68 

 

 

  

 

 

 Some Topological and Polynomial Indices (Hosoya and Schultz) for 

the Intersection Graph of the Subgroup of 𝒁𝒓𝒏 

Alaa J.Nawaf                                    Akram S.Mohammad 

alaagamilnawaf@gmail.com                    akr-tel@tu.edu.iq  

Department of Mathematics, College of Computer and Mathematics, Tikrit University ,Tikrit, Iraq. 

  

 

                                                                                                             
 

 

   Abstract  

         Let  𝑍𝑟𝑛 be any group with identity element (e) . A subgroup intersection graph of  a 

subset 𝑍𝑟𝑛 is the Graph with V (ᴦ𝑆𝐼(𝑍𝑟𝑛)) = 𝑍𝑟𝑛 - e and two separate peaks c and d 

contiguous for c and d if and only if      |〈𝑐〉∩ 〈𝑑〉| > 1, Where 〈𝑐〉 is a Periodic subset of 

  𝑍𝑟𝑛 resulting from  𝑐 ∈   𝑍𝑟𝑛. We find some topological indicators in this paper and Multi-

border (Hosoya and Schultz) of  ᴦ𝑆𝐼(𝑍𝑟𝑛) , where 𝑟 ≥ 2 ,𝑛 > 1 ,𝑟 is aprime number. 

Keyword: Hosoya Polynomial, Schultz Polynomial. , connectivity index, Sum connectivity 

index, Forgotten index, first Zagreb index, Harmonic index.  

1.Introduction 

      A topological index is a real number associated with the graph, which must be 

structurally constant. Topological index sometimes called molecular structure descriptor[1]. 

Many topological indicators have been identified and many applications have been found as 

a means of nemuls chemical, pharmaceutical and other molecular properties. The Weiner 

Index is the first topological indicator used in chemistry. More precisely, in 1947, Harold 

Weiner presented and developed this interesting indicator to determine the physical 

properties of the hens known as paraffins. 

    In this paper we examine some topological indicators that depend on the degree of 

examples of Eccentric connectivity index[2], connectivity index[3], sum connectivity 

index[4], Zagreb index[5], forgotten index[6], The index of geometric- arithmetic [3], Index 

of Atom-Bond Connectivity [3] and Harmonic index[7]. 

Ibn Al Haitham Journal for Pure and Applied Science 

Journal homepage: http://jih.uobaghdad.edu.iq/index.php/j/index 

 

Doi: 10.30526/34.4.2704 

Article history: Received 26 January 2021, Accepted 29 March 20 12 , Published in October 2021. 

 

mailto:nawaf@gmail.com
mailto:akr-tel@tu.edu.iq


Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34(4)2021 
 

69 

 

     In (2019),Abdussakir [8] introduced topological indices about symmetric group graph, 

also in (2020) G. R. Roshini [9] studied topological indices of transformation graphs , and in 

(2021) Alaa .J and Akram.S [10] studied topological indices and (hosoya and Schultz) 

polynomial about subgroup intersection graph of a group  𝑍𝑟 .   

     One of the graphic concepts obtained from the group is the concept of a subset cross chart 

of a group introduced by [11]. In refere to the subgroup intersection graph definition by [11], 

let Graph group be the intersection (G) where G is a graph with V (ᴦ𝑆𝐼(𝐺))= G-e and two 

distinct peaks A and B are adjacent in the ᴦ𝑆𝐼 (G) if 

 |〈a〉∩〈b〉|> 1, where〈a〉is a periodic subset of G resulting from a ∈G. 

2. Method and Materials 

     In an existing article, all graphs are simple, limited, connected and :directed. For 𝐺 =

(𝑉(𝐺),𝐸(𝐺)) graph, the order of 𝐺 is 𝑝(𝐺) = 𝑉(𝐺) and the scale of 𝐺 is 𝑞(𝐺) = 𝐸(𝐺). 

Let 𝑑𝑒𝑔(𝑢) denote the degree of vertex 𝑢 in 𝐺. If 𝑑𝑒𝑔(𝑢) = 0, then u is an isolated vertices. 

Let 𝑑(𝑢,𝑣)Indicatex the distance between the peaks 𝑢 and 𝑣 in 𝐺. The eccentricity 

𝑒𝑐𝑐(𝑢) of the vertex 𝑢 is 𝑒𝑐𝑐(𝑢) = 𝑠𝑢𝑝{𝑑(𝑢,𝑣):𝑣 ∈ 𝑉(𝐺)}. 

The following definition refers to a graph 𝐺 = (𝑉(𝐺),𝐸(𝐺)). 

Eccentric connectivity index of 𝐺 is [2] 

𝜉𝐶(𝐺))) = ∑ 𝑑𝑒𝑔(𝑢).𝑒(𝑢)𝑢∈ 𝑉(𝐺)  

The index of connectivity of 𝐺 is[3] 

𝑋(𝐺) = ∑
1

√𝑑𝑒𝑔(𝑢).𝑑𝑒𝑔(𝑣)𝑢𝑣∈ 𝐸(𝐺)
 

Sum the index of connectivity of 𝐺 is [4] 

𝑆(𝐺) = ∑
1

√𝑑𝑒𝑔(𝑢)+𝑑𝑒𝑔(𝑣)
𝑢𝑣∈ 𝐸(𝐺)  

     A first zagreb index of G is [5] 

𝑀1(𝐺) = ∑ (𝑑𝑒𝑔(𝑢))
2

𝑢∈ 𝑉(𝐺)  

     A second  zagreb index of 𝐺 is [5] 

𝑀2(𝐺) = ∑ 𝑑𝑒𝑔(𝑢).𝑑𝑒𝑔(𝑣)𝑢𝑣 𝐸𝐺)  

The forgotten index of 𝐺 is [6] 

𝐹(𝐺) ) = ∑ (𝑑𝑒𝑔(𝑢))3

𝑢∈𝑉(𝐺)

 

Atom Bond connectivity index of G is [3] 



Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34(4)2021 
 

70 

 

  𝐴𝐵𝐶(𝐺)) = ∑ √
𝑑𝑒𝑔(𝑢)+𝑑𝑒𝑔(𝑣)−2

𝑑𝑒𝑔(𝑢).𝑑𝑒𝑔(𝑣)𝑢𝑣∈ 𝐸(𝐺)
   

Geometric-Arithmetic index of 𝐺 is [3] 

 𝐺𝐴(𝐺) = ∑
2√𝑑𝑒𝑔(𝑢).𝑑𝑒𝑔(𝑣)

𝑑𝑒𝑔(𝑢)+𝑑𝑒𝑔(𝑣)𝑢𝑣∈ 𝐸(𝐺)
   

Harmonic index of 𝐺 is [7] 

𝐻(𝐺) = ∑
2

𝑑𝑒𝑔(𝑢)+ 𝑑𝑒𝑔 (𝑣)𝑢𝑣∈ 𝐸(𝐺)
 

3. The Main Result  

      To get a good look ,:  𝑟 ≥ 2,𝑛 > 1 , a subgroup intersection graph of a group 𝑍𝑟𝑛 is 

(ᴦ𝑆𝐼(𝑍𝑟𝑛))  a graph with V (ᴦ𝑆𝐼(𝑍𝑟𝑛)) = 𝑍𝑟𝑛 − 𝑒 ,  any two different vertices a and b are 

adjacent:|〈a〉 ∩ 〈b〉| >  1, where 〈a〉 is the subset created by a ∈ 𝑍𝑟𝑛. 

 

Theorem 3.1 

         Let 𝑍𝑟𝑛 be a group with 𝑟 ≥ 2 ,𝑛 > 1,:then the Eccentric connectivity index 

of ᴦ𝑆𝐼(𝑍𝑟𝑛) is  𝜉
𝑐(ᴦ𝑆𝐼(𝑍𝑟𝑛)) = (𝑟

𝑛 − 1)(𝑟𝑛 − 2)       

Proof: 

𝑑𝑒𝑔(𝑣) = 𝑟𝑛 − 2  ,∀𝑣 ∈ 𝑉( ᴦ𝑆𝐼(𝑍𝑟𝑛)), 𝑣 = 1,2, . . . ,𝑟
𝑛 − 1   

e (𝑣) = 1  ,∀𝑣 ∈ V( ᴦSI(Z𝑟𝑛)), 𝑣 = 1,2, . . . ,𝑟
𝑛 − 1 

𝜉𝐶(ᴦ𝑆𝐼(𝑍𝑟𝑛))) = ∑ 𝑑𝑒𝑔(𝑢).𝑒(𝑢)
𝑢∈ 𝑉(ᴦ𝑆𝐼(𝑍𝑟𝑛))

 

= 𝑑𝑒𝑔(1).𝑒(1) + ⋯+ 𝑑𝑒𝑔(𝑟𝑛 − 1).𝑒(𝑟𝑛 − 1)⏟                            
(𝑟𝑛−1)𝑡𝑖𝑚𝑒𝑠 

                         

=(𝑟𝑛 − 1)(𝑟𝑛 − 2)                          

Theorem 3.2 

            Let Z𝑟𝑛 be a group with 𝑟 ≥ 2 , n > 1 then the connectivity index of  ᴦSI(Z𝑟𝑛) is 

𝑋(ᴦ𝑆𝐼 ((𝑍𝑟𝑛)) = 1 +
∑ (𝑟𝑛−𝑖)𝑟−1𝑖=3
(𝑟𝑛−2)

          

Proof: 

𝑑𝑒𝑔(𝑣) = 𝑟𝑛 − 2  ,∀𝑣 ∈ 𝑉( ᴦ𝑆𝐼(𝑍𝑟𝑛)), 𝑣 = 1,2, . . . ,𝑟
𝑛 − 1   



Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34(4)2021 
 

71 

 

                  

𝑋(ᴦ𝑆𝐼(𝑍𝑟𝑛)) = ∑
1

√𝑑𝑒𝑔(𝑢).𝑑𝑒𝑔(𝑣)𝑢𝑣∈ 𝐸(ᴦ𝑆𝐼(𝑍𝑟𝑛))
 

 
=

1

√𝑑𝑒𝑔(1).𝑑𝑒𝑔(2)
+ ⋯+

1

√𝑑𝑒𝑔(1).𝑑𝑒𝑔(𝑟𝑛 − 1)⏟                                
(𝑟𝑛−2)𝑡𝑖𝑚𝑒𝑠

 

   +
1

√𝑑𝑒𝑔(2).𝑑𝑒𝑔(3)
+⋯+

1

√𝑑𝑒𝑔(3).𝑑𝑒𝑔(𝑟𝑛 − 1)⏟                                
(𝑟𝑛−3)𝑡𝑖𝑚𝑒𝑠

 

+
1

√𝑑𝑒𝑔(3).𝑑𝑒𝑔(4)
+ ⋯+

1

√𝑑𝑒𝑔(3).𝑑𝑒𝑔(𝑟𝑛 − 1)⏟                                
(𝑟𝑛−4)𝑡𝑖𝑚𝑒𝑠

 

+⋯..+
1

√𝑑𝑒𝑔(𝑟𝑛 − 2).𝑑𝑒𝑔(𝑟𝑛 − 1)
 

 

=
(𝑟𝑛 − 2)

(𝑟𝑛 − 2)
+
(𝑟𝑛 −3)

(𝑟𝑛 −2)
+
(𝑟𝑛 − 4)

(𝑟𝑛 − 2)
+ ⋯+

1

(𝑟𝑛 − 2)
 

= 1 +
∑ (𝑟𝑛 − 𝑖)𝑟

𝑛−1
𝑖=3

(𝑟𝑛 − 2)
 

 
 

 

Theorem 3.3 

          Let 𝑍𝑟𝑛 be a group with 𝑟 ≥ 2,𝑛 > 1,:then the Sum connectivity index of ᴦ𝑆𝐼(𝑍𝑟𝑛) is 

S(ᴦ𝑆𝐼(𝑍𝑟𝑛)) =
∑ (𝑟𝑛−𝑖)𝑟
𝑛−1
𝑖=2

√2𝑟𝑛−4     
       

Proof: 

𝑑𝑒𝑔(𝑣) = 𝑟𝑛 − 2  ,∀𝑣 ∈ 𝑉( ᴦ𝑆𝐼(𝑍𝑟𝑛)), 𝑣 = 1,2, . . . , 𝑟
𝑛 − 1   

         

𝑆(ᴦ𝑆𝐼(𝑍𝑟𝑛)) = ∑
1

√𝑑𝑒𝑔(𝑢)+ 𝑑𝑒𝑔(𝑣)𝑢𝑣∈ 𝐸(ᴦ𝑆𝐼(𝑍𝑟𝑛))
 

 
=

1

√𝑑𝑒𝑔(1)+ 𝑑𝑒𝑔(2)
+ ⋯+

1

√𝑑𝑒𝑔(1) + 𝑑𝑒𝑔(𝑟𝑛 − 1)⏟                                  
(𝑟𝑛−2)𝑡𝑖𝑚𝑒𝑠

 

   +
1

√𝑑𝑒𝑔(2) + 𝑑𝑒𝑔(3)
+ ⋯+

1

√𝑑𝑒𝑔(3)+ 𝑑𝑒𝑔(𝑟𝑛 − 1)⏟                                  
(𝑟𝑛−3)𝑡𝑖𝑚𝑒𝑠

 

+
1

√𝑑𝑒𝑔(3) + 𝑑𝑒𝑔(4)
+ ⋯+

1

√𝑑𝑒𝑔(3)+ 𝑑𝑒𝑔(𝑟𝑛 − 1)⏟                                  
(𝑟𝑛−4)𝑡𝑖𝑚𝑒𝑠

 

+⋯..+
1

√𝑑𝑒𝑔(𝑟𝑛 − 2) + 𝑑𝑒𝑔(𝑟𝑛 − 1)
 

 



Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34(4)2021 
 

72 

 

 

=
(𝑟𝑛 − 2)

√2𝑟𝑛 −4
+
(𝑟𝑛 −3)

√2𝑟𝑛 − 4
+
(𝑟𝑛 − 4)

√2𝑟𝑛 − 4
+ ⋯+

1

√2𝑟𝑛 − 4
 

=
∑ (𝑟𝑛 − 𝑖)𝑟

𝑛−1
𝑖=2

√2𝑟𝑛 − 4
 

 

Theorem 3.4 

        Let 𝑍𝑟𝑛 be a group with 𝑟 ≥ 2,𝑛 > 1 then the first zegrab index of  ᴦ𝑆𝐼(𝑍𝑟𝑛) 

is 𝑀1(ᴦ𝑆𝐼(𝑍𝑟𝑛)) =(𝑟
𝑛 − 1)(𝑟𝑛 − 2)2  

Proof: 

𝑑𝑒𝑔(𝑣) = 𝑟𝑛 − 2  ,∀𝑣 ∈ 𝑉( ᴦ𝑆𝐼(𝑍𝑟𝑛)), 𝑣 = 1,2, . . . ,𝑟
𝑛 − 1   

  𝑀1(ᴦ𝑆𝐼(𝑍𝑟𝑛)) = ∑ (𝑑𝑒𝑔(𝑢))
2

𝑢∈ 𝑉(ᴦ𝑆𝐼(𝑍𝑟𝑛))
                                

   (𝑑𝑒𝑔(1))2 + ⋯+(𝑑𝑒𝑔(𝑟𝑛 − 1))2⏟                      
(𝑟𝑛−1)𝑡𝑖𝑚𝑒𝑠

= 

=  (𝑟𝑛 − 1)(𝑟𝑛 − 2)2  

   

Theorem 3.5 

         Let 𝑍𝑟𝑛 be a group with 𝑟 ≥ 2 ,𝑛 > 1 then the second zegrab index of ᴦ𝑆𝐼(𝑍𝑟𝑛) 

is 𝑀2(ᴦ𝑆𝐼(𝑍𝑟𝑛)) = (𝑟
𝑛 − 2)3 +(𝑟𝑛 − 2)2 ∑ (𝑟𝑛 − 𝑖)𝑟

𝑛−1
𝑖=3   

Proof: 

𝑑𝑒𝑔(𝑣) = 𝑟𝑛 − 2  ,∀𝑣 ∈ 𝑉( ᴦ𝑆𝐼(𝑍𝑟𝑛)), 𝑣 = 1,2, . . . , 𝑟
𝑛 − 1   

  𝑀2(ᴦ𝑆𝐼(𝑍𝑟𝑛)) = ∑ 𝑑𝑒𝑔(𝑢).𝑑𝑒𝑔(𝑣)𝑢𝑣 𝐸(ᴦ𝑆𝐼(𝑍𝑟𝑛))    

 = 𝑑𝑒𝑔(1).𝑑𝑒𝑔(2)+ ⋯+𝑑𝑒𝑔(1).𝑑𝑒𝑔(𝑟𝑛 − 1)⏟                            
(𝑟𝑛−2)𝑡𝑖𝑚𝑒𝑠

+

𝑑𝑒𝑔(2) .𝑑𝑒𝑔(3) + ⋯+𝑑𝑒𝑔(2).𝑑𝑒𝑔(𝑟𝑛 − 1)⏟                            
(𝑟𝑛−3)times

+

𝑑𝑒𝑔(3) .𝑑𝑒𝑔(4) + ⋯+𝑑𝑒𝑔(3).𝑑𝑒𝑔(𝑟𝑛 − 1)⏟                            
(𝑟𝑛−4)𝑡𝑖𝑚𝑒𝑠

+ ⋯+ 𝑑𝑒𝑔(𝑟𝑛 −2).𝑑𝑒𝑔(𝑟𝑛 − 1) 

 

= (𝑟𝑛 − 2)3 + (𝑟𝑛 − 3)(𝑟𝑛 − 2)2 + (𝑟𝑛 − 4)(𝑟𝑛 − 2)2 + ⋯+ (𝑟𝑛 −2)2 

= (𝑟𝑛 − 2)3 + (𝑟𝑛 − 2)2 ∑ (𝑟𝑛 − 𝑖)

𝑟𝑛−1

𝑖=3

 



Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34(4)2021 
 

73 

 

Theorem 3.6 

       Let 𝑍𝑟𝑛 be a group with r ≥ 2 ,𝑛 > 1   then the forgotten index of  ᴦ𝑆𝐼(𝑍𝑟𝑛) is 

F(ᴦ𝑆𝐼(𝑍𝑟𝑛)) = (𝑟
𝑛 − 1)(𝑟𝑛 − 2)3                                  

Proof: 

𝑑𝑒𝑔(𝑣) = 𝑟𝑛 − 2  ,∀𝑣 ∈ 𝑉( ᴦ𝑆𝐼(𝑍𝑟𝑛)), 𝑣 = 1,2, . . . , 𝑟
𝑛 − 1   

F(ᴦ𝑆𝐼(𝑍𝑟𝑛) ) = ∑ (𝑑𝑒𝑔(𝑢))
3

𝑢∈𝑉(ᴦ𝑆𝐼(𝑍𝑟𝑛))
 

= (𝑑𝑒𝑔(1))3 + ⋯+ (𝑑𝑒𝑔(𝑟𝑛 − 1))3⏟                      
(𝑟𝑛−1) 𝑡𝑖𝑚𝑒𝑠

                        

=(𝑟𝑛 − 1)(𝑟𝑛 − 2)3                        

  

Theorem 3.7 

        Let Z𝑟𝑛 be a group with 𝑟 ≥ 2 ,𝑛 > 1,:then the Atom Bond connectivity index 

of  ᴦSI(Z𝑟𝑛) is 

ABC(ᴦSI(Z𝑟𝑛))) = √2𝑟
𝑛 − 6+

√2𝑟𝑛 − 6∑ (𝑟𝑛 − i)𝑟
𝑛−1
i=3

(𝑟𝑛 − 2)
 

Proof:  

𝑑𝑒𝑔(𝑣) = 𝑟𝑛 − 2  ,∀𝑣 ∈ 𝑉( ᴦ𝑆𝐼(𝑍𝑟𝑛)), 𝑣 = 1,2, . . . , 𝑟
𝑛 − 1    

𝐴𝐵𝐶(ᴦ𝑆𝐼(𝑍𝑟𝑛)) = ∑ √
𝑑𝑒𝑔(𝑢)+ 𝑑𝑒𝑔(𝑣) − 2

𝑑𝑒𝑔(𝑢).𝑑𝑒𝑔(𝑣)𝑢𝑣∈ 𝐸(ᴦ𝑆𝐼(𝑍𝑟𝑛))
 

            = √
𝑑𝑒𝑔(1) + 𝑑𝑒𝑔(2) − 2

𝑑𝑒𝑔(1).𝑑𝑒𝑔(2)
+ ⋯+√

𝑑𝑒𝑔(1) + 𝑑𝑒𝑔(𝑟𝑛 − 1) − 2

𝑑𝑒𝑔(1).𝑑𝑒𝑔(𝑟𝑛 −1)
⏟                                      

(𝑟𝑛−2)𝑡𝑖𝑚𝑒𝑠

+√
𝑑𝑒𝑔(2) + 𝑑𝑒𝑔(3) − 2

𝑑𝑒𝑔(2).𝑑𝑒𝑔(3)
+ ⋯+ √

𝑑𝑒𝑔(2)+ 𝑑𝑒𝑔(𝑟𝑛 − 1) − 2

𝑑𝑒𝑔(2) .𝑑𝑒𝑔(𝑟𝑛 − 1)
⏟                                      

(𝑟𝑛−3)𝑡𝑖𝑚𝑒𝑠

+√
𝑑𝑒𝑔(3) + 𝑑𝑒𝑔(4) − 2

𝑑𝑒𝑔(3).𝑑𝑒𝑔(4)
+ ⋯+ √

𝑑𝑒𝑔(3)+ 𝑑𝑒𝑔(𝑟𝑛 − 1) − 2

𝑑𝑒𝑔(3) .𝑑𝑒𝑔(𝑟𝑛 − 1)
⏟                                      

(𝑟𝑛−4)𝑡𝑖𝑚𝑒𝑠

+ ⋯

+√
𝑑𝑒𝑔(𝑟𝑛 − 2) + 𝑑𝑒𝑔(𝑟𝑛 − 1) − 2

𝑑𝑒𝑔(𝑟𝑛 − 2).𝑑𝑒𝑔(𝑟𝑛 − 1)
 



Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34(4)2021 
 

74 

 

                        = √2𝑟𝑛 − 6+
(𝑟𝑛 − 3)√2𝑟𝑛 − 6

(𝑟𝑛 − 2)
+
(𝑟𝑛 −4)√2𝑟𝑛 − 6

(𝑟𝑛 − 2)
+ ⋯+

√2𝑟𝑛 − 6

(𝑟𝑛 − 2)
 

                       = √2𝑟𝑛 − 6 +
√2𝑟𝑛 − 6∑ (𝑟𝑛 − 𝑖)𝑟

𝑛−1
𝑖=3

(𝑟𝑛 − 2)
 

Theorem 3.8 

      Let 𝑍𝑟𝑛 be a group with 𝑟 ≥ 2 ,𝑛 > 1 then the Geometric –Arithmetic index of  ᴦ𝑆𝐼(𝑍𝑟𝑛) 

is 𝐺𝐴(ᴦ𝑆𝐼(𝑍𝑟𝑛)) = (𝑟
𝑛 − 2)+ ∑ (𝑟𝑛 − 𝑖)𝑟

𝑛−1
𝑖=3 

 

Proof: 

𝑑𝑒𝑔(𝑣) = 𝑟𝑛 − 2  ,∀𝑣 ∈ 𝑉( ᴦ𝑆𝐼(𝑍𝑟𝑛)), 𝑣 = 1,2, . . . , 𝑟
𝑛 − 1        

𝐺𝐴(ᴦ𝑆𝐼(𝑍𝑟𝑛))) = ∑
2√𝑑𝑒𝑔(𝑢).𝑑𝑒𝑔(𝑣) 

𝑑𝑒𝑔(𝑢) + 𝑑𝑒𝑔(𝑣)𝑢𝑣∈ 𝐸(ᴦ𝑆𝐼(𝑍𝑟𝑛))
 

=
2√𝑑𝑒𝑔(1).𝑑𝑒𝑔 (2)

𝑑𝑒𝑔(1) + 𝑑𝑒𝑔 (2)
+ ⋯+

2√𝑑𝑒𝑔(1).𝑑𝑒𝑔 (𝑟𝑛 −1)

𝑑𝑒𝑔(1)+ 𝑑𝑒𝑔 (𝑟𝑛 − 1)⏟                                
(𝑟𝑛−2)𝑡𝑖𝑚𝑒𝑠

+
2√𝑑𝑒𝑔(2).𝑑𝑒𝑔 (3)

𝑑𝑒𝑔(2)+ 𝑑𝑒𝑔 (3)
+ ⋯+

2√𝑑𝑒𝑔(2).𝑑𝑒𝑔 (𝑟𝑛 − 1)

𝑑𝑒𝑔(2) + 𝑑𝑒𝑔 (𝑟𝑛 − 1)⏟                                
(𝑟𝑛−3)𝑡𝑖𝑚𝑒𝑠

+
2√𝑑𝑒𝑔(3).𝑑𝑒𝑔 (4)

𝑑𝑒𝑔(3)+ 𝑑𝑒𝑔 (4)
+ ⋯+

2√𝑑𝑒𝑔(3).𝑑𝑒𝑔 (𝑟𝑛 − 1)

𝑑𝑒𝑔(3) + 𝑑𝑒𝑔 (𝑟𝑛 − 1)⏟                                
(𝑟𝑛−4)𝑡𝑖𝑚𝑒𝑠

+ ⋯.+
2√𝑑𝑒𝑔(𝑟𝑛 − 2).𝑑𝑒𝑔 (𝑟𝑛 − 1)

𝑑𝑒𝑔(𝑟𝑛 − 2) + 𝑑𝑒𝑔 (𝑟𝑛 − 1)
 

                 =
2(𝑟𝑛 − 2)2

2𝑟𝑛 − 4
+
2(𝑟𝑛 − 3)(𝑟𝑛 − 2)

2𝑟𝑛 − 4
+
2(𝑟𝑛 − 4)(𝑟𝑛 − 2)

2𝑟𝑛 − 4
+ ⋯+

2(𝑟𝑛 − 2)

2𝑟𝑛 − 4
    

                = (𝑟𝑛 − 2)+ ∑ (𝑟𝑛 − 𝑖)

𝑟𝑛−1

𝑖=3

 

Theorem 3.9 

      Let  𝑍𝑟𝑛 be a group with 𝑟 ≥ 2 ,𝑛 > 1 then the Harmonic index of  ᴦ𝑆𝐼(𝑍𝑟𝑛) is  

1 +
2

2𝑟𝑛−4
∑ (𝑟𝑛 − 𝑖)𝑟

𝑛−1
𝑖=3 𝐻(ᴦ𝑆𝐼(𝑍𝑟𝑛))) =    

Proof: 

𝑑𝑒𝑔(𝑣) = 𝑟𝑛 − 2  ,∀𝑣 ∈ 𝑉( ᴦ𝑆𝐼(𝑍𝑟𝑛)), 𝑣 = 1,2, . . . ,𝑟
𝑛 − 1                                     



Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34(4)2021 
 

75 

 

𝐻(ᴦ𝑆𝐼(𝑍𝑟𝑛))) = ∑
2

𝑑𝑒𝑔(𝑢) +𝑑𝑒𝑔(𝑣)𝑢𝑣∈ 𝐸(ᴦ𝑆𝐼(𝑍𝑟𝑛))
 

                          =
2

𝑑𝑒𝑔(1)+ 𝑑𝑒𝑔 (2)
+ ⋯+

2

𝑑𝑒𝑔(1)+ 𝑑𝑒𝑔 (𝑟𝑛 − 1)⏟                              
(𝑟𝑛−2)𝑡𝑖𝑚𝑒𝑠

+
2

𝑑𝑒𝑔(2) + 𝑑𝑒𝑔 (3)
+ ⋯+

2

𝑑𝑒𝑔(2)+ 𝑑𝑒𝑔 (𝑟𝑛 − 1)⏟                              
(𝑟𝑛−3)𝑡𝑖𝑚𝑒𝑠

 

+
2

𝑑𝑒𝑔(3)+ 𝑑𝑒𝑔 (4)
+ ⋯+

2

𝑑𝑒𝑔(3)+ 𝑑𝑒𝑔 (𝑟𝑛 − 1)⏟                              
(𝑟𝑛−4)𝑡𝑖𝑚𝑒𝑠

+ ⋯.+
2

𝑑𝑒𝑔(𝑟𝑛 − 2) + 𝑑𝑒𝑔 (𝑟𝑛 − 1)
 

                   =
2(𝑟𝑛 − 2)

2𝑟𝑛 − 4
+
2(𝑟𝑛 − 3)

2𝑟𝑛 − 4
+
2(𝑟𝑛 −4)

2𝑟𝑛 −4
+⋯+

2

2𝑟𝑛 − 4
 

                  = 1+
2

2𝑟𝑛 − 4
∑ (𝑟𝑛 − 𝑖)

𝑟𝑛−1

𝑖=3

 

4.( Hosoya and Schultz)  Polynomial of  ᴦ𝑺𝑰(𝒁𝒓𝒏)   

        In this section, we find the (Hosoya  and Schultz ) Polynomial of  ᴦ𝑆𝐼(𝑍𝑟𝑛) . 

Definition 4.1(12): 

      Let G be a connected graph , then a Hosoya Polynomial of graph G is defined by 

H(G;x) = ∑ 𝑑(𝐺,𝑘) xk
𝑑𝑖𝑎𝑚(𝐺)
k=0

 , where 𝑑(𝐺,𝑘) is the number of pairs of vertices of a graph 

G that are at distance k apart , for 𝑘 = 0,1,2,…,𝑑𝑖𝑎𝑚(𝐺),where 

  𝑑𝑖𝑎𝑚(𝐺) = maxu,v∈V(G)d(u,v).  

𝑑(𝐺,0) = 𝑝(𝐺) -1 :Note 4.2(13) 

                        2- 𝑑(𝐺,1) =  𝑞(𝐺)      

Definition 4.3(14): 

      Let G be a connected graph , then a Schultz Polynomial of a graph G is  defined 

by Sc(G;x) = ∑ (𝑑𝑒𝑔(𝑢)+ 𝑑𝑒𝑔(𝑣))xd(u,v)u,v∈V(G)
u≠v         

 , where 𝑑𝑒𝑔(𝑢) is the degree of the 

vertices u and 𝑑𝑒𝑔(𝑣) is the degree of vertices 𝑣, 𝑑(𝑢,𝑣) is the distance between 𝑢 and 𝑣.   

Theorem 4.4: 

     H( ᴦ𝑆𝐼(𝑍𝑟𝑛);𝑥) = 𝑐0 + 𝑐1𝑥 , where 𝑟 ≥ 2 , 𝑛 > 1, 𝑐0 = 𝑟
𝑛 − 1,  

𝑐1 = ∑ (𝑟
𝑛 −𝑖)𝑟

𝑛−1
𝑖=2                                        

 



Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34(4)2021 
 

76 

 

Proof: 

       For every  𝑟 ≥ 2 ,𝑛 > 1 , we note that every vertices of graph ᴦ𝑆𝐼(𝑍𝑟𝑛) is adjacent of all 

vertices of graph  ᴦ𝑆𝐼(𝑍𝑟𝑛) , then diam( ᴦ𝑆𝐼(𝑍𝑟𝑛)) = 1 , its mean is H( ᴦ𝑆𝐼(𝑍𝑟𝑛),𝑥) = 𝑐0 +

𝑐1𝑥  where  

𝑐𝑖 = 𝑑( ᴦ𝑆𝐼(𝑍𝑟𝑛), 𝑖) ,∀ 𝑖 = 0,1  

It is clear that  𝑐0 = 𝑑( ᴦ𝑆𝐼(𝑍𝑟𝑛),0) = | ᴦ𝑆𝐼(𝑍𝑟𝑛)| = 𝑟
𝑛 − 1 

Now we find the size of  ᴦ𝑆𝐼(𝑍𝑟𝑛) , we note that there is 𝑚𝑟𝑛−1 of edges s.t  

𝑚1 = 𝑟
𝑛 − 2 ,𝑚2 = 𝑟

𝑛 − 3,…. ,𝑚𝑟𝑛−1 = 1  

Then,: 𝑐1 = 𝑚1 + 𝑚2 + ⋯+ 𝑚𝑟𝑛−1 

We can write:  

      𝑐1 = ∑ (𝑟
𝑛 − 𝑖

𝑟𝑛−1

𝑖=2

) 

Theorem 4.5:  

     𝑆𝑐( ᴦ𝑆𝐼(𝑍𝑟𝑛);𝑥) = ∑ (𝑟
𝑛 − 𝑖)(2𝑟𝑛 − 4)𝑥𝑟

𝑛−1
𝑖=2   , where  𝑟 ≥ 2 , 𝑛 > 1. 

Proof:  

𝑑𝑒𝑔(𝑣) = 𝑟𝑛 − 2  ,∀𝑣 ∈ 𝑉( ᴦ𝑆𝐼(𝑍𝑟𝑛)), 𝑣 = 1,2, . . . ,𝑟
𝑛 − 1   

 𝑑(𝑢,𝑣) = 1 ,∀𝑢,𝑣 ∈ 𝑉( ᴦ𝑆𝐼(𝑍𝑟𝑛) 

𝑆𝑐( ᴦ𝑆𝐼(𝑍𝑟𝑛);𝑥) = ∑ (𝑑𝑒𝑔(𝑢)+ 𝑑𝑒𝑔(𝑣))𝑥
𝑑(𝑢,𝑣)

𝑢,𝑣∈𝑉( ᴦ𝑆𝐼(𝑍𝑟𝑛))
  

      = (𝑑𝑒𝑔(1)+ 𝑑𝑒𝑔(2))𝑥 + ⋯+(𝑑𝑒𝑔(1) + 𝑑𝑒𝑔 (𝑟𝑛 − 1))𝑥⏟                                    
(𝑟𝑛−2)𝑡𝑖𝑚𝑒𝑠

+ (𝑑𝑒𝑔(2)+ 𝑑𝑒𝑔(3))𝑥 + ⋯+ (𝑑𝑒𝑔(2) + 𝑑𝑒𝑔 (𝑟𝑛 − 1))𝑥⏟                                    
(𝑟𝑛−3)𝑡𝑖𝑚𝑒𝑠

+⋯

+ (𝑑𝑒𝑔(𝑟𝑛 − 1) + 𝑑𝑒𝑔 (𝑟𝑛 − 2))𝑥 

                               = ∑ (𝑟𝑛 − 𝑖)(2𝑟𝑛 − 4)𝑥

𝑟𝑛−1

𝑖=2

 

           

5. Conclusions 

     This article has presented the formulae of some degree-based and eccentric-based 

topological indices of subgroup intersection graph of a group  𝑍𝑟𝑛 , where 𝑟 ≥ 2 ,𝑛 > 1. For 

further research, Revise on subgroup intersection graph of a group  𝑍𝑟𝑛 , where 𝑟 ≥ 2 ,𝑛 > 1, 

𝑟 is a prime number. 

 



Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34(4)2021 
 

77 

 

References  

1. H.Q u ; S. Cao. On the adjacent eccentric distance sum index of graphs  . PLos One , 1-12     

2015 

2. Morgan, M .J., Mukwembi S. and Swart H.C. On the eccentric Connectivity index of 

Graph , Discrete Mathematics . 2011 , 311. 2009, 1229-1234.      

3. Abdelgader, M.S.; Wang, C. ; Mohamed, S.A. Computation of Topological indices of 

some Special Graphs , mathematics . 2018,6(33).  

4. Kinkar, Ch.D; Das, S. ; Zkou, B. Sum-connectivity index of Graph, Frontiers of 

mathematics in china . 2016, 11(1),47-54. 

5. Khalifeh, M.H.; Yousefi- Azari, H. ; Ashrafi, A.R . The first and second Zagreb indices of 

some Graph operations , Discrete applied mathematics  .2009, 157, 2008, 804-811. 

6. Khaksari, A.; Ghorbain, M. The Forgotten Topological index, Iranian Journal of 

mathematical chemistry . 2017, 8(3), 327-338. 

7. Zhong, L.  The Harmonic index for Graphs, applied mathematics letters, 2012,25(3),561-

566. 

8. Abdussakir, Some Topological Indices of Subgroup Graph of Symmetric Group, 

Mathematics and Statistics,2019,7(4),98-105. 

9. Roshini, G. R. Some Degree Based Topological Indices of Transformation Graphs, Bull. 

Int. Math .Virtual Inst. 2020, 10(2),  225-237.  

10. Alaa, J. Nawaf ; Akram, S. Mohammad, Some Topological Indices and ( Hosoya and 

Schultz) Polynomial of Subgroup intersection graph of a group  𝑍𝑟 , Journal of Al-Qadisiyah 

for computer and Mathematics .2021, 13(1),120-130. 

11. Tamizh Chelvam,T. ; Sattanathan  ,  M. Subgroup intersection graph of a group .J . Adv. 

Research in pure Math (doi: 10.5373/ jarpm. 2011, 3(4),44-49. 594.100910, issN:1943-

2380). 

12.Hosoya,H.On Some Counting Polynomials in Chemistry , 𝟏𝟗𝟖𝟓. 

13. Chartrand, G.; Lesniak ., Graph and Diagraphs , 6th ed ., Wadsworth and Brooks/Cole . 

California. 2016.  

14.Gutman.Selected properties of the Schultz molecular Topological index, .J.Chem. 

Inf. Comput. Sci. 𝟏𝟗𝟗𝟒,34, 1087-1089 .