J.
Indones.
Math.
Soc.
Vol.
No.
, pp.
01Ae20.
SOME BOND-ADDITIVE INDICES AND ITS
POLYNOMIAL OF CELLULOSE
Kandan1,2 .
Subramanian2 Department of Mathematics.
Annamalai University.
Annamalai Nagar-608 002.
India, kandan2k@gmail.
Department of Mathematics.
Government Arts College.
Chidambaram-608 102.
India, subramanian87@gmail.
Abstract.
Cellulose is one of the natural bio-polymers which have been extensively used in various fields due to their valuable and remarkable chemical and physical Due to a key ingredients of cellulose in various product, itAos applications have widely been recognized in many industries like pharmaceutical, bio-fuel, textiles, etc.
The study of graphs using chemistry attracts a lot of researchers globally because of its enormous application.
One such application is studying topological indices of a chemical graph associated to a molecular structure.
In this this work we have obtained the exact value of szeged.
Padmakar-Ivan(PI), additively weighted PI index, multiplicatively weighted PI index, additively weighted szeged index, multiplicatively weighted szeged index and its polynomial for cellulose chemical structure.
Moreover we derived the relation between these indices for the cellulose.
Key words and Phrases: Cellulose.
Szeged index.
Padamakar-Ivan index.
PadmakarIvan polynomial.
Szeged polynomial INTRODUCTION Most of the chemical compounds under consideration are carbon-based.
Often one uses the term chemical graphs .
olecular structur.
of a compound is presented with a graph, where the edges indicate the links and the vertices represent the atoms.
Both in the context of complex networks and in more traditional applications of chemical graph theory, it garnered a great deal of interest.
It aids in the modeling of a wide range of systems, the structure and operation of which are influenced by the connection patterns of its constituent parts.
Modern materials science needs have encouraged the creation of a wide range of bio-based materials, in which cellulose and its derivatives play a significant part, as a reaction to environmental concerns in.
, 8, 11, 18, 21, .
Because of celluloseAos exceptional and one-of-a-kind chemical and physical properties, it has received wide recognition in a number of industries.
The main cause of this is because cellulose is extremely rigid due to the tight link between its molecules.
It is the most prevalent organic substance on the planet in terms of chemistry.
The molecules of cellulose are strongly bonded by hydrogen bonds as a result of their intricate intra- and extra molecular As a result, they are insoluble in typical polar solvents like water, 2020 Mathematics Subject Classification: 05C12, 05C05 Received: 22-10-2022, accepted: 10-03-2024.
Kandan.
Subramanian Figure 1.
The molecular structure of Cellulose.
Figure 2.
The chemical graph of Cellulose.
alcohols, and amines, which makes it challenging to process into the proper shape.
The three-dimensional structure of the cellulose network (C6 H10 O5 )d is denoted by CLd , where d is the countable of cellulose units.
Figure 1.
is the molecular structure, represented of cellulose network and Figure 2.
represented the chemical graph of cellulose on d = 1, d = 2 and d = 3 units.
Over the last two decades, a lot of different of numerical values have been suggested and investigated, referred to variably as structural invariants, topological indices, or molecular descriptors.
A moleculeAos topology is expressed numerically as a topological description.
In Quantitative Structure-Property Relationship (QSPR) and Quantitative Structure-Activity Relationship (QSAR) investigations, these topological descriptors are utilized to estimate the physicochemical and/or biological characteristics of molecules.
, .
Several degree, spectrum, matching, and distance-based topological descriptors have been suggested and explored in the literature .
, 42, .
, some of the interesting indices are Sombor index.
Steiner Gutman Index.
Estrada index and Laplacian Estrada index of a graphs, see .
, 38, 39, .
One of the oldest topological indices and most investigated is the Wiener index, after the successful of this index.
Gutman et al.
introduced the generalization of the Wiener index for a acyclic graph known as szeged(S.
Consequentially, another szeged like index called Padmakar-Ivan index(PI) proposed by Kahadikar et al.
Recently DosUlicA et al.
, introduced a distance based topological index called Mostar index, which measure of the global peripherality of a molecular structure.
PI, szeged, and Mostar indices are the interesting bond-additive type indices which quantities the degree of peripherality of particular edge and of the graph as a whole.
Very recently Kandan et al.
, derived some bond-additive topological indices and their polynomial see .
, 23, 24, 25, 26, 27, .
For a connected G = (V (G).
E(G)), the Padmakar-Ivan and szeged indices of G defined as Sz(G) = EAa .
G)EAb .
G) e=abAE(G) P I(G) = (EAa .
G) EAb .
G)) e=abAE(G) respectively, where EAa .
G) denotes the number of vertices of G closer to a than to b and EAb .
G) denotes the number of vertices of G closer to b than to a.
Note that in this definitions the equidistant vertices not counted for any edge of G.
In the literature, many researchers found the applications and were extensively studied for various molecular structure to these indices see.
, 30, 32, .
and for some recent investigation see.
, 12, 36, .
Inspired by this extension of szeged and the PI index.
IliAc and MilosavljeviAc .
proposed weighted version named as the Some Bond-Additive Indices and Its Polynomial of Cellulose additively weighted szeged and the additively weighted PI index, respectively which are defined as SzA (G) = (Oa .
G) Ob .
G))EAa .
G)EAb .
G) e=abAE(G) P IA (G) = (Oa .
G) Ob .
G)) (EAa .
G) EAb .
G)) .
e=abAE(G) Laterly.
Arockiaraj et al.
introduced the multiplicatively weighted szeged and P I index respectively which are defined as SzM (G) = (Oa .
G).
Ob .
G))EAa .
G)EAb .
G) e=abAE(G) P IM (G) = (Oa .
G).
Ob .
G)) (EAa .
G) EAb .
G)) e=abAE(G) where Oa .
G).
Ob .
G) denotes the degree of vertex a, b respectively.
For more works on these weighted indices, see.
, 6, .
Non-isomorphic graphs can be distinguished using a graph polynomial.
Many graph polynomials have been created for quantifying the structural information of molecular graphs related for quantifying the structural information of molecular Graph polynomials were utilized in chemistry in conjunction with the molecular orbital theory of unsaturated compounds, and they were also a valuable source of structural descriptors used in constructing structure property models.
, 14, 15, 33, .
Distance-based and degree-based graph polynomials are useful because they contain a wealth of information about topological indices.
In .
Szeged and P I polynomial of a graph G respectively defined as Sz(G, .
= xEAa .
G)EAb .
G) e=abAE(G) P I(G, .
= xEAa .
G) EAb .
G) .
e=abAE(G) Since many graph parameter are derived from the graph polynomial, it is interesting to study new graph polynomial, which are used to model a the behavior of physical, chemical or biological system.
Various topological indices can be derived from polynomials by taking their value at some point directly, or by taking integrals or In light of the preceding condition and the newly proposed weighted version, very recently.
Kandan et al.
introduced the weighted version polynomial of szeged and PI of a graph and derived it for some graphs.
The Additively weighted szeged polynomial of a graph G is defined as SzA (G, .
= x(Oa .
G) Ob .
G))EAa .
G)EAb .
G) .
e=abAE(G) The Multiplicatively weighted szeged polynomial of G is defined as SzM (G, .
= x(Oa .
G).
Ob .
G))EAa .
G)EAb .
G) .
e=abAE(G) The Additively weighted Padmakar-Ivan polynomial of G is defined as P IA (G, .
= e=abAE(G) x(Oa .
G) Ob .
G))EAa .
G) EAb .
G) .
Kandan.
Subramanian The Multiplicatively weighted padmakar-Ivan polynomial of G is defined as P IM (G, .
= x(Oa .
G).
Ob .
G))EAa .
G) EAb .
G) .
e=abAE(G) Recently.
Kahalaf et al.
derived several graph polynomials of cellulose and for more results on polynomials see .
, 10, .
As the main result of this paper is to compute the exact form of szeged.
PI, weighted szeged, weighted PI and their polynomial of cellulose.
As a main result, in Section 3, we obtain the exact value of szeged index, additively weighted szeged index .
Multiplicatively weighted szeged index and its polynomial.
Moreover using these results we derived their relation and bounds for cellulose.
In Section 4, similar results obtained for the PadmakarIvan index, additively weighted Padmakar-Ivan index.
Multiplicatively weighted Padmakar-Ivan index and its polynomial.
Further we obtained their relations and boundness of cellulose.
MAIN RESULT
By chemical graph structure analysis and observation on CLd , we note that the edge set E(CLd ) can be divided into five groups based on degree, which are summarized as follows see Figure 2.
E1 (CLd ) = .
= abAE(CLd ) : Oa .
CLd ) = 1.
Ob .
CLd ) = .
, |E1 | = 2d E2 (CLd ) = .
= abAE(CLd ) : Oa .
CLd ) = 1.
Ob .
CLd ) = .
, |E2 | = 4d 2 E3 (CLd ) = .
= abAE(CLd ) : Oa .
CLd ) = 2.
Ob .
CLd ) = .
, |E3 | = 10d Oe 2 E4 (CLd ) = .
= abAE(CLd ) : Oa .
CLd ) = 3.
Ob .
CLd ) = .
, |E4 | = 8d It can be observed that in general, |V (CLd )| = 22d 1, |E(CLd )| = 24d.
As a part of the main proof we use the above classification of edge partition of cellulose, now we summarize the value of EAa .
CLd ).
EAb .
CLd ) in the Table 1.
SZEGED INDEX AND ITS POLYNOMIAL
Using the Table 1, we have the following results on szeged related indices of Theorem 3.
Let CLd be the chemical graph of the cellulose, dOe1 Sz(CLd ) = 2d.
22d 4d.
d Oe .
1452 22d 25 935.
Oe d2 ) Oe 1332d 22d.
Oe .
d2 d dOe1 990 902.
Oe d2 ) Oe 1166d.
Proof.
To obtain the Szeged index of cellulose, by the definition of szeged index and from the Table 1.
we have Sz(CLd ) = EAa .
CLd )EAb .
CLd ) e=abOOE(CLd ) EAa .
CLd )EAb .
CLd ) EAa .
CLd )EAb .
CLd ) e=abOOE1 (CLd ) e=abOOE2 (CLd ) EAa .
CLd )EAb .
CLd ) EAa .
CLd )EAb .
CLd ) e=abOOE3 (CLd ) e=abOOE4 (CLd ) Some Bond-Additive Indices and Its Polynomial of Cellulose Table 1.
Edge partitions and EAa .
EAb value of cellulose (Oa .
Ob ) Cellulose CLi=1,2,.
,d .
EAa .
CLd ) EAb .
CLd ) |E| EAa .
CLd )EAb .
CLd ) .
22d 16-22i 22d 17-22i 22d 12-22i 22d 11-22i 22d 18-22i (Oa .
Ob ) |E| Cellulose CLi=1,2,.
,d .
EAa .
CLd ) EAb .
CLd ) 2d.
44d-2 .
dOe1 Oe d2 ) Oe 240d dOe1 Oe d2 ) Oe 272d dOe1 Oe d ) Oe 132d dOe1 Oe d2 ) Oe 110d dOe1 Oe d2 ) Oe 306d EAa .
CLd )EAb .
CLd ) .
22d 17-22i 22d 17-22i 22d 1-22.
22d 16-22i 22d 6-22i 22d 5-22i dOe1 Oe d2 ) Oe 272d Oe .
( 22d 25 Oe .
( 3 ) .
dOe1 Oe d2 ) Oe 272d dOe1 Oe d2 ) Oe 240d .
dOe1 3 .
Oe d2 ) Oe 30d .
dOe1 3 .
Oe d ) Oe 20d For convenient, we have calculated each summation separately to the corresponding edge partition as mentioned early.
For the edge partition E1 :
EAa .
CLd )EAb .
CLd ) = 2d.
e=abOOE1 (CLd ) For the edge partition E2 :
EAa .
CLd )EAb .
CLd ) = .
22d e=abOOE2 (CLd ) For the edge partition E3 :
EAa .
CLd )EAb .
CLd ) = 2d.
d Oe .
e=abOOE3 (CLd ) Kandan.
Subramanian .
i Oe .
d 16 Oe 22.
i Oe .
d 17 Oe 22.
i Oe .
d 12 Oe 22.
i 18 Oe 22.
i Oe .
i Oe .
d 17 Oe 22.
i Oe .
d 11 Oe 22.
dOe1 Oe .
d 1 Oe 22.
Oe .
) dOe1 Oe .
d Oe 22.
Oe .
) dOe1 Oe d2 ) = 2d.
d Oe .
dOe1 dOe1 Oe 240d .
Oe d2 ) Oe 272d .
dOe1 Oe d2 ) Oe 110d Oe d2 ) Oe 132d .
dOe1 dOe1 Oe d2 ) Oe 306d .
242 22d 25 22d 25 176.
Oe d2 ) Oe 272d 11d.
Oe .
Oe .
dOe1 22d 25 = 4d.
d Oe .
1452 1001 935.
Oe d2 ) Oe 1332d 22d.
Oe .
For the edge partition E4 :
EAa .
CLd )EAb .
CLd ) = 3 i Oe .
d 16 Oe 22.
i Oe .
d 17 Oe 22.
e=abOOE4 (CLd ) .
i Oe .
i 6 Oe 22.
i Oe .
d 5 Oe 22.
i Oe i2 ) 374i 352.
Oe .
Oe 272 i Oe i2 ) 352i 330.
Oe .
Oe 240 i Oe i2 ) 132i 110.
Oe .
Oe 30 i Oe i2 ) 110i 88.
Oe .
Oe 20 i Oe i2 ) 374i 352.
Oe .
Oe .
i Oe i2 ) 352i 330.
Oe .
Oe 240 i Oe i2 ) 132i 110.
Oe .
Oe .
i Oe i2 ) 110i 88.
Oe .
Oe .
dOe1 dOe1 = 3 .
Oe d ) Oe 272d .
dOe1 Oe d2 ) Oe 240d 3 .
Oe d2 ) Oe 30d dOe1 Oe d2 ) Oe 20d Some Bond-Additive Indices and Its Polynomial of Cellulose dOe1 = .
Oe d2 ) Oe 1166d Hence by summing up these values we have dOe1 Sz(CLd ) = 2d.
22d 4d.
d Oe .
dOe1 Oe d ) Oe 1332d 22d.
Oe .
Oe d2 ) Oe 1166d Hence the theorem.
n Next, we obtained obtained the weighted version of szeged index in the consecutive theorems.
Theorem 3.
Let CLd be the chemical graph of cellulose, dOe1 SzA (CLd ) = 3.
) 4(.
5 4d.
d Oe .
dOe1 Oe d ) Oe 1332d 22d.
Oe .
Oe d2 ) Oe 1166d Proof.
To obtain the additively weighted szeged index of the CLd , by the definition of Additively weighted szeged index and from the Table 1.
we have SzA (CLd ) = (Oa .
CLd ) Ob .
CLd ))EAa .
CLd )EAb .
CLd ) e=abAE(CLd ) (Oa .
CLd ) Ob .
CLd ))EAa .
CLd )EAb .
CLd ) i=1,2,3,4.
e=abAEi (CLd ) By using Theorem 3.
1, we have SzA (CLd ) = .
(Sz(E1 )) .
(Sz(E2 )) .
(Sz(E3 )) .
(Sz(E4 )) dOe1 = 3.
) 4(.
d Oe .
22d 25 935.
Oe d2 ) Oe 1332d 22d.
Oe .
dOe1 6 d2 d 1936 990 902.
Oe d2 ) Oe 1166d Hence the theorem.
n Theorem 3.
Let CLd be the chemical graph of cellulose.
SzM (CLd ) = 2.
) 3(.
6 4d.
d Oe .
dOe1 1001 935.
Oe d2 ) Oe 1332d 22d.
Oe .
22d 25 dOe1 9(.
1936 990 902.
Oe d2 ) Oe 1166.
Kandan.
Subramanian Proof.
To obtained the Multiplicatively weighted szeged index of CLd , by the definition of Multiplicatively weighted szeged index and from the Table 1.
we have SzM (CLd ) = (Oa .
CLd ).
Ob .
CLd ))EAa .
CLd )EAb .
CLd ) e=abOOE(CLd ) (Oa .
CLd ).
Ob .
CLd ))EAa .
CLd )EAb .
CLd ) i=1,2,3,4.
e=abOOEi (CLd ) By using Theorem 3.
1, we have SzM (CLd ) = .
(Sz(E1 )) .
(Sz(E2 )) .
(Sz(E3 )) .
(Sz(E4 )) dOe1 = 2.
) 3(.
6 4d.
d Oe .
dOe1 Oe d2 ) Oe 1332d 22d.
Oe .
Oe d2 ) Oe 1166.
Hence the theorem.
n To the continuity of the above result, now we derive the exact expression of szeged and its related polynomial Theorem 3.
Let CLd be the chemical graph of cellulose.
Sz(CLd , .
= 2dx22d .
x22d 2dx.
dOe.
d 2 x484.
iOei ) 352i 330.
Oe.
Oe240 5 x 484.
iOei2 ) 374i 352.
Oe.
Oe272 484.
dOei2 ) 264i 242.
Oe.
Oe132 x484.
dOei ) 242i 220.
Oe.
Oe110 x484.
iOei ) 374.
Oe.
396iOe306 3 x484.
iOei ) 132i 110.
Oe.
Oe30 x484.
iOei ) 110i 88.
Oe.
Oe20 dOe1 x22(.
Oe.
Oe22.
Oe.
) dOe1 x22d(.
Oe.
)Oe.
Oe.
Oe.
Proof.
To obtained the szeged polynomial of CLd , by the definition of szeged polynomial and applying the Table 1.
, we have Sz(CLd , .
= xEAa .
CLd )EAb .
CLd ) e=abAE(CLd ) Sz(CLd , .
= e=abAE1 (CLd ) xEAa .
CLd )EAb .
CLd ) xEAa .
CLd )EAb .
CLd ) e=abAE2 (CLd ) xEAa .
CLd )EAb .
CLd ) e=abAE3 (CLd ) xEAa .
CLd )EAb .
CLd ) e=abAE4 (CLd ) For convenient, we have calculated each summation separately to the corresponding edge partitions as mentioned early.
For the edge partition E1 :
e=abAE1 (CLd ) xEAa .
CLd )EAb .
CLd ) = e=abAE1 (CLd ) x22d = 2dx22d Some Bond-Additive Indices and Its Polynomial of Cellulose For edge partition E2 :
xEAa .
CLd )EAb .
CLd ) = e=abAE2 (CLd ) x22d = .
x22d e=abAE2 (CLd ) For edge partition E3 :
xEAa .
CLd )EAb .
CLd ) = 2dx.
dOe.
iOe.
d 16Oe22.
e=abAE3 (CLd ) x.
iOe.
d 17Oe22.
iOe.
d 11Oe22.
iOe.
d 17Oe22.
iOe.
d 12Oe22.
i 18Oe22.
iOe.
Oe.
d 1Oe22.
Oe.
) x.
Oe.
dOe22.
Oe.
) = 2dx.
dOe.
iOei ) 352i 330.
Oe.
Oe240 x484.
iOei ) 374i 352.
Oe.
Oe272 x484.
dOei ) 264i 242.
Oe.
Oe132 x484.
dOei ) 242i 220.
Oe.
Oe.
dOe1 x484.
iOei ) 374.
Oe.
396iOe306 x22(.
Oe.
Oe22.
Oe.
) dOe1 x22d.
Oe.
)Oe.
Oe.
Oe.
= 2dx.
dOe.
iOei ) 352i 330.
Oe.
Oe240 2 x 484.
iOei2 ) 374i 352.
Oe.
Oe272 x484.
dOei ) 264i 242.
Oe.
Oe132 x484.
dOei ) 242i 220.
Oe.
Oe110 x484.
iOei ) 374.
Oe .
396i Oe 306 dOe1 x22(.
Oe.
Oe22.
Oe.
) dOe1 22d.
Oe.
Oe.
Oe.
Oe.
For edge partition E4 :
e=abAE4 (CLd ) xEAa .
CLd )EAb .
CLd ) = 3 x.
iOe.
d 17Oe22.
Kandan.
Subramanian x.
iOe.
d 16Oe22.
3 x.
iOe.
i 6Oe22.
iOe.
d 5Oe22.
iOei ) 374i 352.
Oe.
Oe272 x484.
iOei ) 132i 110.
Oe.
Oe30 x484.
iOei ) 352i 330.
Oe.
Oe240 x484.
iOei ) 110i 88.
Oe.
Oe20 d 3 x484.
iOei ) 374i 352.
Oe.
Oe272 x484.
iOei ) 352i 330.
Oe.
Oe240 3 x484.
iOei ) 132i 110.
Oe.
Oe30 x484.
iOei ) 110i 88.
Oe.
Oe20 Hence by summarizing these values obtained here for the partitions E1 ,E2 ,E3 and E4 , we have Sz(CLd , .
= 2dx22d .
x22d 2dx.
dOe.
d 2 x484.
iOei ) 352i 330.
Oe.
Oe240 5 x 484.
iOei2 ) 374i 352.
Oe.
Oe272 484.
dOei2 ) 264i 242.
Oe.
Oe132 x484.
dOei ) 242i 220.
Oe.
Oe110 x484.
iOei ) 374.
Oe.
396iOe306 3 x 484.
iOei2 ) 132i 110.
Oe.
Oe30 484.
iOei2 ) 110i 88.
Oe.
Oe20 dOe1 x22(.
Oe.
Oe22.
Oe.
) dOe1 x22d(.
Oe.
)Oe.
Oe.
Oe.
Hence the theorem.
n Theorem 3.
Let CLd be the chemical graph of cellulose.
SzA (CLd , .
= 2dx3.
dOe.
iOei ) 352i 330.
Oe.
Oe.
2 x5.
iOei ) 374i 352.
Oe.
Oe.
dOei ) 264i 242.
Oe.
Oe.
dOei ) 242i 220.
Oe.
Oe.
iOei ) 374.
Oe.
396iOe.
3 x6.
iOei ) 374i 352.
Oe.
Oe.
iOei ) 352i 330.
Oe.
Oe.
3 x6.
iOei ) 132i 110.
Oe.
Oe.
dOe1 iOei ) 110i 88.
Oe.
Oe.
Oe.
Oe22.
Oe.
)) dOe1 x5.
Oe.
)Oe.
Oe.
Oe.
) .
Some Bond-Additive Indices and Its Polynomial of Cellulose Proof.
To obtained the Additively weighted szeged polynomial of CLd , by the definition of Additively weighted szeged polynomial and from the Table 1.
, we have SzA (CLd , .
= x(Oa .
CLd ) Ob .
CLd ))EAa .
CLd )EAb .
CLd ) e=abAE(CLd ) SzA (CLd , .
= x(Oa .
CLd ) Ob .
CLd ))EAa .
CLd )EAb .
CLd ) e=abAE1 (CLd ) x(Oa .
CLd ) Ob .
CLd ))EAa .
CLd )EAb .
CLd ) e=abAE2 (CLd ) x(Oa .
CLd ) Ob .
CLd ))EAa .
CLd )EAb .
CLd ) e=abAE3 (CLd ) x(Oa .
CLd ) Ob .
CLd ))EAa .
CLd )EAb .
CLd ) e=abAE4 (CLd ) For our convenient, now we calculate each summation separately to the corresponding edge partition as mentioned early.
For edge partition E1 :
x(Oa .
CLd ) Ob .
CLd ))EAa .
CLd )EAb .
CLd ) = 2dx.
22d e=abAE1 (CLd ) For edge partition E2 :
x(Oa .
CLd ) Ob .
CLd ))EAa .
CLd )EAb .
CLd ) = e=abAE2 (CLd ) x.
22d e=abAE2 (CLd ) = .
22d For edge partition E3 :
x(Oa .
CLd ) Ob .
CLd ))EAa .
CLd )EAb .
CLd ) e=abAE3 (CLd ) .
dOe.
= 2dx .
iOe.
d 16Oe22.
iOe.
d 12Oe22.
iOe.
d 11Oe22.
i 18Oe22.
iOe.
d 17Oe22.
iOe.
Oe.
d 1Oe22.
Oe.
) x.
Oe.
dOe22.
Oe.
) 5.
dOe.
= 2dx 5.
iOei2 ) 352i 330.
Oe.
Oe.
dOe1 x5.
dOei ) 264i 242.
Oe.
Oe.
iOei ) 374.
Oe.
396iOe.
iOei ) 374i 352.
Oe.
Oe.
iOe.
d 17Oe22.
dOei ) 242i 220.
Oe.
Oe.
) x5.
Oe.
Oe22.
Oe.
)) Kandan.
Subramanian dOe1 x5.
Oe.
)Oe.
Oe.
Oe.
) = 2dx5.
dOe.
iOei ) 352i 330.
Oe.
Oe.
2 x 5.
iOei2 ) 374i 352.
Oe.
Oe.
dOei ) 264i 242.
Oe.
Oe.
dOei ) 242i 220.
Oe.
Oe.
iOei ) 374.
Oe.
396iOe.
dOe1 5.
Oe.
Oe22.
Oe.
2 )) dOe1 5.
Oe.
)Oe.
Oe.
Oe.
) For edge partition E4 :
x(Oa .
CLd ) Ob .
CLd ))EAa .
CLd )EAb .
CLd ) e=abAE4 (CLd ) x.
iOe.
d 17Oe22.
iOe.
d 16Oe22.
iOe.
i 6Oe22.
iOe.
d 5Oe22.
iOei ) 374i 352.
Oe.
Oe.
iOei ) 352i 330.
Oe.
Oe.
iOei ) 132i 110.
Oe.
Oe.
iOei ) 110i 88.
Oe.
Oe.
d 3 x6.
iOei ) 374i 352.
Oe.
Oe.
iOei ) 352i 330.
Oe.
Oe.
3 x6.
iOei ) 132i 110.
Oe.
Oe.
iOei ) 110i 88.
Oe.
Oe.
Hence by summarizing these values obtained here for E1 .
E2 ,E3 and E4 , we have SzA (CLd , .
= 2dx3.
dOe.
iOei ) 352i 330.
Oe.
Oe.
2 x5.
iOei ) 374i 352.
Oe.
Oe.
dOei ) 264i 242.
Oe.
Oe.
dOei ) 242i 220.
Oe.
Oe.
iOei ) 374.
Oe.
396iOe.
3 x6.
iOei ) 374i 352.
Oe.
Oe.
iOei ) 352i 330.
Oe.
Oe.
3 x6.
iOei ) 132i 110.
Oe.
Oe.
Some Bond-Additive Indices and Its Polynomial of Cellulose dOe1 iOei ) 110i 88.
Oe.
Oe.
Oe.
Oe22.
Oe.
)) dOe1 x5.
Oe.
Oe.
Oe.
Oe.
)) Hence the theorem.
n Theorem 3.
Let CLd be the chemical graph of cellulose.
SzM (CLd , .
= 2dx2.
dOe.
iOei ) 352i 330.
Oe.
Oe.
2 x6.
iOei ) 374i 352.
Oe.
Oe.
dOei ) 264i 242.
Oe.
Oe.
dOei ) 242i 220.
Oe.
Oe110 x6.
iOei ) 374.
Oe.
396iOe.
3 x9.
iOei ) 374i 352.
Oe.
Oe.
iOei ) 352i 330.
Oe.
Oe.
3 x9.
iOei ) 132i 110.
Oe.
Oe.
dOe1 iOei ) 110i 88.
Oe.
Oe.
Oe.
Oe22.
Oe.
)) dOe1 x6.
Oe.
)Oe.
Oe.
Oe.
) Proof.
To obtained the Multiplicatively weighted szeged polynomial of CLd , by the definition of Multiplicatively weighted szeged polynomial and from the Table 1.
, we SzM (CLd , .
= x(Oa .
CLd ).
Ob .
CLd ))EAa .
CLd )EAb .
CLd ) e=abAE(CLd ) SzM (CLd , .
= x(Oa .
CLd ).
Ob .
CLd ))EAa .
CLd )EAb .
CLd ) e=abAE1 (CLd ) x(Oa .
CLd ).
Ob .
CLd ))EAa .
CLd )EAb .
CLd ) e=abAE2 (CLd ) x(Oa .
CLd ).
Ob .
CLd ))EAa .
CLd )EAb .
CLd ) e=abAE3 (CLd ) x(Oa .
CLd ).
Ob .
CLd ))EAa .
CLd )EAb .
CLd ) e=abAE4 (CLd ) For convenient, we have calculated each summation separately to the corresponding edge partition as mentioned early.
For edge partition E1 :
e=abAE1 (CLd ) x(Oa .
CLd ).
Ob .
CLd ))EAa .
CLd )EAb .
CLd ) = e=abAE1 (CLd ) = 2dx.
22d x.
22d Kandan.
Subramanian For edge partition E2 :
x(Oa .
CLd ).
Ob .
CLd ))EAa .
CLd )EAb .
CLd ) = e=abAE2 (CLd ) x.
22d e=abAE2 (CLd ) = .
22d For edge partition E3 :
x(Oa .
CLd ).
Ob .
CLd ))EAa .
CLd )EAb .
CLd ) e=abAE3 (CLd ) = 2dx.
dOe.
iOe.
d 16Oe22.
iOe.
d 12Oe22.
i 18Oe22.
iOe.
iOe.
d 17Oe22.
dOe1 x.
iOe.
d 11Oe22.
d 17Oe22.
iOe.
Oe.
d 1Oe22.
Oe.
) dOe1 x.
Oe.
dOe22.
Oe.
) = 2dx6.
dOe.
iOei ) 352i 330.
Oe.
Oe.
2 dOe1 x6.
dOei ) 264i 242.
Oe.
Oe.
iOei ) 374.
Oe.
396iOe.
dOei ) 242i 220.
Oe.
Oe.
) x6.
Oe.
Oe22.
Oe.
)) dOe1 x5.
iOei ) 374i 352.
Oe.
Oe.
Oe.
)Oe.
Oe.
Oe.
) = 2dx6.
dOe.
iOei ) 352i 330.
Oe.
Oe.
2 x6.
iOei ) 374i 352.
Oe.
Oe.
dOei2 ) 264i 242.
Oe.
Oe.
dOei ) 242i 220.
Oe.
Oe.
iOei ) 374.
Oe.
396iOe.
dOe1 x6.
Oe.
Oe22.
Oe.
)) dOe1 6.
Oe.
)Oe.
Oe.
Oe.
) For edge partition E4 :
x(Oa .
CLd ).
Ob .
CLd ))EAa .
CLd )EAb .
CLd ) e=abAE4 (CLd ) x.
iOe.
d 17Oe22.
iOe.
d 16Oe22.
Some Bond-Additive Indices and Its Polynomial of Cellulose x.
iOe.
i 6Oe22.
iOe.
d 5Oe22.
iOei ) 374i 352.
Oe.
Oe.
iOei ) 132i 110.
Oe.
Oe.
iOei ) 352i 330.
Oe.
Oe.
iOei ) 110i 88.
Oe.
Oe.
d 3 x9.
iOei ) 374i 352.
Oe.
Oe.
iOei ) 352i 330.
Oe.
Oe.
3 x9.
iOei ) 132i 110.
Oe.
Oe.
iOei ) 110i 88.
Oe.
Oe.
Hence by summarizing these values obtained here for E1 .
E2 ,E3 and E4 , we have SzM (CLd , .
= 2dx2.
dOe.
iOei ) 352i 330.
Oe.
Oe.
2 x6.
iOei ) 374i 352.
Oe.
Oe.
dOei ) 264i 242.
Oe.
Oe.
dOei ) 242i 220.
Oe.
Oe110 x6.
iOei ) 374.
Oe.
396iOe.
3 x9.
iOei ) 374i 352.
Oe.
Oe.
iOei ) 352i 330.
Oe.
Oe.
3 x9.
iOei ) 132i 110.
Oe.
Oe.
iOei2 ) 110i 88.
Oe.
Oe.
dOe1 x6.
Oe.
Oe22.
Oe.
) dOe1 x6.
Oe.
Oe.
Oe.
Oe.
) Hence the theorem.
n Using the results obtained here, the following remarks are easy to observe.
Remark 3.
Sz (CLd , .
= Sz(CLd ) and Sz(CLd , .
= |E(CLd )|.
Remark 3.
SzA (CLd , .
= SzA (CLd ) and SzA (CLd , .
= |E(CLd )|.
Remark 3.
SzM (CLd , .
= SzM (CLd ) and SzM (CLd , .
= |E(CLd )|.
The following corollaries shows the relationship between szeged index.
Additively weigted szeged index and Multiplicatively weigted szeged index for cellulose.
Corollary 3.
SzA (CLd ) = 5Sz(CLd ) Oe 2Sz(E1 ) Oe Sz(E2 ) Sz(E4 ).
Corollary 3.
SzM (CLd ) = 6Sz(CLd ) Oe 4Sz(E1 ) Oe 3Sz(E2 ) 3Sz(E4 ).
Corollary 3.
Sz(CLd ) < SzA (CLd ) < SzM (CLd ).
PADMAKAR-IVAN INDEX AND ITS POLYNOMIAL
In this section, we compute another interesting topological indices based on distance called the Padmakar-Ivan index and its polynomial.
Observe that for any Kandan.
Subramanian edge e = ab of cellulose CLd has no equidistant vertices, thus we have EAa .
CLd ) EAb .
CLd ) = |V (CLd )|.
Theorem 4.
Let CLd be the chemical graph of cellulose, then P I(CLd ) = 24d.
Proof.
To obtained the Padmakar-Ivan index of the CLd , by the definition of P I index and from the Table 1.
, we have P I(CLd ) = (EAa .
CLd ) EAb .
CLd )) e=abAE(CLd ) (EAa .
CLd ) EAb .
CLd )) (EAa .
CLd ) EAb .
CLd )) e=abAE1 (CLd ) e=abAE2 (CLd ) (EAa .
CLd ) EAb .
CLd )) e=abAE3 (CLd ) (EAa .
CLd ) EAb .
CLd )) e=abAE4 (CLd ) Since there is no equidistant vertices exist in CLd , thus we have P I(CLd ) = 2d.
d Oe .
= .
d 4d 2 10d Oe 2 8.
= 24d.
Hence the theorem.
n Theorem 4.
Let CLd be the chemical graph of cellulose, then P IA (CLd ) = .
d Oe .
Proof.
To obtained the Additively weighted Padmakar-Ivan index of the CLd , by the definition of Additively weighted P I index and from the Table 1.
, we have P IA (CLd ) = (Oa .
CLd ) Ob .
CLd ))(EAa .
CLd ) EAb .
CLd )) e=abAE(CLd ) .
(EAa .
CLd ) EAb .
CLd )) .
(EAa .
CLd ) EAb .
CLd )) e=abAE1 (CLd ) e=abAE2 (CLd ) .
(EAa .
CLd ) EAb .
CLd )) e=abAE3 (CLd ) .
(EAa .
CLd ) EAb .
CLd )) e=abAE4 (CLd ) = 2d.
|V (CLd )| .
|V (CLd )| .
d Oe .
|V (CLd )| .
|V (CLd )| = .
d 16d 8 50d Oe 10 48.
|V (CLd )| = .
d Oe .
|V (CLd )|, since |V (CLd )| = 22d 1 = .
d Oe .
Hence the theorem.
n It should be noted that a similar method can also be used in the study of more general Gaussian-type indices see .
on generalized distance Gaussian Estrada index of graph.
Theorem 4.
Let CLd be the chemical graph of cellulose, then P IM (CLd ) = .
d Oe .
Some Bond-Additive Indices and Its Polynomial of Cellulose Proof.
To obtained the Multiplicatively weighted Padmakar-Ivan index of the CLd , by the definition of Multiplicatively weighted P I index and from the Table 1.
, we P IM (CLd ) = (Oa .
CLd ).
Ob .
CLd ))(EAa .
CLd ) EAb .
CLd )) e=abAE(CLd ) .
(EAa .
CLd ) EAb .
CLd )) .
(EAa .
CLd ) EAb .
CLd )) e=abAE1 (CLd ) e=abAE2 (CLd ) .
(EAa .
CLd ) EAb .
CLd )) e=abAE3 (CLd ) .
(EAa .
CLd ) EAb .
CLd )) e=abAE4 (CLd ) = 2d.
|V (CLd )| .
|V (CLd )| .
d Oe .
|V (CLd )| .
|V (CLd )| = .
d 12d 6 60d Oe 12 72.
|V (CLd )| = .
d Oe .
|V (CLd )|, since |V (CLd )| = 22d 1 = .
d Oe .
Hence the theorem.
n The relationships between the Padmakar-Ivan index.
Additively weighted Padmakar-Ivan index, and Multiplicatively weighted Padmakar-Ivan index for the cellulose graphs are shown in the following corollaries.
Corollary 4.
P IA (CLd ) = 4P I(CLd ) |V (CLd )||E(CLd )| Oe .
V (CLd )|.
Corollary 4.
P IM (CLd ) = 3P I(CLd ) 2.
d Oe .
|V (CLd )|.
Corollary 4.
P I(CLd ) < P IA (CLd ) < P IM (CLd ).
Example 4.
One can check the above corollaries for, d = 3, with |V (CL3 )| = 67, |E(CL3 )| = 72.
P I(CL3 ) = 4824.
P IA (CL3 ) = 23986.
P IM (CL3 ) = 29346.
One may generalize the above corollaries for any molecular structure.
the continuity of the above result, now we derive the P I related polynomialAos of Theorem 4.
Let CLd be the chemical graph of cellulose, then P I(CLd , .
= 24dx.
Proof.
To obtained the Padmakar-Ivan polynomial of the CLd , by the definition of Padmakar-Ivan polynomial and from the Table 1.
, we have P I(CLd , .
= x(EAa .
CLd ) EAb .
CLd )) e=abAE.
CLd ) x(EAa .
CLd ) EAb .
CLd )) e=abAE1 (CLd ) = 2dx x(EAa .
CLd ) EAb .
CLd )) e=abAE2 (CLd ) x(EAa .
CLd ) EAb .
CLd )) e=abAE3 (CLd ) |V (CLd )| x(EAa .
CLd ) EAb .
CLd )) e=abAE4 (CLd ) .
x |V (CLd )| .
d Oe .
V (CLd )| 8d.
V (CLd )| = .
d 4d 2 10d Oe 2 8.
, since |V (CLd )| = 22d 1 Kandan.
Subramanian = 24dx.
Hence the theorem.
n In this connection now we can obtained the weighted P I related polynomial using the Table 1.
Theorem 4.
Let CLd be the chemical graph of cellulose, then P IA (CLd , .
= 2dx3.
d Oe .
| .
Proof.
To obtained the Additively weighted P I polynomial of the CLd , by the definition of Additively weighted P I polynomial and from the Table 1.
, we have P IA (CLd , .
= x(Oa .
CLd ) Ob .
CLd ))(EAa .
CLd ) EAb .
CLd )) e=abAE(CLd ) x.
(EAa .
CLd ) EAb .
CLd )) e=abAE1 (CLd ) x.
(EAa .
CLd ) EAb .
CLd )) e=abAE2 (CLd ) .
(EAa .
CLd ) EAb .
CLd )) x.
(EAa .
CLd ) EAb .
CLd )) e=abAE4 (CLd ) e=abAE3 (CLd ) = 2dx .
|V (CLd )| .
|V (CLd )| .
d Oe .
|V (CLd )| .
|V (CLd )| , since |V (CLd )| = 22d 1 = 2dx3.
d Oe .
Hence the theorem.
n Theorem 4.
Let CLd be the chemical graph of cellulose, then P IM (CLd , .
= 2dx2.
d Oe .
Proof.
To obtained the Multiplicatively weighted P I polynomial of the CLd , by the definition of Multiplicatively weighted P I polynomial and from the Table 1.
we have P IM (CLd , .
= x(Oa .
CLd ).
Ob .
CLd ))(EAa .
CLd ) EAb .
CLd )) e=abAE(CLd ) x.
(EAa .
CLd ) EAb .
CLd )) e=abAE1 (CLd ) x.
(EAa .
CLd ) EAb .
CLd )) e=abAE2 (CLd ) x.
(EAa .
CLd ) EAb .
CLd )) e=abAE3 (CLd ) .
|V (CLd )| = 2dx x.
(EAa .
CLd ) EAb .
CLd )) e=abAE4 (CLd ) .
|V (CLd )| .
d Oe .
|V (CLd )| .
|V (CLd )| , since |V (CLd )| = 22d 1 = 2dx2.
d Oe .
Hence the theorem.
n Finally we observe the following remarks, which follows easily from the results discussed here.
Remark 4.
P I (CLd , .
= P I(CLd ) and P I(CLd , .
= |E(CLd )|.
Remark 4.
P IA (CLd , .
= P IA (CLd ) and P IA (CLd , .
= |E(CLd )|.
Remark 4.
P IM (CLd , .
= P IM (CLd ) and P IM (CLd , .
= |E(CLd )|.
Some Bond-Additive Indices and Its Polynomial of Cellulose CONCLUSION In this study, we primarily calculate the bond-additive based indices such as szeged.
PI, weighted Szeged, weighted PI and their polynomials of cellulose graphs using chemical graph analysis are distance calculation.
Our theoretical formulations demonstrate the great potential for practical implementation in pharmacy and chemical engineering.
As directed networks often predict complex dynamical behaviors more faithfully than undirected networks, it is desirable to bring directness to higher-order structures in the future.
REFERENCES