Order Sum Graph of a Group

: The concept of the order sum graph associated with a finite group based on the order of the group and order of group elements is introduced. Some of the properties and characteristics such as size, chromatic number, domination number, diameter, circumference, independence number, clique number, vertex connectivity, spectra, and Laplacian spectra of the order sum graph are determined. Characterizations of the order sum graph to be complete, perfect, etc. are also obtained


Introduction:
Investigating graphs associated with different algebraic structures has fascinated researchers around the world. Many research articles developed new concepts by linking graph theory and algebra. The coprime graph of a finite group is a graph consisting of elements of the group as vertices and there is an edge between two vertices say and if (| |, | |) = 1. Some of the properties of the coprime graph were also investigated 1 .
The distance coprime graph of some standard graphs such as path, star, wheel, cycle, and complete bipartite graph were obtained 2 . Some properties and characterizations for the distance coprime graph were also obtained. Recently, the notion of the non-inverse graph of a group (in short, * -graph) of , denoted by Γ, has been introduced as a simple graph with vertex set consisting of elements of and there is an edge between two vertices , v ∈ Γ if and are not inverses of one another. Certain structural properties of non-inverse graphs of groups have been discussed.
Motivated by the studies mentioned above, in this paper, a graph called order sum graph based on the properties of groups is introduced. The definition of the order sum graph is given in the next section. Different types of domination are studied in the recent past such as tadpole domination 3 , independent domination 4 , -domination 5, etc. The variants of domination are investigated in this paper for order sum graphs.

Order Sum Graph of a Group
Recall that the order of a group ( , * ) is the number of elements of its underlying set and the order of an element, say , of is the least positive integer , such that = , the identity element of the group. In this paper, denotes the order of the group . The notion of the order sum graph of a given finite group is given below:

Definition 1:
The order sum graph, Γ ( ) associated with a group , is a graph whose vertices are the elements of and two vertices say , ∈ Γ ( ) are adjacent if ( ) + ( ) > ( ). For example, consider the group of fourth roots of unity under multiplication, that is, = {1, −1, , − } and ( ) = 4. Here, the identity element of is 1 and hence (−1) = 2, (1) = 1, ( ) = 4, (− ) = 4. The order sum graph associated with is given in Fig. 1. The finite group is represented by and its order sum graph by Γ . In this context, the questions about the structural properties of the order sum graphs of groups arise much interest. The following establish a necessary and sufficient for Γ ( ) to be connected. Theorem 1: The order sum graph of a group is connected if and only if is a cyclic group. Proof. Let be a group of order . Now, let the order sum graph associated with be connected. Then in Γ , there is a path connecting any two vertices. Assume is not a cyclic group now. Then, there is no element in whose order is the same as . Let a, b ∈ . Since the order of is , (a) ≠ , (a)| . Therefore, (a) ≤ 2 . Similarly, (b)| , (b) ≠ and hence ( ) ≤ 2 . Thus, for any two elements in , (a) + (b) ≤ . Therefore, by Definition 1, there is no path between any two vertices in Γ ( ). This is in direct opposition to the hypothesis that Γ ( ) is connected. Therefore, there exists at least one element in whose order is equal to ( ). Thus, is a cyclic group. □ Conversely, let be a cyclic group. Then, there is at least one element, say a in which generates . Here, (a) = ( ) = . Then, ( ) + ( ) > , where , = 2,3, … , are the remaining elements of . Therefore, by Definition-1, a is adjacent to all 's, = 2,3, … , in Γ ( ). Therefore, any two vertices of Γ ( ) have a path connecting them through . Hence, Γ is connected.
The order sum graph associated with ( 6 , +) is given in Fig. 2. ( 1 ) = ( ) = . Then, must be a cyclic group with exactly one generator and ( 1 ) + ( ) > , where = 2,3, … , . By Definition 1, 1 will be adjacent to all , = 2,3, … , in Γ ( ). Since 1 is the only element in such that ( 1 ) = , 's, = 2,3, … , are not adjacent to each other in Γ ( ). Therefore, Γ ( ) is a star graph 1, −1 . But, by Theorem 3, such a group can have at most two elements. Hence, Γ ( ) ≅ 2 . □ In this context, it is interesting to check the case when there exists no element in with the order ( ). Consider the following example: Let 8 be a group of elements less than 8 and relatively prime to 8 under the operation of multiplication modulo 8. That is, 8 Here, ( (8)) = 4, (3) = 2, (5) = 2, (1) = 1, (7) = 2. The order sum graph associated with the group 8 is given in Fig. 3: At this point, it is interesting to investigate the conditions under which the order sum graphs of groups are null graphs and hence the following result: Proposition 5: The ( ) of order is a null graph if and only if is not a cyclic group. Proof. Consider a cyclic group with elements then, there is no element in whose order is equal to . Let , ∈ , then, ( )| , ( ) ≠ and ( )| , ( ) ≠ . This implies that ( ) ≤ 2 and ( ) ≤ 2 . Hence, ( ) + ( ) ≤ for any two elements in , then, by Definition 1, any two pairs of vertices and in Γ ( ) have no edge. Therefore, Γ ( ) is a null graph of vertices. On the contrary, let Γ ( ) be a null graph. Assuming is a cyclic group, there is at least one element, say in whose order is equal to . Then, by Definition 1, there is an edge between and all the other vertices in Γ ( ). Thus, there is a path connecting any two vertices of Γ ( ) through . Therefore, Γ ( ) is connected which is a contradiction to the fact that it is a null graph. Hence, is not a cyclic group. □ The following theorem discusses the case where the order sum graph of a group becomes a complete graph.

Theorem 6:
The order sum graph associated with a group of order ≥ 3, is complete if and only if all the elements of except the identity element have order equal to . Proof. Consider a group with ≥ 3 elements. Let Γ ( ) be a complete graph. Assume that there are − 2 elements, say , = 1,2, … , − 2 in whose order is equal to . Out of the remaining two elements of , one is the identity element whose order is equal to 1 and the other element say −1 has to order less than . Hence, ( −1 ) + ( ) ≤ . Therefore, by Definition 1, there is no edge between and −1 , which is a contradiction that Γ ( ) is complete. Now, let all the elements of say , = 1,2, … , − 1 other than the identity element have order equal to . Thus, ( ) + ( ) > , = 1,2, … , , = 1,2, … , ; ≠ . Therefore, there is an edge between any two pair of vertices in Γ ( ).
Hence, Γ ( ) is complete. □ As an illustration of the above result, consider the group  Consider the following result on a group of prime order. Theorem 7: Let be a group with a prime number of elements say . Then, is cyclic and has − 1 generators 6 . In view of Theorem 7, the following theorem examines the structural properties of the order sum graph of a group of prime order. Theorem 8: The order sum graph associated with a group of prime order is complete. Proof. Consider a group with elements, where is prime. Then, by Theorem 7, is cyclic and has − 1 generators. By Theorem 2, the order of all the − 1 generators of is equal to and the order of the identity element is 1. Thus, all the elements of except the identity element have order equal to . Therefore, by Theorem 6, Γ ( ) is complete. □ The following theorem discusses the diameter of an order sum graph of a group.
□ In general, colors are required to color the vertices 1 , 2 , … , , since there are vertices with degree − 1 in Γ ( ). The remaining − vertices with degree say +1 , +2 , … , +( − ) can be colored with one color. Hence, (Γ ( )) = + 1. The size of the order sum graph of a finite group is investigated below: Theorem 11: The size of the order sum graph associated with a cyclic group of order is , where < is the number of generators of . Proof. Consider a cyclic group of order ≥ 2 and let < be the number of generators of , then there are vertices with degree − 1 and − vertices with degree in Γ ( ). Therefore, the sum of degrees of the vertices in Γ ( ) is ( − 1) + ( − )( ) = (2 − − 1). Hence, by the handshaking lemma, the size of Γ ( ) is . □ The following theorem establishes the isomorphism between the order sum graphs of two isomorphic groups. Theorem 12: The order sum graphs associated with two cyclic groups with the same order are isomorphic. Proof. Let 1 and 2 be cyclic groups of the same order. Then, both 1 and 2 are isomorphic groups 9 . That is, the number of generators of 1 is the same as that of 2 . Note that a vertex in the order sum graph, corresponding to a generator of the group will have an edge with all the other vertices of the graph. Also, it is observed that no two vertices of the order sum graph corresponding to the non-generators of the group will be adjacent to each other (see Definition 1). Therefore, both the order sum graphs Γ ( 1 ) and Γ ( 2 ) have the same number of edges from the corresponding vertices. It is clear from the above-mentioned facts that the adjacency will also be preserved in the graphs Γ ( 1 ) and Γ ( 2 ). Hence, Γ ( 1 ) and Γ ( 2 ) are also isomorphic. □ A perfect graph is a graph in which for every induced subgraph of , the clique number is equal to its chromatic number. 9 Recall the following theorem which characterizes the perfect graphs. Theorem 13: A graph is perfect if and only if neither nor its complement contains an induced odd cycle of length greater than or equal to five. 10 It is interesting to check when the order sum graph of a group is perfect. Theorem 14 Let be a cyclic group with generators. Then, the order sum graph associated with is perfect if and only if any one of the following is true: i. = 1 and < 6; ii.
Proof. Let be a cyclic group with generators. Let Γ ( ) be perfect. Now, if ≥ 4, then must be at least + 1, including the generators and identity element of . Therefore, there is an induced odd cycle of length 5 consisting of four generator vertices and one non-generator vertex in the order sum graph with at least 4 generators. By Theorem 13, this is a contradiction to the hypothesis that Γ ( ) is perfect. Hence, < 4. There exist three cases:   1). By Theorem 13, this is a contradiction to the hypothesis that Γ ( ) is perfect. Hence, < 5 but since there are three generators, can be a minimum 4 including the identity element of . Therefore, = 4.
Hence, the order sum graph associated with is perfect if and only if either = 1 and < 6 or = 2 and ≤ 6 or = 3 and = 4. □ Some Parameters of Order Sum Graphs Naturally, investigation on different graph parameters of the order sum graphs sounds interesting at this point. In this context, the following result discusses the domination in order sum graph of groups. Proposition 15: The domination number of the order sum graph associated with a group of order is either 1 or . Proof. By Theorem 1 and Proposition 5, Γ ( ) is either connected or a null graph. Hence, either there is a minimum one vertex in Γ ( ) which is adjacent to all other vertices in Γ or between any two pairs of vertices in Γ ( ), there is no edge. Therefore, (Γ ( )) = 1 or . □ It is observed that the order sum graph is connected if and only if the group is cyclic. Otherwise, the graph will be a null graph. Hence, in this section, some parameters of the order sum graphs associated with cyclic groups are discussed. An independent set of a graph is the set of vertices that are not adjacent to each pair. The independence number of , denoted by ( ), is the maximum cardinality of a maximal independent set of . The following theorem discusses the independence number of the order sum graph of a cyclic group. Theorem 16: The independence number of the order sum graph is equal to − , where is the number of generators of the group . Proof. Let be a cyclic group with generators. By Definition 1, two vertices , ∈ Γ ( ) are adjacent if ( ) + ( ) > . This is possible only when either ( ) or ( ) or both are equal to . That is, either or or both are generators of . Therefore, if a vertex in Γ ( ) is corresponding to a generator, then it is adjacent to all the other vertices of Γ ( ). Similarly, a vertex in Γ ( ) corresponding to a non-generator is adjacent only to all the vertices corresponding to generators. Hence, the set of vertices corresponding to all nongenerators is the largest independent set of Γ ( ). Therefore, the independent number of Γ ( ) is − . □ The vertex cover of a graph is the vertex set of so that each edge of has at least one end vertex in . The covering number of , is the minimum cardinality of a minimal vertex cover of . The covering number of the order sum graph of a group is discussed in the following result. Corollary 17: The vertex covering the number of Γ ( ) is equal to the number of generators of the group. Proof. The proof follows immediately from the fact that ( ) + ( ) = , the order of the graph. □ Recall that the clique number of is the order of a maximum clique in . The clique number of the order sum graph is determined in the theorem given below: Theorem 18: The clique number of the order sum graph associated with a cyclic group of order is equal to + 1, where is the number of generators of . Proof. Let be a cyclic group with generators. By Definition 1, it is clear that a generator is adjacent to all the other vertices in Γ ( ) and a non-generator is adjacent only to generators. Now, if has exactly one generator, the clique number of Γ ( ) is 2. If has two generators, then maximum clique in Γ ( ) should be 3 whose vertex set consists of two generators and one non-generator. Therefore, the clique number of Γ ( ) is 3. If proceeded similarly, when there are generators, a maximum clique in Γ ( ) by joining the vertices corresponding to generators and one nongenerator is obtained. Therefore, the clique number of Γ ( ) is + 1.
□ If is a graph whose vertex set is given by ( ). Then, the set ⊆ ( ) is an independent dominating set if is a dominating set and an independent set. The minimum cardinality of such a set is the independent domination number and is represented by ( ). The following theorem determines the independent domination number of an order sum graph of a cyclic group.

Proposition 19
The independent domination number of Γ ( ) is equal to 1. Proof. Consider to be a cyclic group. Therefore, has at least one generator, say . By Definition 1, is adjacent to all the other vertices in Γ ( ). Therefore, the singleton set { } is a minimal dominating set as well as an independent set with minimum cardinality. Hence, (Γ ( )) = 1. □ The following theorem determines the upper domination number of the order sum graph of a cyclic group. Theorem 20 The upper domination number of the order sum graph of a cyclic group with generators is − . Proof. Consider a cyclic group with generators. Then, the minimal dominating sets possible in the order sum graph of are the singleton sets each consisting of a generator and the set containing all the non-generators of . It is known that the set of all non-generators of is a minimal dominating set with maximum cardinality − . Therefore, the upper domination number of Γ ( ) is − .

Distance Related Parameters of Order Sum Graphs
A vertex cutset of a connected graph is a set of its vertices whose deletion makes the graph disconnected. Similarly, a cutset or an edge cutset is a set of edges whose deletion makes the graph disconnected. The connectivity of a graph, denoted by ( ) is the minimum cardinality of a minimal vertex-cut of , whereas the edge connectivity ′( ) is the minimum cardinality of a minimal cutset. The following result discusses the relation between vertex connectivity and the edge connectivity of the order sum graph of a group. Theorem 21: For the order sum graph associated with a cyclic group , ( ( )) = ′( ( )). Proof. By Definition 1, a vertex corresponding to a generator of is adjacent to all the other vertices in Γ ( ). Since there are generators, a minimum number of vertices whose removal makes the order sum graph disconnected or trivial is . Therefore, the vertex connectivity (Γ ( )) = . □ Furthermore, by Definition-1, a vertex corresponding to a non-generator is adjacent only to all generators. Therefore, minimum all the edges which are incident on a fixed non-generator vertex have to be removed to make Γ ( ) disconnected or trivial. Since these edges are all coming from different generators and the no. of generators is equal to , the edge connectivity ′(Γ ( )) = .
Hence, (Γ ( )) = ′(Γ ( )) = , completing the proof. □ The center of a graph is the set of all vertices whose eccentricity is equal to the radius of the graph. The following theorem discusses the center of the order sum graph of a given finite group. Theorem 22: The center of the order sum graph of a group is the set of its vertices corresponding to the generators of . Proof. Consider the order sum graph Γ ( ). A vertex corresponding to a generator of is adjacent to all other vertices in Γ ( ). Therefore, the eccentricity of each of the vertices corresponding to generators of is 1. Further, a vertex corresponding to a non-generator of is adjacent to all vertices corresponding to generators and is at a distance 2 from other vertices corresponding to nongenerators. Therefore, the eccentricity of each of the vertices corresponding to non-generators of is 2.
Since the eccentricity of vertices corresponding to generators of is minimum, the set of vertices in Γ ( ) corresponding to all the generators of the group forms the center of Γ ( ). Hence the result. □ The periphery of a graph is the set of all vertices whose eccentricity is equal to the diameter of the graph. Since the vertices of Γ ( ) corresponding to the non-generators of a finite cyclic group are not adjacent to each other, the distance between such vertices will be 2, which is equal to the diameter of the graph. Theorem 23 The periphery of the order sum graph of a (cyclic) group is the set of its vertices corresponding to the non-generators of . Theorem 24 Let be a cyclic group with generators. Then, the circumference of the order sum graph associated with is equal to Proof. To find the circumference of the order sum graph associated with , the following steps are undertaken: Step 1: Visit a generator vertex say .
Step 2: Traverse through a non-generator vertex and then to a generator vertex that is not visited.
Step 3: Repeat step 2 till all the generator vertices are visited.
Step 4: Traverse to a non-generator vertex that is not visited and then go back to initial vertex .
Now, the following cases arise: Then, the number of generators is less than the number of non-generators in . The circumference of Γ ( ) is 2 since there are generators and the equal number of nongenerators required to form the largest cycle are obtained using the above steps. is odd. In both cases, either number of generators is greater than or equal to the number of non-generators in . Therefore, the circumference of Γ obtained by following the above steps is equal to as all the vertices are traversed. Since there are vertices in , for ≥ ⌈ 2 ⌉, the circumference is equal to . □
Therefore, the characteristic polynomial of (Γ ( )) is given by

Conclusion:
In this paper, some properties and characteristics of a new algebraic graph, called an order sum graph associated with a group have been discussed. Some of the parameters such as size, diameter, chromatic number, clique number, independence number, vertex connectivity, edge connectivity, domination number, spectra, and Laplacian spectra of the order sum graph have been obtained. Some characterizations of the order sum graph have also been obtained. As a newly introduced notion, many structural characteristics of order sum graphs are yet to be investigated. Moreover, determining the order sum graphs corresponding to different standard groups are also much promising for future studies.