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Symmetry 2018, 10(7), 237; https://doi.org/10.3390/sym10070237

Article
On Eccentricity-Based Topological Indices and Polynomials of Phosphorus-Containing Dendrimers
1
Department of Mathematics and Research Institute of Natural Science, Gyeongsang National University, Jinju 52828, Korea
2
Center for General Education, China Medical University, Taichung 40402, Taiwan
3
School of Natural Sciences, National University of Sciences and Technology, Sector H-12, Islamabad 44000, Pakistan
4
University of Engineering and Technology, Lahore 54000, Pakistan (RCET)
5
Department of Mathematics, Division of Science and Technology, University of Education, Lahore 54000, Pakistan
*
Author to whom correspondence should be addressed.
Received: 26 May 2018 / Accepted: 21 June 2018 / Published: 24 June 2018

Abstract

:
In the study of the quantitative structure–activity relationship and quantitative structure-property relationships, the eccentric-connectivity index has a very important place among the other topological descriptors due to its high degree of predictability for pharmaceutical properties. In this paper, we compute the exact formulas of the eccentric-connectivity index and its corresponding polynomial, the total eccentric-connectivity index and its corresponding polynomial, the first Zagreb eccentricity index, the augmented eccentric-connectivity index, and the modified eccentric-connectivity index and its corresponding polynomial for a class of phosphorus containing dendrimers.
Keywords:
eccentric-connectivity index; augmented eccentric-connectivity index; molecular graph; phosphorus containing dendrimers
MSC:
05C90

1. Introduction

Dendrimers are synthetic polymers with highly branched structures, consisting of multiple branched monomers radiating from a central core. Layers of monomers are attached stepwise during synthesis, with the number of branch points defining the generation of a dendrimer [1]. Different kinds of experiments have proved that these polymers with well-defined dimensional structures and topological architectures have an array of applications in medicine [2]. Nowadays, dendrimers are currently attracting the interest of a great number of scientists because of their unusual physical and chemical properties and their wide range of potential applications in different fields, such as physics, biology, chemistry, engineering, and medicine [3]. A topological index, sometimes known as a graph theoretic index, is a numerical invariant of a chemical graph. Topological indices are the mathematical measures associated with molecular graph structures that correlate a chemical structure with various physical properties, biological activities or chemical reactivities. A topological index is an invariant of a graph, G 1 ; that is, if T o p ( G 1 ) denotes a topological index of graph G 1 , and if G 2 is another graph such that G 1 G 2 , then T o p ( G 1 ) = T o p ( G 2 ) . In chemistry, biochemistry and nanotechnology, distance-based topological indices of graphs are useful in isomer discrimination, structure–property relationships and structure–activity relationships.

2. Definitions and Notations

Let G be a connected and simple molecular graph with vertex set, V ( G ) , and edge set, E ( G ) . The vertices of G correspond to atoms, and an edge between two vertices corresponds to the chemical bond between these vertices. In graph G, two vertices, u and v, are adjacent, if and only if, they are the end vertices of an edge, e E ( G ) , and we write e = u v or e = v u . For a vertex, u, the set of neighbor vertices is denoted by N u and is defined as N u = { v V ( G ) : u v E ( G ) } . The degree of vertex u V ( G ) is denoted by d u and is defined as d u = | N u | . Let S u denote the sum of the degrees of all neighbors of vertex u, that is S u = v N u d v . A ( u 1 , u n ) -path on n vertices is defined as a graph with vertex set, { u i : 1 i n } , and edge set, { u i u i + 1 : 1 i n 1 } . The distance, d ( u , v ) , between two vertices, u , v V ( G ) , is defined as the length of the shortest ( u , v ) -path in G. For a given vertex, v V ( G ) , the eccentricity, ε ( v ) , is defined as the largest distance between v and any other vertex, u in G. In 1947, Harold Wiener published a paper entitled “Structural Determination of Paraffin Boiling Points” [4]. In this work, the quantity, W e , eventually named the Wiener index or Wiener number, was introduced for the first time, and he showed that there are excellent correlations between W e and a variety of physico-chemical properties of organic compounds. Another distance-based topological index of the graph G is the eccentric-connectivity index, ξ ( G ) , which is defined as [5]
ξ ( G ) = u V ( G ) ε ( u ) d u .
Different applications and mathematical properties of this index were discussed in [6,7,8,9]. For a graph, G, the eccentric-connectivity polynomial in variable y is defined as [10]
E C P ( G , y ) = u V ( G ) d u y ε ( u ) .
The total eccentricity index of a graph, G, is expressed as follows:
ς ( G ) = u V ( G ) ε ( u ) .
The total eccentric-connectivity polynomial in variable y of a graph, G, is defined as [10]
T E C P ( G , y ) = u V ( G ) y ε ( u ) .
The first Zagreb index of a graph, G, in terms of eccentricity was given by Ghorbani and Hosseinzadeh [11], as follows:
M 1 * * ( G ) = u V ( G ) ( ε ( u ) ) 2 .
Gupta and his co-authors [12] introduced the augmented eccentric-connectivity index of a graph, G, and it is defined as
A ε ( G ) = u V ( G ) M ( u ) ε ( u ) ,
where M ( u ) denotes the product of degrees of all neighbors of vertex u. Various properties of this index have been studied in [13,14]. For a graph, G, the modified versions of the eccentric-connectivity index and its polynomial are defined as follows
Λ ( G ) = u V ( G ) S u ε ( u ) ,
M E C P ( G , y ) = u V ( G ) S u y ε ( u ) .
Several mathematical and chemical properties of the modified eccentric-connectivity index and its polynomial were studied in [10,15]. Some major types of topological indices of graphs are degree-based, distance-based, and counting-related. Some degree-based topological indices have been computed for some classes of dendrimers, see for instance [16,17,18]. For a study of distance-based topological indices, see [19,20,21]. In this paper, we compute several distance-based indices, namely, the eccentric-connectivity index, the total eccentric-connectivity index, and the modified eccentric-connectivity index for the phosphorus-containing dendrimer Cyclotriphosphazene ( N 3 P 3 ) [22]. We also compute the corresponding polynomials of these indices for the same dendrimer. We also compute the first Zagreb eccentricity index and the augmented eccentric-connectivity index for the said dendrimer.

3. The Eccentricity-Based Indices and Polynomials for the Molecular Graph

Let the molecular graph of this dendrimer be D ( n ) , where the generation stage of D ( n ) is represented by n. The first and second generations are shown in Figure 1 and Figure 2 respectively.
The size and order of graph D ( n ) are 6 ( 9 × 2 n + 2 13 ) and 9 ( 8 + 11 × 2 n ) , respectively. To compute the eccentricity-based indices and polynomials of D ( n ) , it is enough to compute the required information for a set of representatives of V ( D ( n ) ) . We will compute the required information by using computational arguments. We make three sets of representatives of V ( D ( n ) ) , say A = { α 1 , α 2 } , B = { β 1 , β 2 , , β 13 } and C = { a i , b i , c i , d i , e i , f i , g i , h i , j i , k i , l i } where 1 i n , as shown in Figure 1 and Figure 2. The degree, S u , M ( u ) , and eccentricity for each u for the sets A, B, and C are shown in Table 1 and Table 2. For simplicity, we assume γ = 9 n + 9 i throughout the paper. By using Table 1 and Table 2, we calculate the different eccentricity-based indices and their corresponding polynomials. In the following theorem, we determine the eccentric-connectivity index of D ( n ) .
Theorem 1.
For graph D ( n ) , the eccentric-connectivity index is given by
ξ ( D ( n ) ) = 18 ( 2 n + 2 × 79 78 n + 2 n × 303 n + 1 ) .
Proof. 
By putting the values of Table 1 and Table 2 into Equation (1), the eccentric-connectivity index of D ( n ) can be written as follows:
ξ ( D ( n ) ) = ξ ( A ) + ξ ( B ) + ξ ( C ) = u A ε ( u ) d u + u B ε ( u ) d u + u C ε ( u ) d u = ( 2 × 3 ) ( 9 n + 15 ) + ( 3 × 4 ) ( 9 n + 14 ) + ( 3 × 2 n + 1 × 2 ) ( 9 n + 15 ) + ( 3 × 2 n + 1 × 3 ) ( 9 n + 16 ) + ( 2 × 2 n + 2 × 3 ) ( 9 n + 17 ) + ( 2 × 2 n + 2 × 3 ) ( 9 n + 18 ) + ( 3 × 2 n + 1 × 3 ) ( 9 n + 19 ) + ( 2 × 2 n + 1 × 3 ) ( 9 n + 20 ) + ( 2 × 2 n + 1 × 3 ) ( 9 n + 21 ) + ( 2 × 2 n + 2 × 3 ) ( 9 n + 23 ) + ( 4 × 2 n + 2 × 3 ) ( 9 n + 24 ) + ( 1 × 2 n + 3 × 3 ) ( 9 n + 25 ) + ( 3 × 2 n + 1 × 3 ) ( 9 n + 22 ) + ( 3 × 2 n + 1 × 3 ) ( 9 n + 25 ) + ( 1 × 2 n + 1 × 3 ) ( 9 n + 26 ) + i = 1 n ( ( 2 × 2 i × 3 ) ( γ + 6 ) + ( 3 × 2 i × 3 ) ( γ + 7 ) + ( 2 × 2 i + 1 × 3 ) ( γ + 8 ) + ( 2 i + 2 × 3 ) ( γ + 9 ) + ( 3 × 2 i × 3 ) ( γ + 10 ) + ( 3 × 2 i + 1 ) ( γ + 11 ) + ( 2 i + 1 × 3 ) ( γ + 12 ) + ( 3 × 2 i × 3 ) ( γ + 13 ) + ( 2 i × 3 ) ( γ + 14 ) + ( 4 × 2 i × 3 ) ( γ + 14 ) + ( 2 i × 3 ) ( γ + 15 ) ) .
After some calculations, we get
ξ ( D ( n ) ) = 18 ( 2 n + 2 × 79 78 n + 2 n × 303 n + 1 ) ,
which completes the theorem. ☐
When the degrees of vertices are not taken into account, then by using the values of Table 1 and Table 2 in (3), we have the following result.
Corollary 1.
For graph D ( n ) , the total eccentric-connectivity index is given by
ς ( D ( n ) ) = 9 ( 2 n + 2 × 69 n + 2 n + 1 × 149 72 n 3 ) .
In the next theorem, the eccentric-connectivity polynomial for the molecular graph is derived.
Theorem 2.
For graph D ( n ) , the eccentric-connectivity polynomial is given by
E C P ( D ( n ) , y ) = 6 y 9 n + 14 ( y + 2 ) + 3 × 2 n + 1 y 9 n + 15 ( y 11 + 7 y 10 + 8 y 9 + 4 y 8 + 3 y 7 + 2 y 6 + 2 y 5 + 3 y 4 + 4 y 3 + 4 y 2 + 3 y + 2 ) + 6 ( y 3 + 5 y 2 + 3 y + 2 ) × y 9 n + 21 ( 2 n y 9 n 1 ) 2 y 9 1 + 6 ( 2 y 5 + 3 y 4 + 4 y 3 + 4 y 2 + 3 y + 2 ) × y 9 n + 15 ( 2 n y 9 n 1 ) 2 y 9 1 .
Proof. 
By using Table 1 and Table 2 in (2), we have
E C P ( D ( n ) , y ) = E C P ( A , y ) + E C P ( B , y ) + E C P ( C , y ) = u A d u y ε ( u ) + u B d u y ε ( u ) + u C d u y ε ( u ) = ( 2 × 3 ) y 9 n + 15 + ( 4 × 3 ) y 9 n + 14 + ( 3 × 2 n + 2 ) y 9 n + 15 + ( 3 × 3 × 2 n + 1 ) y 9 n + 16 + ( 2 × 3 × 2 n + 2 ) y 9 n + 17 + ( 2 × 3 × 2 n + 2 ) y 9 n + 18 + ( 3 × 3 × 2 n + 1 ) y 9 n + 19 + ( 2 × 3 × 2 n + 1 ) y 9 n + 20 + ( 2 × 3 × 2 n + 1 ) y 9 n + 21 + ( 3 × 3 × 2 n + 1 ) y 9 n + 22 + ( 2 × 3 × 2 n + 2 ) y 9 n + 23 + ( 4 × 3 × 2 n + 2 ) y 9 n + 24 + ( 1 × 3 × 2 n + 3 ) y 9 n + 25 + ( 3 × 3 × 2 n + 1 ) y 9 n + 25 + ( 1 × 3 × 2 n + 1 ) y 9 n + 26 + i = 1 n ( ( 2 × 3 × 2 i ) y γ + 6 + ( 2 × 3 × 2 i + 1 ) y γ + 8 + ( 2 × 3 × 2 i + 1 ) y γ + 9 + ( 3 × 3 × 2 i ) y γ + 10 + ( 2 × 3 × 2 i ) y γ + 11 + ( 3 × 3 × 2 i ) y γ + 7 + ( 2 × 3 × 2 i ) y γ + 12 + ( 3 × 3 × 2 i ) y γ + 13 + ( 3 × 2 i ) y γ + 14 + ( 4 × 3 × 2 i ) y γ + 14 + ( 3 × 2 i ) y γ + 15 ) .
After some calculations, we get the required result. ☐
By putting the values of Table 1 and Table 2 into (4), we get the following result.
Corollary 2.
For graph D ( n ) , the total eccentric-connectivity polynomial is given by
T E C P ( D ( n ) , y ) = 3 y 9 n + 14 ( y + 1 ) + 3 × 2 n + 1 y 9 n + 15 ( y 11 + 5 y 10 + 2 y 9 + 2 y 8 + y 7 + y 6 + y 5 + y 4 + 2 y 3 + 2 y 2 + y + 1 ) + 6 ( y 3 + 2 y 2 + y + 1 ) × y 9 n + 21 ( 2 n y 9 n 1 ) 2 y 9 1 + 6 ( y + 1 ) ( y 2 + 1 ) 2 × y 9 n + 15 ( 2 n y 9 n 1 ) 2 y 9 1 .
In the next theorem, we compute the closed formula for the first Zagreb eccentricity index.
Theorem 3.
For graph D ( n ) , the first Zagreb eccentricity index is given by
M 1 * * ( D ( n ) ) = 3 ( 2 n + 4 × 7295 n 2 + 2 n + 3 × 2097 n 1944 n 2 162 n + 2 n + 1 × 11641 4053 ) .
Proof. 
By using the values of Table 1 and Table 2 in (5), we compute the first Zagreb eccentricity index of D ( n ) as follows:
M 1 * * ( D ( n ) ) = M 1 * * ( A ) + M 1 * * ( B ) + M 1 * * ( C ) = v A [ ε ( v ) ] 2 + v B [ ε ( v ) ] 2 + v C [ ε ( v ) ] 2 = 3 ( 9 n + 15 ) 2 + 3 ( 9 n + 14 ) 2 + ( 3 × 2 n + 1 ) ( 9 n + 15 ) 2 + ( 3 × 2 n + 1 ) ( 9 n + 16 ) 2 + ( 3 × 2 n + 2 ) ( 9 n + 17 ) 2 + ( 3 × 2 n + 2 ) ( 9 n + 18 ) 2 + ( 3 × 2 n + 1 ) ( 9 n + 19 ) 2 + ( 3 × 2 n + 1 ) ( 9 n + 20 ) 2 + ( 3 × 2 n + 1 ) ( 9 n + 21 ) 2 + ( 3 × 2 n + 1 ) ( 9 n + 22 ) 2 + ( 3 × 2 n + 2 ) ( 9 n + 23 ) 2 + ( 3 × 2 n + 2 ) ( 9 n + 24 ) 2 + ( 3 × 2 n + 3 ) ( 9 n + 25 ) 2 + ( 3 × 2 n + 1 ) ( 9 n + 25 ) 2 + ( 3 × 2 n + 1 ) ( 9 n + 26 ) 2 + i = 1 n ( ( 3 × 2 i ) ( γ + 6 ) 2 + ( 3 × 2 i ) ( γ + 7 ) 2 + ( 3 × 2 i + 1 ) ( γ + 8 ) 2 + ( 3 × 2 i + 1 ) ( γ + 9 ) 2 + ( 3 × 2 i ) ( γ + 10 ) 2 + ( 3 × 2 i ) ( γ + 11 ) 2 + ( 3 × 2 i ) ( γ + 12 ) 2 + ( 3 × 2 i ) ( γ + 13 ) 2 + ( 3 × 2 i ) ( γ + 14 ) 2 + ( 3 × 2 i ) ( γ + 14 ) 2 + ( 3 × 2 i ) ( γ + 15 ) 2 ) .
After some calculations, we obtain
M 1 * * ( D ( n ) ) = 3 ( 2 n + 4 × 7295 n 2 + 2 n + 3 × 2097 n 1944 n 2 162 n + 2 n + 1 × 11 , 641 4053 ) ,
which finishes the theorem. ☐
We determine the augmented eccentric-connectivity index in the next theorem.
Theorem 4.
For graph D ( n ) , the augmented eccentric-connectivity index is given by
A ε ( D ( n ) ) = 48 9 n + 15 + 48 9 n + 14 + 36 × 2 n + 1 9 n + 15 + 24 × 2 n + 1 9 n + 16 + 18 × 2 n + 2 9 n + 17 + 18 × 2 n + 2 9 n + 18 + 24 × 2 n + 1 9 n + 19 + 18 × 2 n + 1 9 n + 20 + 18 × 2 n + 1 9 n + 21 + 24 × 2 n + 1 9 n + 22 + 36 × 2 n + 2 9 n + 23 + 18 × 2 n + 2 9 n + 24 + 12 × 2 n + 3 9 n + 25 + 48 × 2 n + 1 9 n + 25 + 9 × 2 n + 1 9 n + 26 + 72 9 n + 15 + + 36 × 2 n 18 n + 6 + 48 9 n + 16 + + 24 × 2 n 18 n + 7 + 72 9 n + 17 + + 18 × 2 n + 1 18 n + 8 + 72 9 n + 18 + + 18 × 2 n + 1 18 n + 9 + 48 9 n + 19 + + 24 × 2 n 18 n + 10 + 36 9 n + 20 + + 18 × 2 n 18 n + 11 + 36 9 n + 21 + + 18 × 2 n 18 n + 12 + 48 9 n + 22 + + 24 × 2 n 18 n + 13 + 18 9 n + 23 + + 9 × 2 n 18 n + 14 + 72 9 n + 23 + + 36 × 2 n 18 n + 14 + 24 9 n + 24 + + 12 × 2 n 18 n + 15 .
Proof. 
By using the values of Table 1 and Table 2 in (6), we compute the augumented eccentric-connectivity index of D ( n ) in the following way:
A ε ( D ( n ) ) = A ε ( A ) + A ε ( B ) + A ε ( C ) = u A M ( u ) ε ( u ) + u B M ( u ) ε ( u ) + u C M ( u ) ε ( u ) = 3 × 16 9 n + 15 + 3 × 16 9 n + 14 + 3 × 2 n + 1 × 12 9 n + 15 + 3 × 2 n + 1 × 8 9 n + 16 + 3 × 2 n + 2 × 6 9 n + 17 + 3 × 2 n + 2 × 6 9 n + 18 + 3 × 2 n + 1 × 8 9 n + 19 + 3 × 2 n + 1 × 6 9 n + 20 + 3 × 2 n + 1 × 6 9 n + 21 + 3 × 2 n + 1 × 8 9 n + 22 + 3 × 2 n + 2 × 12 9 n + 23 + 3 × 2 n + 2 × 6 9 n + 24 + 3 × 2 n + 3 × 4 9 n + 25 + 3 × 2 n + 1 × 16 9 n + 25 + 3 × 2 n + 1 × 3 9 n + 26 + i = 1 n ( 3 × 2 i × 12 γ + 6 + 3 × 2 i × 8 γ + 7 + 3 × 2 i + 1 × 6 γ + 8 + 3 × 2 i + 1 × 6 γ + 9 + 3 × 2 i × 8 γ + 10 + 3 × 2 i × 6 γ + 11 + 3 × 2 i × 6 γ + 12 + 3 × 2 i × 8 γ + 13 + 3 × 2 i × 3 γ + 14 + 3 × 2 i × 12 γ + 14 + 3 × 2 i × 4 γ + 15 ) .
After some calculations, we obtain the required result. ☐
Now, we compute the closed formula for the modified eccentric-connectivity index.
Theorem 5.
For graph D ( n ) , the modified eccentric-connectivity index is given by
Λ ( D ( n ) ) = 6 ( 2 n × 2277 n 567 n + 2 n + 1 × 1229 + 21 ) .
Proof. 
By using the values of Table 1 and Table 2 in (7), we compute the modified eccentric-connectivity index of D ( n ) in the following way:
Λ ( D ( n ) ) = Λ ( A ) + Λ ( B ) + Λ ( C ) = u A S u ε ( u ) + u B S u ε ( u ) + u C S u ε ( u ) = ( 8 × 3 ) ( 9 n + 15 ) + ( 8 × 3 ) ( 9 n + 14 ) + ( 7 × 3 × 2 n + 1 ) ( 9 n + 15 ) + ( 5 × 3 × 2 n + 2 ) ( 9 n + 17 ) + ( 5 × 3 × 2 n + 2 ) ( 9 n + 18 ) + ( 6 × 3 × 2 n + 1 ) ( 9 n + 19 ) + ( 5 × 3 × 2 n + 1 ) ( 9 n + 20 ) + ( 5 × 3 × 2 n + 1 ) ( 9 n + 21 ) + ( 6 × 3 × 2 n + 1 ) ( 9 n + 22 ) + ( 7 × 3 × 2 n + 2 ) ( 9 n + 23 ) + ( 7 × 3 × 2 n + 2 ) ( 9 n + 24 ) + ( 4 × 3 × 2 n + 3 ) ( 9 n + 25 ) + ( 9 × 3 × 2 n + 1 ) ( 9 n + 25 ) + ( 3 × 3 × 2 n + 1 ) ( 9 n + 26 ) + ( 6 × 3 × 2 n + 1 ) ( 9 n + 16 ) + i = 1 n ( ( 7 × 3 × 2 i ) ( γ + 6 ) + ( 6 × 3 × 2 i ) ( γ + 7 ) + ( 5 × 3 × 2 i + 1 ) ( γ + 8 ) + ( 5 × 3 × 2 i + 1 ) ( γ + 9 ) + ( 6 × 3 × 2 i ) ( γ + 10 ) + ( 5 × 3 × 2 i ) ( γ + 11 ) + ( 5 × 3 × 2 i ) ( γ + 12 ) + ( 7 × 3 × 2 i ) ( γ + 13 ) + ( 3 × 3 × 2 i ) ( γ + 14 ) + ( 8 × 3 × 2 i ) ( γ + 14 ) + ( 4 × 3 × 2 i ) ( γ + 15 ) ) .
After some calculations, we obtain
Λ ( D ( n ) ) = 6 ( 2 n × 2277 n 567 n + 2 n + 1 × 1229 + 21 ) ,
which completes the proof. ☐
Finally, we compute the closed formula for the modified eccentric-connectivity polynomial.
Theorem 6.
For graph D ( n ) , the modified eccentric-connectivity polynomial is given by
M E C P ( D ( n ) , y ) = 24 y 9 n + 14 ( y + 1 ) + 2 n + 1 × y 9 n + 15 ( 9 y 11 + 75 y 10 + 42 y 9 + 42 y 8 + 18 y 7 + 15 y 6 + 15 y 5 + 18 y 4 + 30 y 3 + 30 y 2 + 18 y + 21 ) + 6 ( 5 y 5 + 6 y 4 + 10 y 3 + 10 y 2 + 6 y + 7 ) y 9 n + 15 ( 2 n y 9 n 1 ) 2 y 9 1 + 6 ( 4 y 3 + 11 y 2 + 7 y + 5 ) y 9 n + 21 ( 2 n y 9 n 1 ) 2 y 9 1 .
Proof. 
By using the values of Table 1 and Table 2 in (8), we compute the modified eccentric-connectivity polynomial of D ( n ) in the following way:
M E C P ( D ( n ) , y ) = M E C P ( A , y ) + M E C P ( B , y ) + M E C P ( C , y ) = u A S u y ε ( u ) + u B S u y ε ( u ) + u C S u y ε ( u ) = ( 8 × 3 ) y 9 n + 15 + ( 8 × 3 ) y 9 n + 14 + ( 7 × 3 × 2 n + 1 ) y 9 n + 15 + ( 6 × 3 × 2 n + 1 ) y 9 n + 16 + ( 5 × 3 × 2 n + 2 ) y 9 n + 17 + ( 5 × 3 × 2 n + 2 ) y 9 n + 18 + ( 6 × 3 × 2 n + 1 ) y 9 n + 19 + ( 5 × 3 × 2 n + 1 ) y 9 n + 20 + ( 5 × 3 × 2 n + 1 ) y 9 n + 21 + ( 6 × 3 × 2 n + 1 ) y 9 n + 22 + ( 7 × 3 × 2 n + 2 ) y 9 n + 23 + ( 7 × 3 × 2 n + 2 ) y 9 n + 24 + ( 4 × 3 × 2 n + 3 ) y 9 n + 25 + ( 9 × 3 × 2 n + 1 ) y 9 n + 25 + ( 3 × 3 × 2 n + 1 ) y 9 n + 26 + i = 1 n ( ( 7 × 3 × 2 i ) ( y γ + 6 ) + ( 6 × 3 × 2 i ) ( y γ + 7 ) + ( 5 × 3 × 2 i + 1 ) ( y γ + 8 ) + ( 5 × 3 × 2 i + 1 ) ( y γ + 9 ) + ( 6 × 3 × 2 i ) ( y γ + 10 ) + ( 5 × 3 × 2 i ) ( y γ + 11 ) + ( 5 × 3 × 2 i ) ( y γ + 12 ) + ( 7 × 3 × 2 i ) ( y γ + 13 ) + ( 3 × 3 × 2 i ) ( y γ + 14 ) + ( 8 × 3 × 2 i ) ( y γ + 14 ) + ( 4 × 3 × 2 i ) ( y γ + 15 ) ) .
After some calculations, we obtain the required result. ☐

4. Conclusions

In this paper we discussed the theoretical topics in molecular science and computed the eccentric topological indices for a class of phosphorus-containing dendrimers in regard to their molecular structure analysis, distance computing and mathematical derivation. Phosphorus-containing dendrimers have various applications in nanomedicine and materials science; therefore, these theoretical results could have applications in medical science.

Author Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Funding

This research received no external funding.

Acknowledgments

This work was supported by Higher Education Commission Pakistan.

Conflicts of Interest

The authors declare no conflict of interest. We are thankful to both reviewers and editor for positive suggestions that improve the quality of this paper.

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Figure 1. First generation.
Figure 1. First generation.
Symmetry 10 00237 g001
Figure 2. Second generation.
Figure 2. Second generation.
Symmetry 10 00237 g002
Table 1. Sets A and B with their degrees, S u , M ( u ) , eccentricities, and frequencies.
Table 1. Sets A and B with their degrees, S u , M ( u ) , eccentricities, and frequencies.
RepresentativeDegree S u M ( u ) EccentricityFrequency
α 1 2816 9 n + 15 3
α 2 4816 9 n + 14 3
β 1 2712 9 n + 15 3 × 2 n + 1
β 2 368 9 n + 16 3 × 2 n + 1
β 3 256 9 n + 17 3 × 2 n + 2
β 4 256 9 n + 18 3 × 2 n + 2
β 5 368 9 n + 19 3 × 2 n + 1
β 6 256 9 n + 20 3 × 2 n + 1
β 7 256 9 n + 21 3 × 2 n + 1
β 8 368 9 n + 22 3 × 2 n + 1
β 9 2712 9 n + 23 3 × 2 n + 2
β 10 476 9 n + 24 3 × 2 n + 2
β 11 144 9 n + 25 3 × 2 n + 3
β 12 3916 9 n + 25 3 × 2 n + 1
β 13 133 9 n + 26 3 × 2 n + 1
Table 2. Set C with degrees, S u , M ( u ) , eccentricities, and frequencies.
Table 2. Set C with degrees, S u , M ( u ) , eccentricities, and frequencies.
RepresentativeDegree S u M ( u ) EccentricityFrequency
a i 2712 9 n + 9 i + 6 = γ + 6 3 × 2 i
b i 368 γ + 7 3 × 2 i
c i 256 γ + 8 3 × 2 i + 1
d i 256 γ + 9 3 × 2 i + 1
e i 368 γ + 10 3 × 2 i
f i 256 γ + 11 3 × 2 i
g i 256 γ + 12 3 × 2 i
h i 378 γ + 13 3 × 2 i
j i 133 γ + 14 3 × 2 i
k i 4812 γ + 14 3 × 2 i
l i 144 γ + 15 3 × 2 i

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