1. Introduction
We define an undirected power graph
for a group G as follows. Let us denote the cylic subgroup genarated by
by
, that is,
, where
denotes the set of naturel numbers. The graph
is an undirected graph where vertex set is G and two vertices
are adjacent if and only if
and
or
(which is equivalent to say
and
or
for some positive integer m.) [1] [2] [3] [4].
For a graph G, let
and
denote the degree of a vertex
and the distance between vertices
, respectively. Let
denote the line graph of G, that is, the graph with vertex set
and two distinct edges
adjacent in
whenever they share an end-vertex in G. Furthermore, for,
, we let
denote the distance between e and f in the line graph
.
We consider the power graph
for the additive group
of integers modulo n. The diameter of a graph G is the greatest distance between any pair of vertices, and denoted by
. In
, the distance is one if the vertices is adjacent and the distance is two if the vertices is non adjacent. Therefore,
. The order an element
in
is denoted by (
) or
. For a positive integer n,
denotes the Euler’s totient function of n.
In this paper, the wiener index and the edge-wiener index, denoted by
and
, respectively and they are defined as follows:
Now, we give some theorem and corollary in literature. Using our main theorems;
Theorem 1. ( [5] ) For each finite group, the number of edges of the undirected power graph
is given by the formula
Corollary 2. ( [6] ) The number of edges of the undirected power graph
is given by
.
Theorem 3. ( [3] ) Let G be connected graph with n vertices and m edges. If
, Then
.
Theorem 4. ( [5] ) A finite group has a complete undirected power graph if and only if it is cyclic and has order equal to pk, where p is a prime and k is a nonnegative integer.
2. Main Results
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
for
, or
, where p and q are distinct prime numbers and k is a nonnegative integer.
Theorem 5. Let
be an undirected power graph of with n vertices and m edges. Then
Proof. Let
be a set. In
, for
, there are two cases; If
then
. Otherwise, i.e.
, then
. Therefore
For definition of R, we obtain. Thus
the proof is complete.
Corollary 6. Let p and k is prime number and nonnegative integer, respectively. For
power graph of order
and m edges,
.
Proof. In [2], If
then
. For any
,
.
Thus
Therefore the proof is proved.
Theorem 7. Let
be a power graph of with n vertices and m edges. Then
Proof. If we consider Theorem 3. for
, we write
.
If we put the value of m into the formula, we obtain
Thus, the proof is complete.
Corollary 8. Let
be a power graph of with
, where p is a prime number. Then
.
Proof. Let
be a prime number. Then
Theorem 9. Let
be a power graph of with n vertices and m edges. Then
Proof. Where
is power graph
, using theorem 3. And corollary 2, we obtain
If we write this m in formula for
End of proof.
Corollary 10. Let
be a power graph of with
vertices and m edges, wherep and q are distinct prime numbers. Then
or equiently
.
Proof. If we write
in theorem 9., we obtain
(*)
On the other hand;
where
(**)
(**) equation put in (*) equation, we obtain,
.
This completes the proof.
On the other hand using m in (**), we obtain
This completes the proof.
Theorem 11. If
is a power graph of order
or
and m edges, where p and q are distinct prime and k is a nonnegative integer. Then
and
Proof. If
in Corollary 6.
.
And so
And if
in Corollary 10.
therefore
.
Also
.
We write
.
And so,
.
Theorem 12. If
is a power graph of order
and m edges, where p is prime and k is a nonnegative integer. Then
.
Proof. For
power graph,
and
,
.
Let’s consider to this figure in
power graph any
. For
power graph of Line graph as shown in Figure 1.
Choose the random
edge and this corner in neighborhood
line graph in Figure 2. In the same way, with
point neighborhood amount of points
. In the same way
neighborhood with corner amount of point
and therefore
if each elements for calculated and if edge-Wiener index identified we have the following result.
In edge-Wiener index
Concluded, namely the prove end.
Theorem 13. If
is a power graph of order
and m edges, where p is prime and k is a nonnegative integer. Then
Proof.
(
) is in
. In the same way,
Case 1. for
and according to
,
, therefore
ve
, namely this equation the proof.
Case 2. For
is
in theorem 12.,
Thus the proof is completed.
3. Conclusion
We will show the undirected power graph of a Group G with P(G). Here, the undirected P(Zn) Power graph of the group (Zn, +) according to N = pk 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.
Acknowledgements
This paper is derived from the first author’s PH’s thesis.