paper, we introduce the higher-order Sombor index of a molecular In addtition, we compute the second order Sombor index of some standard class of graphs and line graph of subdivision graph of 2D-lattice, nanotube and nanotorus of T U C4 C8 .
, .
and also we obtain the expressions of the second order Sombor index of the line graph of subdivision graph of tadpole graph, wheel graph, ladder graph and chain silicate network CSn .
Further, we study the linear regression analysis of the second order Sombor index with the entropy, acentric factor, enthalpy of vaporization and standard enthalpy of vaporization of an octane isomers.
Key words and Phrases: Topological indices, line graph, subdivision graph, nanostructure, tadpole graph.
Introduction and Preliminaries A topological index is a molecular descriptor that is calculated based on the molecular graph of a chemical compound.
Chemical graph theory is a branch of mathematical chemistry, which has an major effect on the development of the chemical sciences.
In molecular graph, graph is used to represent a molecule by considering the atoms as the vertices and molecular bonds as the edges.
A graphical invariant is a number related to a graph.
In other words, it is a fixed number under graph automorphisms.
In chemical graph theory, these invariants are also called the topological indices.
There are several topological indices available today, some of which are used in chemistry.
Chemical Data Bases have registered over 3000 topological graph indices.
Chemists and mathematicians both investigate this Two-dimensional topological indices have been a successful method in recent years for the discovery of several novel medications, including anticonvulsants, anineoplastics, antimalarials, antiallergics, and silico generation .
, .
The use of topological indices and quantitative structure-activity relationships (QSAR) has 2020 Mathematics Subject Classification: 05C09, 05C38, 05C90.
Received: 04-03-2022, accepted: 16-03-2023.
Basavanagoud and G.
Veerapur therefore demonstrated that they have evolved from being a promising potential to serving as a cornerstone in the process of drug development and other research fields .
, 11, .
Most crucially, three-dimensional molecular characteristics .
opographic indice.
and molecular chirality are also provided with the further research of chemical indices and drug design and discovery .
Studying three-dimensional quantitative structure-activity relationships, such as molecular chirality, is becoming more and more important.
Nevertheless, there have only been a few outcomes thus far, with the exception of one related term that is commonly cited in .
Boiling points, solubilities, densities, anaesthetics, narcotics, toxicities, resistance etc.
just a few examples of the impressive range of physical, chemical, and biological properties to which higher order topological indices have been successfully applied.
These results have been published in more than two books and several hundred scientific journals .
, .
The literature has revealed findings about these indices mathematical properties .
, .
Let G = (V.
E) be such graph with V as vertex set and E as edge set and |V | = n, |E| = m.
The degree dG .
of a vertex v OO V (G) is the number of edges incident to it in G.
Li and Zhao introduced the first general Zagreb index .
as M (G) = .
u ) .
uOOV (G) The connectivity index ( or RandicA index ) of a graph G , denoted by N(G), was introduced by RandicA .
in the study of branching properties of alkanes.
It is defined as .
N (G) =
dG .
A dG .
uvOOE (G) In .
, .
with the intention of extending the applicability of the connectivity index.
Kier.
Hall.
Murray and RandicA considered the higher-order connectivity index of a graph G as N(G) = .
dG .
1 )dG .
2 ) A A A dG .
1 ) u u au
OOE (G)
The first and second Zagreb .
indices of a graph G are defined as M1 (G) = dG .
2 , vOOv(G) M2 (G) = dG .
A dG .
uvOOE(G) The first Zagreb index .
can be written also as M1 (G) = dG .
dG .
uvOOE(G) .
Chemical applicability of second order Sombor index Basavanagoud et.
considered higher-order first Zagreb index as M1 (G) = .
G .
1 ) dG .
2 ) A A A dG .
1 )].
u1 u2 au 1 OOE (G) Basavanagoud et.
defined the second order first Zagreb index as M1 (G) = .
G .
1 ) dG .
2 ) dG .
3 )].
u1 u2 u3 OOE (G) The Sombor index was introduced by I.
Gutman .
to be described as X p
SO(G) =
dG .
2 dG .
uvOOE(G) Bearing in mind Eqs.
, .
, .
, ( 1.
, we can consider the higher-order Sombor index of Eq.
as SO(G) = dG .
1 )2 dG .
2 )2 A A A dG .
1 )2 .
u1 u2 au 1 OOE (G) Here.
E (G) denote the path of length in a graph G, for example E1 (G) and E2 (G) are path of length 1 and 2 in a graph G respectively.
By Eq.
, it is consistent to define the second order Sombor index as dG .
1 )2 dG .
2 )2 dG .
3 )2 .
SO(G) =
u1 u2 u3 OOE2 (G) Estimating the second order Sombor index of graphs In this section, we compute the second order Sombor index of some standard class of graphs viz.
, path graph Pn , wheel graph Wn 1 , complete bipartite graph Kr,s and r Oe regular graph.
The following Remark, which is needed to prove main Remark 2.
For a graph G on m edges, the number of paths of length 2 in G is Oem 21 M1 (G).
Theorem 2.
Let Pn be the path graph on n Ou 4 vertices.
Then Oo SO(Pn ) = 2.
Oe .
Proof.
For a path graph Pn on n Ou 4 vertices each vertex is of degree either 1 or Based on the degree of vertices on the path of length 2 in Pn we can partition E2 (Pn ).
In Pn , path .
, 2, .
appears 2 times and path .
, 2, .
Oe .
Hence by Eq.
we get the required result.
Theorem 2.
Let Wn 1 be the wheel graph n Ou 4 vertices.
Then Oo Oo n2 3n( n2 .
SO(Wn 1 ) = n3 3 Basavanagoud and G.
Veerapur Proof.
For a wheel graph Wn 1 on n Ou 4 vertices each vertex is of degree either 3 or Based on the degree of vertices on the path of length 2 in Wn 1 we can partition E2 (Wn ).
In Wn 1 , path .
, 3, .
appears n times and path .
, 3, .
appears n 3n Therefore by Eq.
, we get the required result.
Theorem 2.
Let Kr,s be the complete bipartite graph on r Ou 2, s Ou 3, vertices.
Then sr.
Oe .
p 2 rs.
Oe .
p 2 2r s2 2s r2 .
SO(Kr,s ) = Proof.
For a complete bipartite graph Kr,s with r s vertices each vertex is of degree either r or s.
Based on the degree of vertices on the path of length 2 in Kr,s we can partition .
, 2 times and path .
, r, .
appears s 2 times.
Therefore by Eq.
, we get the required result.
Theorem 2.
Let G be a r Oe regular graph on n vertices Oo nr2 3.
Oe .
SO(G) = Proof.
Since G is a r Oe regular graph, the path appears nr.
Oe.
times in G.
There2 fore by Eq.
, we get the required result.
Corollary 2.
For a cycle graph Cn , n Ou 3.
Oo SO(Cn ) = 2n 3.
Corollary 2.
For a complete graph Kn , n Ou 4.
Oo n 3.
Oe .
Oe .
2 SO(Kn ) = Lemma 2.
Let G be a graph with n vertices and m edges.
Then M1 (G) O m nOe2 .
nOe1 .
Lemma 2.
Let G be a graph with n vertices and m edges, m > 0.
Then the M1 (G) = m nOe2 .
nOe1 holds if and only if G is isomorphic to star graph Sn or Kn or KnOe1 O K1 .
Theorem 2.
Let G be a graph with n vertices and m edges.
Then Oo nOe4 SO(G) O .
Oe .
3 A m nOe1 the equality holds if and only if G is isomorphic to Kn .
Chemical applicability of second order Sombor index Proof.
SO(G)
dG .
2 dG .
2 dG .
2 uvwOOE2 (G) O Oo .
Oe .
uvwOOE2 (G) Oo = .
Oe .
3 Oem M1 (G) Oo O .
Oe .
3 Oem m nOe2 nOe1 Oo nOe4 = .
Oe .
3 A m nOe1 .
Relations .
were obtained by taking into account for each vertices v OO V (G), we have dG .
O n Oe 1 and Eq.
, respectively.
Suppose that equality in .
Then inequalities .
become equalities.
From .
we conclude that for every vertex v, dG .
= .
Oe .
Then from .
and Lemma 2.
8 it follows that G is a complete graph.
Conversely, let G be a complete Then it is easily verified that equality holds in .
Lemma 2.
Let G be a graph with n vertices, m edges.
Then M1 (G) Ou 2m.
Oe pn.
where p = b and the equality holds if and only if the difference of the degrees of any two vertices of graph G is at most one.
Theorem 2.
Let G be a graph with n vertices, m edges and the minimum vertex degree .
Then SO(G) Ou mp Oe pn.
) where p = b and the equality holds if and only if G is a regular graph.
Basavanagoud and G.
Veerapur Proof.
SO(G)
uvwOOE (G) Ou dG .
2 dG .
2 dG .
2 Oo uvwOOE2 (G) Oo = 3 Oem M1 (G) Oo Ou 3 Oem .
Oe pn.
) .
mp Oe pn.
Relations .
were obtained by taking into accounting for each vertices v OO V (G), we have dG .
Ou and Eq.
, respectively.
Suppose now that equality in .
Then inequalities .
become equalities.
From .
we conclude that for every vertex v, dG .
= .
Then from Eq.
and Lemma 2.
10 it follows that G is a regular graph.
Conversely, let G be a regular Then it is easily verified that equality holds in .
Computing the the second order Sombor index of some families of In .
Nadeem et al.
obtained expressions for certain topological indices of the line graphs of subdivision graphs of 2D-lattice, nanotube, and nanotorus of T U C4 C8 .
, .
, where p and q denote the number of squares in a row and the number of rows of squares, respectively in 2D-lattice, nanotube and nanotorus as shown in Figure 1 .
, .
The numbers of vertices and edges of 2D-lattice, nanotube and nanotorus of T U C4 C8 .
, .
are given in Table 1.
Figure 1.
2D-lattice of T U C4 C8 .
, .
T U C4 C8 .
, .
T U C4 C8 .
, .
Chemical applicability of second order Sombor index Table 1.
Number of vertices and edges.
Graph 2D-lattices of T U C4 C8 .
, .
T U C4 C8 .
, .
nanotube T U C4 C8 .
, .
nanotorus Number of vertices Number of edges 6pq Oe p Oe q 6pq Oe p In .
, .
Ranjini et al.
presented explicit formula for computing the Shultz index and Zagreb indices of the subdivision graphs of the tadpole, wheel and ladder In 2015.
Su and Xu .
calculated the general sum-connectivity index and coindex of the L(S(Tn,k )).
L(S(Wn )) and L(S(Ln )).
In .
Nadeem et al.
some exact formulas for ABC4 and GA5 indices of the line graphs of the tadpole, wheel and ladder graphs by using the notion of subdivision.
Figure 2.
Subdivision graph of 2D-lattice of T U C4 C8 .
, .
Line graph of the subdivision graph of 2D-lattice of T U C4 C8 .
, .
Table 2.
Partition of paths of length 2 of the graph X.
X .
, dX .
, dX .
) where uvw OO E2 (X) Number of paths of length 2 in X , 2, .
, 2, .
q Oe .
, 3, .
q Oe .
, 3, .
pq Oe 26p Oe 26q .
Lemma 3.
Let X be the line graph of the subdivision graph of 2D Oe lattice of T U C4 C8 .
, .
Then M1 (X) = 108pq Oe 38p Oe 38q.
Theorem 3.
Let X be the line graph of the subdivision graph of 2D Oe lattice of T U C4 C8 .
, .
Then Oo Oo Oo Oo SO(X) = 16 3 4 17.
q Oe .
q Oe .
pq Oe 26p Oe 26q .
Basavanagoud and G.
Veerapur Proof.
The subdivision graph of 2D-lattice of T U C4 C8 .
, .
and the graph X are shown in Fig.
, respectively.
In X there are total 2.
pq OepOe.
vertices each vertex is of degree either 2 or 3 and 18pq Oe 5p Oe 5q edges.
From Remark 2.
and Lemma 3.
1, we get 36pq Oe 14p Oe 14q of paths of length 2 in X.
Based on the degree of vertices on the path of length 2 in X we can partition E2 (X) as shown in Table 2.
Apply Eq.
to Table 2 and get the required result.
Figure 3.
Subdivision graph of T U C4 C8 .
, .
of nanotube.
line graph of the subdivision graph of T U C4 C8 .
, .
of nanotube.
Table 3.
Partition of paths of length 2 of the graph Y .
Y .
, dY .
, dY .
) where uvw OO E2 (Y ) Number of paths of length 2 in Y , 2, .
, 3, .
, 3, .
pq Oe 26.
Lemma 3.
Let Y be the line graph of the subdivision graph of T U C4 C8 .
, .
Then M1 (Y ) = 108pq Oe 38p.
Theorem 3.
Let Y be the line graph of the subdivision graph of T U C4 C8 .
, .
Then Oo Oo Oo SO(Y ) = 4p 17 8p 22 3 2.
pq Oe 26.
Proof.
The subdivision graph of T U C4 C8 .
, .
nanotube and the graph Y are shown in Fig.
, respectively.
In Y there are total 12pq Oe 2p vertices in which each vertex is of degree either 2 or 3 and 18pq Oe 5p edges.
From Remark 2.
1 and Lemma 3.
3, we get 36pq Oe 14p number of paths of length 2 in Y.
Based on the degree of vertices on the paths of length 2 in Y we can partition E2 (Y ) as shown in Table 3.
Apply Eq.
to Table 3 and get the required result.
Chemical applicability of second order Sombor index Figure 4.
Subdivision graph of T U C4 C8 .
, .
of nanotorus.
Line graph of the subdivision graph of T U C4 C8 .
, .
of nanotorus.
Theorem 3.
Let Z be the line graph of the subdivision graph of T U C4 C8 .
, .
Then Oo SO(Z) = 9n 3.
Proof.
The subdivision graph of T U C4 C8 .
, .
nanotorus and the graph Z are shown in Fig.
, respectively.
Since Z is a 3-regular graph with 12pq vertices and 18pq edges.
Therefore by Theorem 2.
4, we get the required result.
Table 4.
Partition of paths of length 2 of the graph A = L(S(Tn,k )) for k = 1.
A .
, dA .
, dA .
) where uvw OO E2 (A) Number of paths of length 2 in A .
, 3, .
, 3, .
, 2, .
, 3, .
, 2, .
2n Oe 4 Table 5.
Partition of paths of length 2 of the graph A = L(S(Tn,k )) for k > 1.
A .
, dA .
, dA .
) where uvw OO E2 (A) Number of paths of length 2 in A .
, 2, .
, 3, .
, 2, .
, 3, .
, 2, .
2n 2k Oe 8 Basavanagoud and G.
Veerapur Lemma 3.
, .
Let A be the line graph of the subdivision graph of the tadpole graph Tn,k .
Then M1 (A) = 8n 8k 12.
Theorem 3.
Let A be a line graph of the subdivision graph of the tadpole graph Tn,k .
Then ( Oo Oo Oo Oo Oo 2 19 4 22 2 17 9 3 2.
n Oe .
3 for k = 1.
SO(A) =
2 22 17 3 .
n 2k Oe .
3 for k > 1.
Proof.
First of all, we consider graph A for n Ou 3 and k > 1.
In this graph there are total 2.
vertices and 2n 2k 1 edges.
From Remark 2.
1 and Lemma 6, we get 2k 2n 5 of paths of length 2 in A.
Based on the degree of vertices on the paths of length 2 in A we can partition E2 (A) as shown in Table 5.
Apply Eq.
to Table 5 and get the required result.
By similar arguments we can obtain the expression of 2 SO(A) for k = 1 from Table 4.
Lemma 3.
Let B be a line graph of the subdivision graph of the wheel graph Wn 1 .
Then M1 (B) = n3 27n.
Table 6.
Partition of paths of length 2 of the graph B.
B .
, dB .
, dB .
) where uvw OO E2 (B) Number of paths of length 2 in B , 3, .
, 3, .
, n, .
Oe .
Oe.
Oe.
, n, .
Theorem 3.
Let B be a line graph of the subdivision graph of the wheel graph Wn 1 .
Then Oo Oo n2 .
Oe .
Oe .
3 SO(B) = 21n 3 2n 18 n n.
Oe .
9 2n Proof.
The graph L(S(Wn 1 )) contains 4.
vertices and n 9n From n3 Oen2 18n Remark 2.
1 and Lemma 3.
8, we get number of paths of length 2 in B.
Based on the degree of vertices on the paths of length 2 in B we can partition E2 (B) as shown in Table 6.
Apply Eq.
to Table 6 and get the required result.
Lemma 3.
, .
Let C be a line graph of the subdivision graph of a ladder graph with order n.
Then M1 (C) = 54n Oe 76.
Chemical applicability of second order Sombor index Table 7.
Partition of paths of length 2 of the graph C.
C .
, dC .
, dC .
) where uvw OO E2 (C) Number of paths of length 2 in C , 2, .
, 2, .
, 3, .
, 3, .
18n Oe 44 Theorem 3.
Let C be a line graph of the subdivision graph of a ladder graph with order n.
Then Oo Oo Oo Oo SO(C) = 8 3 4 17 8 22 3 3.
n Oe .
Proof.
The graph L(S(Ln ) contains 6nOe4 vertices and 18nOe20 From Remark 1 and Lemma 3.
10, we get 18n Oe 28 number of paths of length 2 in C.
Based on the degree of vertices on the paths of length 2 in C we can partition E2 (C) as shown in Table 7.
Apply Eq.
to Table 7 and get the required result.
Figure 5.
CSn .
A subdivision graph of CSn .
A line graph of subdivision graph of CSn .
A chain silicate network CSn of dimension n is obtained by linearly arranging n The number of vertices and the number of edges in CSn with n > 1 are 3n 1 and 6n, respectively .
The number of vertices and number of edges in the Basavanagoud and G.
Veerapur line graph of the subdivision graph L(S(CSn )) = G of CSn are 12n and 27n Oe 9.
Figure 5 shows the line graph of subdivision graph of CSn .
Table 8.
Partition of paths of length 2 of the graph L(S(CSn )) = G.
G .
, dG .
, dG .
) where uvw OO E2 (G) Number of paths of length 2 in G , 6, .
n Oe .
, 6, .
n Oe .
, 3, .
n Oe .
, 3, .
Lemma 3.
Consider the line graph of the subdivision graph G of CSn .
Then M (G) = 2.
A 3 1 .
Oe .
6 1 .
By substituting = 2 in above Lemma we get the first Zagreb index of L(S(CSn )) M1 (G) 18.
n Oe .
Theorem 3.
Let G be the line graph of the subdivision graph of CSn .
Then Oo Oo Oo SO(G) = 90.
n Oe .
n Oe .
n Oe .
Proof.
The graph G contains 12n vertices and 27n Oe 9 edges.
From Remark 2.
and Eq.
, we get 108n Oe 72 number of paths of length 2 in G.
Based on the degree of vertices on the paths of length 2 in G we can partition E2 (G) as shown in Table 8.
Apply Eq.
to Table 8 and get the required result.
Chemical Applicability of the second order Sombor index In this section, a linear regression model of four physical properties is presented for the second order Sombor index 2 SO(G).
The physical properties such as entropy(S), acentric factor (AF), enthalpy of vaporization (HVAP) and standard enthalpy of vaporization (DHVAP) of octane isomers have shown good correlation with the index considered in the study.
The second order Sombor index 2 SO(G) is tested for the octane isomers database available at https://w.
eu/dataset.
SO(G) index are computed and tabulated in column 6 of Table 9.
Using the method of least squares, the linear regression models for S.
AF.
HVAP, and DHVAP are fitted using the data of Table 9.
The fitted models for the SO(G) index are 03814(A2.
Oe 0.
48111(A0.
SO(G))
Acentric F actor 4773309(A0.
Oe 0.
0041000(A0.
SO(G))
HV AP
20523(A1.
Oe 0.
2038(A0.
SO(G))
DHV AP
524133(A0.
Oe 0.
040454(A0.
SO(G))
From Table 10 and Figure 6, it is obvious that the SO(G) index highly correlates with the acentric factor and the correlation coefficient .
=0.
Also, the Chemical applicability of second order Sombor index Table 9.
Experimental values of S.
AF .
HV AP and DHV AP and the corresponding values of the 2 SO of octane isomers.
Alkane
n-Octane
2-Methylheptane 3-Methylheptane 4-Methylheptane 3-Ethylhexane 2, 2-Dimethylhexane 2, 3-Dimethylhexane 2, 4-Dimethylhexane 2, 5-Dimethylhexane 3, 3-Dimethylhexane 3, 4-Dimethylhexane 2-Methyl-3-ethylpentane 3-Methyl-3-ethylpentane 2, 2, 3-Trimethylpentane 2, 2, 4-Trimethylpentane 2, 3, 3-Trimethylpentane 2, 3, 4-Trimethylpentane 2, 2, 3, 3-Trimethylpentane HV AP
DHV AP
SO(G)
Table 10.
Parameters of regression models for the 2 SO(G) index.
Physical properties Entropy Acentric factor
HVAP
DHVAP
Value of the correlation coefficient Residual standard error SO(G) index has good correlation coefficient .
= 0.
8802677 with entropy, .
= 8311469 with HVAP, and .
= 0.
8721235 with DHVAP.
Note: In equations .
- .
, the errors of the regression coefficients are represented within brackets.
Table 10 and Figure 6 show the correlation coefficient and residual standard error for the regression models of four physical properties with SO(G) index.
REFERENCES