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The undirected power graph
P(
Z_{n}) of a finite group
Z_{n} is the graph with vertex set G and two distinct vertices u and v are adjacent if and only if
u ≠
v and
_{n})) of an undirected power graph
P(
Z_{n}) is defined to be sum
_{n}). Similarly, the edge-Wiener index
W_{e}(
P(
Z_{n})) of
P(
Z_{n}) is defined to be the sum
_{n}). In this paper, we concentrate on the wiener index of a power graph
_{pq}) and
P(
Z_{p}). Firstly, we obtain new results on the wiener index and edge-wiener index of power graph
P(
Z_{n}), using
m,n and Euler function. Also, we obtain an equivalence between the edge-wiener index and wiener index of a power graph of
Z_{n}.

We define an undirected power graph P ( G ) for a group G as follows. Let us denote the cylic subgroup genarated by u ∈ G by 〈 u 〉 , that is, 〈 u 〉 = { u m | m ∈ ℕ } , where ℕ denotes the set of naturel numbers. The graph P ( G ) is an undirected graph where vertex set is G and two vertices u , v ∈ G are adjacent if and only if u ≠ v and 〈 u 〉 ⊆ 〈 v 〉 or 〈 v 〉 ⊆ 〈 v 〉 (which is equivalent to say u ≠ v and u m = v or v m = u for some positive integer m.) [

For a graph G, let deg ( u ) and d ( u , v ) denote the degree of a vertex u ∈ V ( G ) and the distance between vertices u , v ∈ V ( G ) , respectively. Let L ( G ) denote the line graph of G, that is, the graph with vertex set E ( G ) and two distinct edges e , f ∈ E ( G ) adjacent in L ( G ) whenever they share an end-vertex in G. Furthermore, for, f ∈ E ( G ) , we let d ( e , f ) denote the distance between e and f in the line graph L ( G ) .

We consider the power graph P ( Z n ) for the additive group Z n of integers modulo n. The diameter of a graph G is the greatest distance between any pair of vertices, and denoted by d i a m ( G ) . In P ( Z n ) , the distance is one if the vertices is adjacent and the distance is two if the vertices is non adjacent. Therefore, d i a m ( P ( Z n ) ) = 2 . The order an element g ¯ in Z n is denoted by ( g ¯ ) or | g | . For a positive integer n, ϕ ( n ) denotes the Euler’s totient function of n.

In this paper, the wiener index and the edge-wiener index, denoted by W ( G ) and W e ( G ) , respectively and they are defined as follows:

W ( G ) = 1 2 ∑ { u , v } ⊆ V ( G ) d ( u , v )

W e ( G ) = 1 2 ∑ { e , f } ⊆ E ( G ) d ( e , f )

Now, we give some theorem and corollary in literature. Using our main theorems;

Theorem 1. ( [

E ( P ( G ) ) = 1 2 ∑ g ∈ G { 2 o ( g ) − ϕ ( o ( g ) ) − 1 }

Corollary 2. ( [

Theorem 3. ( [

Theorem 4. ( [^{k}, where p is a prime and k is a nonnegative integer.

In this section, our aim is to give our main results on the Wiener index and the edge-Wiener index of an undirected power graph P ( Z n ) for n = p k , or n = p q , where p and q are distinct prime numbers and k is a nonnegative integer.

Theorem 5. Let P ( Z n ) be an undirected power graph of with n vertices and m edges. Then

W ( P ( Z n ) ) = 1 2 ∑ { u , v } ⊆ V ( P ( Z n ) ) { 1 , u ∼ v 2 , u ≁ v

Proof. Let

R = { { u , v } ⊆ V ( P ( Z n ) ) | u ~ v if only if u ≠ v , 〈 u 〉 ⊆ 〈 v 〉 or 〈 v 〉 ⊆ 〈 u 〉 } be a set. In P ( Z n ) , for { u , v } ⊆ V ( P ( Z n ) ) , there are two cases; If u ≁ v then d ( u , v ) = 2 . Otherwise, i.e. u ∼ v , then d ( u , v ) = 1 . Therefore

W ( P ( Z n ) ) = 1 2 ∑ { u , v } ⊆ V ( P ( Z n ) ) d ( u , v ) = 1 2 ( ∑ { u , v } ⊆ R d ( u , v ) + ∑ { u , v } ⊈ R d ( u , v ) ) = 1 2 ∑ { u , v } ⊆ R 1 + 1 2 ∑ { u , v } ⊈ R 2 = 1 2 ∑ { u , v } ⊆ V ( P ( Z n ) ) { 1 , { u , v } ⊆ R 2 , { u , v } ⊈ R

For definition of R, we obtain. Thus

W ( G ) = 1 2 ∑ { u , v } ⊆ V ( P ( Z n ) ) { 1 , u ∼ v 2 , u ≁ v

the proof is complete.

Corollary 6. Let p and k is prime number and nonnegative integer, respectively. For P ( Z p k ) power graph of order p k and m edges,

W ( P ( Z p k ) ) = ( p k 2 ) .

Proof. In [

R c = { { u , v } ⊆ V ( P ( Z n ) ) | u ≁ v } = ∅

Thus

W ( P ( Z p k ) ) = 1 2 ∑ { u , v } ⊆ V ( P ( Z n ) ) { 1 , u ∼ v 2 , u ≁ v = 1 2 ( ∑ { u , v } ⊆ R d ( u , v ) + ∑ { u , v } ⊈ ∅ d ( u , v ) ) = 1 2 ( ∑ { u , v } ⊆ R 1 + ∑ { u , v } ⊆ ∅ d ( u , v ) ) = 1 2 ∑ { u , v } ⊆ R 1 = 1 2 p k ( p k − 1 ) = ( p k 2 )

Therefore the proof is proved.

Theorem 7. Let P ( Z n ) be a power graph of with n vertices and m edges. Then

W ( P ( Z n ) ) = 1 2 { ( 2 n 2 ) + ∑ g = 0 n − 1 ( ϕ ( | g ¯ | ) − 2 | g ¯ | ) }

Proof. If we consider Theorem 3. for = P ( Z n ) , we write

W ( P ( Z n ) ) = n ( n − 1 ) − m

m = 1 2 ∑ g ∈ Z n { 2 o ( g ) − ϕ ( o ( g ) ) − 1 } .

If we put the value of m into the formula, we obtain

W ( P ( Z n ) ) = n ( n − 1 ) − m = n ( n − 1 ) − 1 2 ∑ g ∈ Z n { 2 o ( g ) − ϕ ( o ( g ) ) − 1 } = n 2 − n + 1 2 ∑ g ∈ Z n { ϕ ( o ( g ) ) − 2 o ( g ) } − 1 2 ∑ g ∈ Z n 1 = n 2 − n + n 2 + 1 2 ∑ g ∈ Z n { ϕ ( o ( g ) ) − 2 o ( g ) } = { n 2 − n 2 + 1 2 ∑ g = 0 n − 1 ( ϕ ( | g ¯ | ) − 2 | g ¯ | ) }

W ( P ( Z n ) ) = 1 2 { ( 2 n 2 ) + ∑ g = 0 n − 1 ( ϕ ( | g ¯ | ) − 2 | g ¯ | ) }

Thus, the proof is complete.

Corollary 8. Let P ( Z n ) be a power graph of with n = p , where p is a prime number. Then

W ( P ( Z n ) ) = ( P 2 ) .

Proof. Let n = p be a prime number. Then

W ( P ( Z p ) ) = 1 2 { ( 2 p 2 ) + ∑ g = 0 p − 1 ( ϕ ( | g ¯ | ) − 2 | g ¯ | ) } = 1 2 [ 2 p ( 2 p − 1 ) 2 + ϕ ( | 0 ¯ | ) + ϕ ( | 1 ¯ | ) + ⋯ + ϕ ( | p − 1 ¯ | ) − 2 ( | 0 ¯ | + | 1 ¯ | + ⋯ + | p − 1 ¯ | ) ] = 1 2 [ 2 p 2 − p − 1 + ( ϕ ( | 1 ¯ | ) + ⋯ + ϕ ( | p − 1 ¯ | ) ) − 2 ( | 1 ¯ | + ⋯ + | p − 1 ¯ | ) ] = 1 2 [ 2 p 2 − p − 1 + ( p − 1 ) ϕ ( p ) − 2 ( p − 1 ) p ] = 1 2 [ 2 p 2 − p − 1 + ( p − 1 ) 2 − 2 p 2 + 2 p ] = ( p 2 )

Theorem 9. Let P ( Z n ) be a power graph of with n vertices and m edges. Then

W ( P ( Z n ) ) = 1 2 { ( 2 n 2 ) + ∑ d | n ϕ ( d ) ( ϕ ( d ) − 2 d ) } .

Proof. Where P ( Z n ) is power graph = P ( Z n ) , using theorem 3. And corollary 2, we obtain

W ( P ( Z n ) ) = n ( n − 1 ) − m

m = 1 2 ∑ d | n { 2 d − ϕ ( d ) − 1 } ϕ ( d )

If we write this m in formula for W ( P ( Z n ) )

W ( P ( Z n ) ) = n ( n − 1 ) − m = n ( n − 1 ) − 1 2 ∑ d | n { 2 d − ϕ ( d ) − 1 } ϕ ( d ) = n 2 − n + 1 2 ∑ d | n ϕ ( d ) 2 + 1 2 ∑ d | n ϕ ( d ) − ∑ d | n d ϕ ( d ) = n 2 − n 2 + 1 2 ∑ d | n ϕ ( d ) ( ϕ ( d ) − 2 d )

W ( P ( Z n ) ) = 1 2 { ( 2 n 2 ) + ∑ d | n ϕ ( d ) ( ϕ ( d ) − 2 d ) } .

End of proof.

Corollary 10. Let P ( Z n ) be a power graph of with n = p q vertices and m edges, wherep and q are distinct prime numbers. Then

W ( P ( Z p q ) ) = m + 2 ϕ ( p q )

or equiently

W ( P ( Z p q ) ) = ( p q 2 ) + ϕ ( p q ) .

Proof. If we write n = p q in theorem 9., we obtain

W ( P ( Z p q ) ) = 1 2 { ( 2 p q 2 ) + ∑ d | p q ϕ ( d ) ( ϕ ( d ) − 2 d ) } = 1 2 [ p q ( 2 ⋅ p q − 1 ) + ϕ ( 1 ) ( ϕ ( 1 ) − 2 ⋅ 1 ) + ϕ ( p ) ( ϕ ( p ) − 2 ⋅ p ) + ϕ ( q ) ( ϕ ( q ) − 2 ⋅ q ) + ϕ ( p q ) ( ϕ ( p q ) − 2 ⋅ p q ) ] = 1 2 [ p 2 q 2 + p q − 2 ⋅ p − 2 ⋅ q + 2 ] = [ p 2 q 2 − p q 2 + p q − p − q + 1 ] = [ ( p q 2 ) − ϕ ( p q ) ] + 2 ⋅ ϕ ( p q ) (*)

On the other hand;

W ( P ( Z p q ) ) = p q ( p q − 1 ) − m = ( p q 2 ) + ϕ ( p q )

where

m = ( p q 2 ) − ϕ ( p q ) (**)

(**) equation put in (*) equation, we obtain,

W ( P ( Z p q ) ) = m + 2 ϕ ( p q ) .

This completes the proof.

On the other hand using m in (**), we obtain

W ( P ( Z p q ) ) = m + 2 ϕ ( p q ) = ( p q 2 ) − ϕ ( p q ) + 2 ϕ ( p q ) = ( p q 2 ) + ϕ ( p q )

This completes the proof.

Theorem 11. If P ( Z n ) is a power graph of order n = p k or n = p q and m edges, where p and q are distinct prime and k is a nonnegative integer. Then

m a k s { W ( P ( Z n ) ) } = ( n + 1 2 )

and

min { W ( P ( Z n ) ) } = ( n 2 )

Proof. If n = p k in Corollary 6.

W ( P ( Z p k ) ) = ( p k 2 ) .

And so

min { W ( P ( Z n ) ) } = ( n 2 )

And if n = p q in Corollary 10.

W ( P ( Z p q ) ) = ( p q 2 ) + ϕ ( p q )

therefore

W ( P ( Z n ) ) ≤ ( n 2 ) + ϕ ( n ) .

Also

ϕ ( n ) ≤ n .

We write

W ( P ( Z n ) ) ≤ ( n 2 ) + ϕ ( n ) ≤ ( n 2 ) + n .

And so,

m a k s { W ( P ( Z n ) ) } = ( n + 1 2 ) .

Theorem 12. If P ( Z n ) is a power graph of order n = p k and m edges, where p is prime and k is a nonnegative integer. Then

W e ( P ( Z n ) ) = 3 { ( n 3 ) + d i a m ( L ( P ( Z n ) ) ) ( n 4 ) } .

Proof. For P ( Z p k ) power graph, E ( P ( Z n ) ) = ( n 2 ) and ∀ u ∈ V ( P ( Z n ) ) , d ( u ) = n − 1 .

Let’s consider to this figure in P ( Z p k ) power graph any e n ¯ , n − 1 ¯ ∈ E ( P ( Z p k ) ) . For P ( Z p k ) power graph of Line graph as shown in

Choose the random e n ¯ , n − 1 ¯ ∈ E ( P ( Z p k ) ) edge and this corner in neighborhood L ( P ( Z n ) ) line graph in

In edge-Wiener index

W e ( P ( Z p k ) ) = 1 2 ∑ { e , f } ⊆ E ( P ( Z n ) ) d ( e , f ) = 1 2 { ∑ u v = e [ ( d ( u ) + d ( v ) − 2 ) ] + ∑ u v = e [ d i a m ( L ( P ( Z n ) ) ) ⋅ ( ( m − 1 ) − ( d ( u ) + d ( v ) − 2 ) ) ] } = 1 2 { ( n 2 ) [ 2 ( n − 2 ) + d i a m ( L ( P ( Z n ) ) ) ( ( n 2 ) − 1 − 2 ( n − 2 ) ) ] } = [ n ( n − 1 ) ( n − 2 ) 2 + n ( n − 1 ) 4 d i a m ( L ( P ( Z n ) ) ) ( n 2 − 5 n − 6 2 ) ] = 3 ( n 3 ) + n ( n − 1 ) ( n − 2 ) ( n − 3 ) 8 d i a m ( L ( P ( Z n ) ) )

W e ( P ( Z p k ) ) = 3 [ ( n 3 ) + d i a m ( L ( P ( Z n ) ) ) ( n 4 ) ]

Concluded, namely the prove end.

Theorem 13. If P ( Z n ) is a power graph of order n = p k and m edges, where p is prime and k is a nonnegative integer. Then

W e ( P ( Z n ) ) = ( n − 1 2 ) W ( P ( Z n ) )

Proof. n = p k ( ∈ Z + ) is in W ( P ( Z n ) ) = ( n 2 ) . In the same way,

Case 1. for n = 2 , 3 and according to d i a m ( L ( P ( Z n ) ) ) = 1 , W e ( P ( Z 2 ) ) = 0 , therefore W e ( P ( Z 3 ) ) = W ( P ( Z 3 ) ) ve ( 3 − 1 2 ) = 1 , namely this equation the proof.

Case 2. For n ≠ 2 , 3 is d i a m ( L ( P ( Z n ) ) ) = 2 in theorem 12.,

W e ( P ( Z n ) ) = 3 [ ( n 3 ) + d i a m ( L ( P ( Z n ) ) ) ( n 4 ) ] = 3 [ ( n 3 ) + 2 ( n 4 ) ] = 1 2 n ( n − 1 ) [ ( n − 2 ) + ( ( n − 2 ) ( n − 3 ) 2 ) ] = ( n 2 ) ( n − 2 ) [ 1 + n − 3 2 ] = ( n − 1 2 ) W ( P ( Z n ) )

Thus the proof is completed.

We will show the undirected power graph of a Group G with P(G). Here, the undirected P(Z_{n}) Power graph of the group (Z_{n}, +) according to N = p^{k} and n = pq, with p, q being different primes and k being positive integers, is considered and new theorems and results on the Wiener index calculations of these power graphs with the help of Euler function are have been obtained.

This paper is derived from the first author’s PH’s thesis.

The authors declare no conflicts of interest regarding the publication of this paper.

Aşkin, V. and Büyükköse, S. (2021) The Wiener Index of an Undirected Power Graph. Advances in Linear Algebra & Matrix Theory, 11, 21-29. https://doi.org/10.4236/alamt.2021.111003