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Abstract
Let G be a finite abelian group with identity 0. For an integer the additive power graph
of G is the simple undirected graph with vertex set G in which two distinct vertices x and y are adjacent if and only if x + y = nt for some
with
When
the additive power graph has been studied in the name of square graph of finite abelian groups. In this paper, we study the additive power graph of G with n = 3 and name the graph as the cubic power graph. The cubic power graph of G is denoted by
More specifically, we obtain the diameter and the girth of the graph
and its complement
Using these, we obtain a condition for
and its complement
to be self-centered. Also, we obtain the independence number, the clique number and the chromatic number of
and its complement
and hence we prove that
and its complement
are weakly perfect. Also, we discuss about the perfectness of
At last, we obtain a condition for
and its complement
to be vertex pancyclic.
1. Introduction
The definition of Cayley graph was introduced by Arthur Cayley in 1878 to explain the concept of abstract groups which are described by a set of generators. Cayley graphs from finite groups are well studied in several aspects. Especially, Cayley graphs are used as communication grid in the routing problem due to its varied properties like regular and vertex transitive [Citation6]. Several aspects of Cayley graphs are studied in [Citation4, Citation7, Citation8, Citation13–17]. There are several other graph constructions from finite groups and rings [Citation1, Citation5]. For a finite commutative ring R, the set of squares of R (elements of the form x2 for some ) is a very interesting subset from algebraic point of view. Sen Gupta and Sen [Citation2, Citation10] introduced and studied about the square element graph
of R. In fact, the square element graph
of R is the simple undirected graph with vertex set
and two vertices a and b are adjacent if and only if
for some
Subsequently Sen Gupta and Sen [Citation10, Citation11] studied about the square element graph of arbitrary rings (not necessarily commutative) and they also studied about the square element graph of semigroups in [Citation3]. Snowden [Citation12] studied about square roots in finite full transformation semigroups. For a finite abelian group G with identity 0, Raveendra Prathap and Tamizh Chelvam [Citation9] introduced and studied about the square graph of G. The square graph
of G is the simple undirected graph with vertex set G and two distinct vertices x and y are adjacent if and only if
for some
and
Having introduced the square graph
Raveendra Prathap and Tamizh Chelvam [Citation9] studied several properties of its complement. Now, the concept of the square graph is generalized as the additive power graph
of G as the simple undirected graph with vertex set G in which two distinct vertices x and y are adjacent if and only if x + y = nt for some
positive integer n and
Note that when
the additive power graph
becomes the square graph of
Actually the square graph and cubic power graph of a finite abelian group G are generalizations of the Cayley graph on G [Citation6].
In this paper, we study about the additive power graph of a finite abelian group G where
That is we study about the cubic power
graph of finite abelian groups. More specifically, we obtain the diameter and the girth of
and its complement
Further we obtain a necessary and sufficient condition for
and its complement
to be self-centered. Also, we obtain the independence number, the clique number and the chromatic number of
and its complement
and hence we prove that
and its complement
are weakly perfect. Also, we discuss about the perfectness of
At last, we obtain a condition for
and its complement
to be vertex pancyclic.
2. Preliminaries
First let us recollect some basic definitions of graph theory which are essential for this paper. By a graph we mean Γ is a finite graph with vertex set V and edge set E. A graph Γ is said to be complete if each pair of distinct vertices is joined by an edge. We use Kn to denote the complete graph with n vertices. A graph Γ is said to be connected if there exists a path between every pair of distinct vertices in
The distance d(u, v) between the vertices u and v in Γ is the length of the shortest path between u and v. If no path exists between u and v in
then d(a, b) =
For a vertex
the eccentricity of v is the maximum distance from v to any vertex in
That is,
The radius of Γ is the minimum eccentricity among the vertices of
i.e., radius
The diameter of Γ is the maximum eccentricity among the vertices of
i.e., diam
The girth of Γ is the length of a shortest cycle in Γ and is denoted by
A graph Γ is said to be self-centered if the eccentricity of every vertex of Γ is the same. In other words, a graph is a self-centered graph if radius and diameter of the graph are equal. A graph Γ with n vertices is called a refinement of the star graph if
is a subgraph of Γ and Γ is not complete. For a vertex
N(x) is the set of all vertices in G which are adjacent to v and
The degree of a vertex v is the number of the edges in Γ which are incident with v. A clique of Γ is a maximal complete subgraph of Γ and the number of vertices in the largest clique of Γ is called the clique number of Γ and is denoted by A graph Γ with n vertices is called pancyclic if it contains a cycle of length k for every
and it is called vertex pancyclic if every vertex is contained in a cycle of length k for every
An independent set is a set of vertices in a graph, in which no two vertices are adjacent and cardinality of a maximal independent set is called the independent number. A vertex coloring of Γ is a mapping
The elements of S are called colors. If
we say that c is a k-coloring. A coloring is said to be proper if adjacent vertices have different colors. A graph is k-colorable if it has a proper k-coloring. The chromatic number is the least k such that Γ is k-colorable [Citation18, Citation19].
Let G be a finite abelian group and where 3 and
are distinct primes and
for
Then G is the direct product of finite cyclic groups. i.e.,
A partition
of a positive integer
also called an integer partition, is a way of writing
as the sum of positive integers. We write a partition of
as
with
If
is divisible by 3, then we have
where
is a subgroup of G of order
and
Remark 2.1.
For a finite abelian group G, let and
Suppose
Then
and
Note that two distinct vertices x and y are adjacent in
if and only if
Remark 2.2.
If
is not divisible by 3, then G = G1 and
Assume that
is divisible by 3 and so
Consider a partition
of
Here
Now, with respect to a partition of
one can associate
k-tuples of 0’s and 1’s. For a k-tuple,
let
One can check that in either case,
One can verify that
3. Properties of ![](//:0)
![](//:0)
In this section, we list out some basic properties of the cubic power graph of a finite abelian group G. In particular, we prove that
is connected and obtain the diameter, the girth, the independence number, the chromatic number and the clique number of
Using these parameters, we conclude that
is weakly perfect. At the end of the section, we prove that
is vertex pancyclic and not Eulerian.
Lemma 3.1.
Let G be a finite abelian group,
and t be relatively prime to 3. If
and
for
then x is not adjacent to y in
Proof.
Let and
for
and
Then and
with
for some
Let
and
If then
As per the definition of
we get that
and
and so
which implies that
Hence x is not adjacent to y in
□
In this following, we obtain the diameter and the girth of where
is not divisible by 3.
Theorem 3.2.
Let G be a finite abelian group and is not divisible by 3. Then
is connected.
is a refinement of the star graph
Proof.
Assume that is not divisible by 3. As observed in Remark 2.2(1), we have G = G1.
Assume that
for
For every
we have
and so x is adjacent to all y in
Thus deg
Let
be such that
Then
for all
Thus x is adjacent to all
in
From this, we have deg
This is also true for
By (1),
is connected.
Let
Then
for every
Thus
and so diam
Now, assume that
and
is an element in G. Then there exists an element
such that
and
Thus x is not adjacent to y but there exists a path
of length two in G. Thus diam
If
then
and so gr
If
then
as
is not divisible by 3. Let
with
Then
is a cycle of length 3 in G and so gr
Follow from the proof (1) and (3), if
then
is complete and if
then
is a refinement of the star graph
with center either 0 or x in G of order is 2. □
In the following theorem, we obtain the diameter and girth of where
is divisible by 3.
Theorem 3.3.
Let G be a finite abelian group. Assume that t is relatively prime to 3 and
Then
If
then
Otherwise
is a refinement of the star graph
is a regular graph of degree
is isomorphic to disjoint union of
and
Proof.
Note that
Let
for
Then, for every
we have
and so x is adjacent to all y in
Thus deg
Let
for
In this case
is adjacent to all
Thus deg
Let
Then
for all
This implies that, when
and
then one must have
From this, for
we have
where
Hence deg
As observed in the proof of (1),
is either
or a refinement of the star graph with center 0 or x in G of order 2.
Follows from the Case (iii) of (1).
Note that
and
By Lemma 3.1,
is isomorphic to disjoint union of
and
□
From Lemma 3.1 and Theorem 3.3 we have the following corollary.
Corollary 3.4.
Let G be a finite abelian group. Assume that t is relatively prime to 3 and
Then
If
then
Otherwise
is a refinement of the star graph
with center0.
The induced subgraph
is a regular graph of degree
Further
is a regular graph of degree
is the disjoint union of
for
By substituting t = 1 in the statement of Corollary 3.4, we have the following corollary.
Corollary 3.5.
Let G be a finite abelian group of order with
and
Then
If
then
Otherwise
is a refinement of the star graph
with center 0.
The induced subgraph
is a regular graph of degree
Further
is a regular graph of degree
is the disjoint union of
induced subgraphs
Recall that a graph Γ is self-centered if the eccentricity of every vertex in Γ is same. In the following theorem, we obtain a condition for to be self-centered.
Theorem 3.6.
Let G be a finite abelian group with and t is relatively prime to 3. Then
is self-centered if and and only if
and
for some
Proof.
If then
G = G1 and
for every
Thus
and
is self-centered.
Conversely assume that is self-centered. If
is divisible by 3, then by Theorem 3.3(4),
is disconnected, a contradiction to
is self-centered. Hence
is not divisible by 3 and so
Assume that
Then G contains elements x, y such that
and
Now we have,
and
which is contradiction to
is self-centered. Hence
□
In the following theorem, we obtain the independence number of
Theorem 3.7.
Let G be a finite abelian group with and t is relatively prime to 3. Then the independence number
Proof.
Case 1. Assume that i.e.,
is not divisible by 3 and hence
If then
and so
If
then
is adjacent to all vertices except
Hence
is a maximal independent set and so
Case 2. Assume that i.e.,
is divisible by 3 and so by Theorem 3.3(4),
and
are disjoint induced subgraphs of
Hence
Assume that
Case 2.1. Assume that t = 1 and This implies that
and
Case 2.1.1. If then
and so
This gives that
Case 2.1.2. If then there exists
such that
Hence
is a maximal independent set of
and so
Further in this case
is a maximal independent set of
and so
Thus
Case 2.2. Assume that t > 1 or
Case 2.2.1. Suppose In this case
Also
and so
Case 2.2.2. In the remaining cases where
and t is relatively prime to 3. In this case also
is a proper refinement of a star graph and there exists
such that
Hence
is a maximal independent set of
and so
If then
is a maximal independent set in
and so
In general, if and
is a maximal independent set in
then
is a maximal independent set of
in
Thus
in either of the cases.
Hence according to Cases 2.1 and 2.2, we have or
□
In the following theorem, we obtain the clique number of .
Theorem 3.8.
Let G be a finite abelian group. Assume that t is relatively prime to 3 and
Then the clique number
Proof.
Case 1. Let Then
is not divisible by 3. By Theorem 3.2, any
with
is adjacent to other all vertices in
On the other hand, x with
is adjacent with all but
Then the subgraph induced by
is a maximal complete subgraph of
and
Hence
Case 2. Assume that Then
is divisible by 3. By Theorem 3.3,
and
are disjoint induced subgraphs of
From this, we get that
If then
Applying Case 1 for
we get that
In general, if then applying successively the above fact, we get that the clique number
Let For
by Theorem 3.3, we have deg
We claim that K3 is not an induced subgraph of
Suppose not, let
From this, we have that
and
and
b + b and c + c are not 3 multiplies. At the same time, we have
and so
which is a contradiction to a + a is not a 3 multiple. Thus
and so
In general, if
then
Hence □
In the following theorem, we obtain the chromatic number of
Theorem 3.9.
Let G be a finite abelian group. Assume that t is relatively prime to 3 and
Then the chromatic number
Proof.
In general we have
Case 1. Assume that is not divisible by 3 and
In this case
By Theorem 3.2, any
with
is adjacent to other all vertices in
Assigning different colors to all vertices in T and assign a color to each x and – x for those with
This assignment gives a proper coloring for
and so
Thus, by Theorem 3.8, we get that
Case 2. Assume that is divisible by 3. By Theorem 3.3,
and
are disjoint induced subgraphs of
and so
If as observed in the proof of Case 1 of Theorem 3.3, 0 is adjacent to other all vertices in
and so one has to assign separate colors to 0 and all vertices. Any
is not adjacent to
Hence the color of x can be assigned to –x. This assignment shall give a proper coloring for
Hence
In general if then
Thus
If as observed in the proof of Case 2.2.2 of Theorem 3.7,
and
are independent sets in
and hence same color can be assigned to each of these vertices. Further, the remaining vertices can be colored by another color and so
In general if then
and so
Hence □
Note that, a graph Γ is said to be weakly perfect if clique and chromatic numbers of Γ are same. From Theorems 3.8 and 3.9, one can conclude that is weakly perfect and the same is stated below.
Theorem 3.10.
Let G be a finite abelian group. Then the cubic power graph is weakly perfect.
A graph Γ with n vertices is called pancyclic if it contains a cycle of length k for every Further Γ is said to be vertex pancyclic if every vertex of Γ is contained in a cycle of length k for every
In the following theorem, we prove that
is vertex pancyclic where G is a finite abelian group and
is not divisible by 3.
Theorem 3.11.
Let G be a finite abelian group. If is not divisible by 3, then the cubic power graph
is vertex pancyclic.
Proof.
By Theorem 3.2, Let
and
Note that
is a complete subgraph of
and every
is adjacent to all vertices in
except
Hence the cycle
contains a cycle a length i for
Hence
is vertex pancyclic. □
A famous characterization for Eulerian graph is as follows: Any connected graph Γ is an Eulerian graph if and only if all its vertices are of even degree. Now, we observe that is not Eulerian.
Theorem 3.12.
For any finite abelian group G, is not Eulerian.
Proof.
If is divisible by 3, by Theorem 3.3,
is disconnected and hence
is not Eulerian. Assume that
is not divisible by 3. Then
By Theorem 3.2, there exists vertices of odd degree and so
is not Eulerian. □
4. Properties of the complement ![](//:0)
![](//:0)
In this section, we study about the complement of the cubic power graph
of finite abelian groups.
Theorem 4.1.
Let G be a finite abelian group and is not divisible by 3 and
Then
Proof.
If is not divisible by 3, then
For every
there exists
such that
Hence x and y are adjacent in
if and only if
if and only if
When we get K1 and when
we get
Hence
where
□
Theorem 4.2.
Let G be a finite abelian group. Then is connected if and only if
is divisible by 3.
Proof.
If is connected, by Theorem 4.1,
is divisible by 3.
Conversely, assume that is divisible by 3. By Theorem 3.3,
is disconnected and so
is connected. □
Now, let us observe some properties of the induced subgraphs of induced by the subsets
of G.
Theorem 4.3.
Let G be a finite abelian group. Assume that , t is relatively prime to 3 and
Then the following hold.
The induced subgraph
of
is either K1 or
or
For
and
let
and
Then x and y are adjacent in
For
the induced subgraph
is a regular subgraph of
degree
Further
is a regular subgraph of
of degree
diam
gr
is self-centered.
Proof.
First let us consider the case that
is divisible by 3. Note that
Let
Then x adjacent to y in
if and only if
By Theorem 3.3(1.1), if
then
Hence
in
By Theorem 3.3(1.1), if
and
then
Hence
in
In the remaining possibilities, by Theorem 3.3(1.1), for
with
we have
and
for all
with
Hence
in
By Lemma 3.1,
and
are not adjacent in
and so they are adjacent in
Note that
For every
there exists a
such that
By Theorem 3.3((1.3) and (1.4)), deg
of the induced subgraph
This gives that deg
of the induced subgraph
Further with regard to
in
we have deg
Assume that
Then
and so diam
In the remaining possibilities,
for
From (2),
and
are adjacent in
and so diam
Consider
and
Then
Hence gr
Assume that
Then
and so
for all
In the remaining possibilities, is not adjacent to 0 in
From (2),
for all
For
from (3), there exists at least one
which is not adjacent to x. Hence
for all
Thus is self-centered in both the cases. □
In the following theorem, we obtain the independence number of
Theorem 4.4.
Let G be a finite abelian group. Assume that , t is relatively prime to 3 and
Then the independence number
Proof.
Case 1. Let Then
is not divisible by 3. By Theorem 4.1,
Hence
Case 2. Let and
Then
is divisible by 3.
If then
and so
In the remaining possibilities, for
By Theorem 4.3(2), any independent set S is contained in one
for some
Suppose S is an independent set in
for some
and
Let
Then
and so
and
for some
which implies
Thus x2 is adjacent to x3 in
which is contradiction to our assumption that S is independent. Hence
or
Assume that
is an independent set. By Theorem 4.3(1),
Thus
Combining the possibilities of (a) and (b), we get that the maximal independent set S of is a subset of
From this
in
Hence
□
In the following theorem, we obtain the clique number of the
Theorem 4.5.
Let G be a finite abelian group. Assume that , t is relatively prime to 3 and
Then the clique number
Proof.
Case 1. Assume that is not divisible by 3.
Case 1.1. If then
and so
Case 1.2. If by Theorem 4.1,
and so
Case 2. Assume that is divisible by 3.
Case 2.1. If then
and so
Case 2.2. In the remaining possibilities
Case 2.2.1. By Theorem 4.3(1), we get Thus
Case 2.2.2. As observed the proof of Case 2.2.2 in Theorem 3.7, if then
is a maximal clique of
and so
Let Let
be the first index such that
and
Then
where
is a maximal clique of
and so
By Remark 2.2 (4),
Hence from the Cases 2.1, 2.2.1 and 2.2.2, we have or
or
□
In the following theorem, we obtain the chromatic number of
Theorem 4.6.
Let G be a finite abelian group. Assume that , t is relatively prime to 3 and
Then the chromatic number
Proof.
Case 1. Assume that is not divisible by 3.
Case 1.1. If then
and so
Case 1.2. If then
and so
Case 2. Assume that is divisible by 3.
Case 2.1. If then
and so
Case 2.2. In the remaining possibilities,
By Theorem 4.3 (2), is a complete bipartite graph with vertex partitions
and
Then
Case 2.2.1. By Theorem 4.3(1), we get Thus
Case 2.2.2. Assume that If
and
then there exists
such that x and y are not adjacent in
Thus the same color can be assigned to both x and y in
Hence number of colors needed to color
is
Thus we get that
and so
In general, let and
Applying selection of yi as above, for each xi for each
we get that
such that x and y are not adjacent in
Hence the same color can be assigned to both x and y and so number of colors needed to color G2 is
Thus
and so
From the Cases 2.1, 2.2.1 and 2.2.2, we have
From Theorems 4.5 and 4.6, one can conclude that is weakly perfect and the same is stated below.
Theorem 4.7.
For any finite abelian group G, the graph is weakly perfect.
In the following theorem, we give a necessary and sufficient condition for to be Eulerian.
Theorem 4.8.
Let G be a finite abelian group. Then is Eulerian if and only if
Proof.
Assume that By Theorem 4.2,
is connected. To prove that
is Eulerian, it is enough to show that each of the vertex in
is of even degree.
If then
is odd and
Hence each vertex of
is of even degree.
If then both
and
are even. By Theorem 4.3(1),
By Theorem 4.3(3), the induced subgraph
of
is a regular graph of degree
Then each vertex of the induced subgraph
is of even degree. By Theorem 4.3(2), each vertex in G1 is adjacent to all the vertices in
Thus each vertex of
is of even degree.
Conversely, assume that is Eulerian. Then each vertex of
is of even degree. Suppose
and
Then deg
and their exists an element
such that deg
an odd number. Hence it is a contradiction. From this
□
In the following theorem, we give a necessary and sufficient condition for to be vertex pancyclic.
Theorem 4.9.
Let G be a finite abelian group whose order is divisible by 3. Then is vertex pancyclic.
Proof.
Assume that is divisible by 3. By Theorem 4.2,
is connected.
For let
One can arrange the sets
such that
By Theorem 4.5, is a maximal clique of
for
and
is a maximal clique of
and so
Further, by Theorem 4.3(2), for
and
and
are adjacent in
Let
for
By Remark 2.2(4),
and by Theorem 4.13,
These together imply that
and so
Let Note that
and the elements of the set X can be arranged such that
and
Let
for
Now we have the spanning cycle
Let m be an integer such that Select m vertices starting from
in C. If the mth vertex
then choose a different
The first vertex is in X1 and the mth vertex
is in a different
and so the first and last vertices are adjacent. Suppose
obviously the first vertex
and the last vertex
are in two different
and so they are adjacent. This gives a cycle of length m in
Hence
is pancyclic. □
We shall conclude this section by proving that is perfect. Note that a graph Γ is perfect if neither Γ nor
contain an induced odd cycle of degree at least five. So to conclude
is perfect, one has to prove that neither
nor
contains an induced cycle
Assume that
is divisible by 3. By Theorem 4.3(1),
contains no cycle. Hence we consider the induced subgraphs
for
in the following theorem.
Lemma 4.10.
Let G be a finite abelian group such that is divisible by 3. For
the induced subgraphs
of
contain no induced cycle of odd length greater than 3.
Proof.
Let G be a finite abelian group and be divisible by 3. Assume that
contains an induced cycle of odd length greater than 3, for some
Case 1. Assume that Then
which implies that K3 is a subgraph of any cycle C in
of length grader than 3, which is a contradiction. Hence
contains no induced cycle of odd length greater than 3 for any
Case 2. In the remaining possibilities,
(a). Let As observed in the proof of Case 2.2.2 in Theorem 4.5,
and
are maximal cliques of
Let
be a cycle of length 5 in
Then either
or
and so K3 is a subgraph of C in
which is a contradiction. Hence
of
contains no induced cycle of odd length greater than 3.
One can conclude the proof of general case as in the proof of later part of Case 2.2.2 in Theorem 4.5. Hence of
contains no induced cycle of odd length greater than 3 for
□
In the following theorem, we prove that is perfect.
Theorem 4.11.
Let G be a finite abelian group. Then the cubic power graph is perfect.
Proof.
Case 1. Assume that is divisible by 3.
Claim 1. contains no induced cycle of odd length greater than 3. Suppose
contains an induced cycle C of odd length greater than 3. By Corollary 3.4(4),
is the disjoint union of
for
Thus
for some
Suppose By Theorem 3.3(2) and Corollary 3.5(2),
is either complete or refinement of a star graph with center 0 in
and so
in
which is a contradiction.
Suppose for some
Let
and so
Then
for some
From this, we get that
and
This in turn implies that a is adjacent to d and b is adjacent to e. Thus
in
in
for all
which is a contradiction. Hence
contains no induced cycle of length odd order greater than 3.
Claim 2. contains no induced cycle of odd length greater than 3.
By Theorem 4.3(1), contains no cycle. By Lemma 4.10,
contains no induced cycle of length odd order greater than 3 for
Hence
contains no induced cycle of length odd order greater than 3.
Case 2. Assume that is not divisible by 3.
By Theorem 3.2(5), is either complete or a refinement of a star graph with center 0 and so
contains no induced cycle of odd length greater than 3.
By Theorem 4.1, Hence
contains no induced cycle of odd length greater than 3.
Hence is a perfect graph. □
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References
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