Here we consider only finite and simple graphs. The vertex set and the edge set of a graph G are denoted by V ( G ) and E ( G ) respectively. For various graph theoretic notations and terminology we follow [1] . A graph labeling is an assignment of integers to the vertices or edges or both, subject to certain conditions. The concept of mean labeling was introduced in [2] . Since then, several results have been published on mean labeling and its variations [3] . In 2014, the concept of skolem difference mean labeling, one of the variations of mean labeling was due to Murugan et al. [4] . A graph G = ( V , E ) with p vertices and q edges is said to have skolem difference mean labeling if it is possible to label the vertices x ∈ V with distinct elements f ( x ) from { 1,2,3, ⋯ , p + q } in such a way that for each edge e = u v ,

let f * ( e ) = ⌈ | f ( u ) − f ( v ) | 2 ⌉ and the resulting labels of the edges are distinct

and are 1,2,3, ⋯ , q . A graph that admits a skolem difference mean labeling is called skolem difference mean graph. The concept of one modulo N-graceful labeling was introduced by Ramachandran et al. [5] . A function f is called a graceful labeling of a graph G with q edges if f is an injection from the vertices of G to the set { 0,1,2, ⋯ , q } such that, when each edge xy is assigned with the label | f ( x ) − f ( y ) | , the resulting edge labels are distinct. A graph G is said to be one modulo N graceful (where N is a positive integer) if there is a function φ from the vertex set of G to { 0,1, N , ( N + 1 ) ,2 N , ( 2 N + 1 ) , ⋯ , N ( q − 1 ) , N ( q − 1 ) + 1 } in such a way that 1) φ is 1-1; 2) φ induces a bijection φ * from the edge set of G to { 1, N + 1,2 N + 1, ⋯ , N ( q − 1 ) + 1 } where φ * ( u v ) = | φ ( u ) − φ ( v ) | .

Motivated by the concepts of skolem difference mean labeling and one modulo N-graceful labeling and the results in [4] [5] , we introduced a new labeling namely “one modulo N-difference mean labeling” in [6] and established that the graphs B m , n , S m , n , P n @ P m , B ( l , m , n ) , T ( n , m ) , shrub, caterpillar and K 1, n are one modulo N-difference mean graphs. In addition, we showed that the graph C 3 is not a one modulo N-difference mean graph. In this paper, we further study on one modulo N-difference mean labeling and show that some more graphs admit one modulo N-difference mean labeling.

We use the following definitions in the subsequent sequel.

Definition 1.1. Let G = ( V , E ) be a graph and G ′ = ( V ′ , E ′ ) be the copy of G. Then the graph M 2 ( G ) of G is obtained from G and G ′ by joining each vertices in V to its corresponding vertices in V ′ by an edge.

Definition 1.2. The slanting ladder graph S L n is obtained from two paths u 1 , u 2 , u 3 , ⋯ , u n and v 1 , v 2 , v 3 , ⋯ , v n by joining u i with v i + 1 for 1 ≤ i ≤ n − 1 .

Definition 1.3. Let G = ( V , E ) be a bipartite graph with V = V 1 ∪ V 2 . Let G ′ = ( V ′ , E ′ ) be the copy of G with V ′ = V ′ 1 ∪ V ′ 2 such that V ′ 1 and V ′ 2 be the copies of V 1 and V 2 . Then the graph D U P 2 ( G ) is obtained from G and G ′ such that V ( D U P 2 ( G ) ) = V ∪ V ′ and E ( D U P 2 ( G ) ) = E ( G ) ∪ E ( G ′ ) ∪ { v ′ i v j / v i v j ∈ E ( G )   where   v ′ i ∈ V ′ , v j ∈ V } . That is, D U P 2 ( G ) is obtained from G and G ′ by joining each v ′ i ∈ V ′ to v j ∈ V if v i is adjacent to v j in G.

Definition 1.4. A quadrilateral snake graph Q n is obtained from a path u 1 , u 2 , ⋯ , u n by joining u i and u i + 1 to two new vertices x i , y i respectively and then joining x i and y i .

Definition 1.5. An alternate quadrilateral snake is obtained from a path u 1 , u 2 , ⋯ , u n by joining u i and u i + 1 to new vertices x i and y i respectively and then joining the vertices x i and y i for i ≡ 1 ( mod 2 ) and 1 ≤ i ≤ n − 1 . That is, every alternate edge of a path is replaced by cycle C 4 .

Definition 1.6. Let P 3 be a path of length 2 with vertices v 0 , v 1 , v 2 . The graph J l n ( P 3 ) is obtained by taking n copies of P 3 and then identifying the left end vertices v 0 i ( 1 ≤ i ≤ n ) with u and the right end vertices v 2 i ( 1 ≤ i ≤ n ) with v.

Definition 1.7. Two graphs G and H are isomorphic (written G ≃ H ) if there exists a one-to-one correspondence between their vertex sets which preserves adjacency.

Definition 1.8. The union of two graphs G 1 and G 2 is a graph G 1 ∪ G 2 with V ( G 1 ∪ G 2 ) = V ( G 1 ) ∪ V ( G 2 ) and E ( G 1 ∪ G 2 ) = E ( G 1 ) ∪ E ( G 2 ) .

Definition 1.9. The corona G 1 ⊙ G 2 of the graphs G 1 and G 2 is obtained by taking one copy of G 1 (with p vertices) and p copies of G 2 and then joining the ith vertex of G 1 to every vertex of the ith copy of G 2 .

Definition 1.10. Let C n be the cycle with vertices v 1 , v 2 , ⋯ , v n . The graph C n ( t ) is obtained by taking t copies of C n and then identifying the vertices v 1 ( i ) for 1 ≤ i ≤ t .

Theorem 2.1. The disjoint union of paths ∪ P n i ( n i ≥ 2 , is an integer) is a one modulo N-difference mean graph.

Proof. Let n i be the vertices of the path P n i for 1 ≤ i ≤ m and n = n 1 + n 2 + ⋯ + n m .

Define f : V ( ∪ P n i ) → { 0 , 1 , N , N + 1 , 2 N , 2 N + 1 , ⋯ , 2 N ( n − m − 1 ) + 1 } as follows:

f ( u i , j ) = N [ ∑ k = 1 ( n k   is   odd ) i − 1 ( n k − 1 ) + ∑ k = 1 ( n k   is   even ) i − 1 ( n k − 2 ) + i + j − 2 ] if j is odd,

f ( u i , j ) = [ 2 ( n − m − 1 ) + i − j − 1 − ∑ k = 1 ( n k   is   odd ) i − 1 ( n k − 1 ) − ∑ k = 1 ( n k   is   even ) i − 1     n k ] N + 1 if j is even.

Let e i , j = u i , j u i , j + 1 for 1 ≤ i ≤ m and 1 ≤ j ≤ n i − 1 .

The corresponding edge label f * is

f * ( e i , j ) = N ( n − m − ∑ k = 1 i − 1     n k + i − j − 1 ) + 1 for 1 ≤ i ≤ m and 1 ≤ j ≤ n i − 1 .

Therefore, f is a one modulo N-difference mean labeling. Hence, ∪ P n i is a one modulo N-difference mean graph.

Figure 1 shows a one modulo N-difference mean labeling of P 4 ∪ P 5 ∪ P 6 ∪ P 2 ∪ P 3 ∪ P 8 .

Theorem 2.2. The graph M 2 ( P n ) ( n ≥ 2 ) is a one modulo N-difference mean graph.

Proof. Let { v i , v ′ i : 1 ≤ i ≤ n } be the vertices and { e i , e ′ i , a i = v i v ′ i : 1 ≤ i ≤ n } be the edges of the graph M 2 ( P n ) . Then the graph has 2n vertices and 3 n − 2 edges.

Define f : V ( M 2 ( P n ) ) → { 0,1, N , N + 1,2 N ,2 N + 1, ⋯ ,2 N ( 3 n − 3 ) + 1 } by

f ( v 1 ) = 0 ,

For 2 ≤ i ≤ n ,

f ( v i ) = { ( 3 i − 5 ) N if   i   is   odd 3 N ( 2 n − i ) + 1 if   i   is   even

For 1 ≤ i ≤ n ,

f ( v ′ i ) = { ( 6 n − 3 i − 2 ) N + 1 if   i   is   odd 3 i N if   i   is   even

Then the induced edge labels are

f * ( e 1 ) = 3 ( n − 1 ) N + 1 ,

f * ( e i ) = [ 3 ( n − i ) + 1 ] N + 1 for 2 ≤ i ≤ n ,

f * ( e ′ i ) = [ 3 ( n − i ) − 4 ] N + 1 for 1 ≤ i ≤ n ,

f * ( v 1 v ′ 1 ) = [ 3 ( n − 4 ) ] N + 1 ,

f * ( a i ) = [ 3 ( n − i ) ] N + 1 for 2 ≤ i ≤ n .

Therefore, f is a one modulo N-difference mean labeling and hence M 2 ( P n ) is a one modulo N-difference mean graph.

Figure 2 shows a one modulo N-difference mean labeling of M 2 ( P 5 ) .

Corollary 2.3. The ladder graph P n × P 2 is a one modulo N-difference mean graph.

Theorem 2.4. The slanting ladder S L n ( n ≥ 2 ) is a one modulo N-difference mean graph.

Proof. Let u 1 , u 2 , u 3 , ⋯ , u n and v 1 , v 2 , v 3 , ⋯ , v n be the vertices of the path of length n − 1 .

Then E ( S L n ) = { u i u i + 1 , v i v i + 1 , u i v i + 1 : 1 ≤ i ≤ n − 1 } .

Define f : V ( S L n ) → { 0,1, N , N + 1,2 N ,2 N + 1, ⋯ ,2 N ( 3 n − 4 ) + 1 } by

For 1 ≤ i ≤ n ,

f ( u i ) = { 2 ( i − 1 ) N if   i   is   odd 2 N [ 3 n − 2 ( i + 1 ) ] + 1 if   i   is   even

f ( v 1 ) = { 2 ( 3 n − 4 ) N if   n   is   odd 2 ( 3 n − 5 ) N if   n   is   even

For 2 ≤ i ≤ n ,

f ( v i ) = { 2 ( i − 2 ) N if   i   is   odd 2 N [ 3 n − 2 i ] + 1 if   i   is   even

Then the induced edge labels are

For 1 ≤ i ≤ n − 1 ,

f * ( u i u i + 1 ) = { 3 N ( n − i − 1 ) + 1 if   i   is   odd N [ 3 ( n − i ) − 2 ] + 1 if   i   is   even

f * ( v 1 v 2 ) = { 1 if   n   is   odd N + 1 if   n   is   even

For 2 ≤ i ≤ n − 1 ,

f * ( v i v i + 1 ) = { 3 N ( n − i ) + 1 if   i   is   odd N [ 3 ( n − i ) + 1 ] + 1 if   i   is   even

f * ( u i v i + 1 ) = N [ 3 ( n − i ) − 1 ] + 1 for 1 ≤ i ≤ n − 1 .

Therefore, f is a one modulo N-difference mean labeling and hence S L n is a one modulo N-difference mean graph.

Figure 3 shows a one modulo N-difference mean labeling of S L 10 .

Theorem 2.5. The diamond snake graph D S ( n ) ( n ≥ 1 ) is a one modulo N-difference mean graph.

Proof. Let { v 0 , v i , a i , b i : 1 ≤ i ≤ n } be the vertices and { v 0 a 1 , v 0 b 1 , v i a i + 1 , a i v i , v i b i + 1 , b i v i : 1 ≤ i ≤ n } be the edges of the diamond snake graph which has 4 n − 4 vertices and 4n edges.

Define f : V ( D S ( n ) ) → { 0,1, N , N + 1,2 N ,2 N + 1, ⋯ ,2 N ( 4 n − 1 ) + 1 } by

f ( v 0 ) = 0 ,

f ( v i ) = 4 i N for 1 ≤ i ≤ n ,

f ( a i ) = 2 N ( 4 n − 2 i + 1 ) + 1 for 1 ≤ i ≤ n ,

f ( b i ) = 4 N ( 2 n − i ) + 1 for 1 ≤ i ≤ n .

Then the induced edge labels are

f * ( v 0 a 1 ) = ( 4 n − 1 ) N + 1 ,

f * ( v i a i + 1 ) = ( 4 n − 4 i − 1 ) N + 1 for 1 ≤ i ≤ n − 1 ,

f * ( a i v i ) = ( 4 n − 4 i + 1 ) N + 1 for 1 ≤ i ≤ n ,

f * ( v 0 b 1 ) = ( 4 n − 2 ) N + 1 ,

f * ( v i b i + 1 ) = ( 4 n − 4 i − 2 ) N + 1 for 1 ≤ i ≤ n − 1 ,

f * ( b i v i ) = ( 4 n − 4 i ) N + 1 for 1 ≤ i ≤ n .

Therefore, f is a one modulo N-difference mean labeling and hence D S ( n ) is a one modulo N-difference mean graph. A one modulo N-difference mean labeling of D S ( 5 ) is shown in Figure 4.

Theorem 2.6. The quadrilateral snake Q n ( n > 1 ) is a one modulo N-difference mean graph.

Proof. Let u 1 , u 2 , ⋯ , u n be the vertices of the path P n of length n − 1 .

Then { u i , x j , y j : 1 ≤ i ≤ n , 1 ≤ j ≤ n − 1 } be the vertices of and { u i u i + 1 , u i x i , u i + 1 y i , x i y i : 1 ≤ i ≤ n − 1 } be the edges of Q n .

Define f : V ( Q n ) → { 0,1, N , N + 1,2 N ,2 N + 1, ⋯ ,2 N ( 4 n − 5 ) + 1 } by

f ( u 1 ) = 0 ,

For 2 ≤ i ≤ n ,

f ( u i ) = { ( 3 i − 1 ) N if   i   is   odd ( 8 n − 5 i − 2 ) N + 1 if   i   is   even

f ( x 1 ) = 2 ( 4 n − 5 ) N + 1 ,

For 2 ≤ i ≤ n − 1 ,

f ( x i ) = { ( 8 n − 5 i − 3 ) N + 1 if   i   is   odd 3 i N if   i   is   even

For 1 ≤ i ≤ n − 1 ,

f ( y i ) = { ( 3 i + 1 ) N if   i   is   odd ( 8 n − 5 i − 6 ) N + 1 if   i   is   even

Then the induced edge labels are

f * ( u 1 u 2 ) = [ 4 n − 6 ] N + 1

For 2 ≤ i ≤ n − 1 ,

f * ( u i u i + 1 ) = { [ 4 ( n − i ) − 3 ] N + 1 if   i   is   odd [ 4 ( n − i ) − 2 ] N + 1 if   i   is   even

f * ( x 1 y 1 ) = ( 4 n − 7 ) N + 1 ,

For 2 ≤ i ≤ n − 1 ,

f * ( x i y i ) = { [ 4 ( n − i ) − 2 ] N + 1 if   i   is   odd [ 4 ( n − i ) − 3 ] N + 1 if   i   is   even

f * ( u i x i ) = [ 4 ( n − i ) − 1 ] N + 1 for 1 ≤ i ≤ n − 1 ,

f * ( u i + 1 y i ) = [ 4 ( n − i − 1 ) ] N + 1 for 1 ≤ i ≤ n − 1 .

Therefore, f is a one modulo N-difference mean labeling and hence Q n is a one modulo N-difference mean graph.

Figure 5 shows a one modulo N-difference mean labeling of Q 7 .

Theorem 2.7. The alternately quadrilateral snake A ( Q n ) ( n > 1 ) is a one modulo N-difference mean graph.

Proof. Let u 1 , u 2 , ⋯ , u n be the vertices of the path P n of length n − 1 .

Let n = 2 m .

Then { u i , x j , y j : 1 ≤ i ≤ n ,1 ≤ j ≤ m } be the vertices of and { u i u i + 1 , u i x j , u i y j , x j y j : 1 ≤ i ≤ n ,1 ≤ j ≤ m } be the edges of A ( Q n ) .

Define f : V ( A ( Q n ) ) → { 0,1, N , N + 1,2 N ,2 N + 1, ⋯ ,2 N ( 2 n + n 2 − 2 ) + 1 } by

f ( u 1 ) = 0 ,

For 2 ≤ i ≤ n ,

f ( u i ) = { 2 N i if   i   is   odd ( 5 n − 3 i ) N + 1 if   i   is   even

f ( x 1 ) = ( 5 n − 4 ) N + 1 ,

f ( x j ) = ( 5 n − 6 j + 4 ) N + 1 for 2 ≤ j ≤ m ,

f ( y j ) = 4 N j for 1 ≤ j ≤ m .

Then the induced edge labels are

For 1 ≤ i ≤ n − 1 ,

f * ( u i u i + 1 ) = { [ 2 n + n − 5 i − 3 2 ] N + 1 if   i   is   odd [ 2 n + n − 5 i − 2 2 ] N + 1 if   i   is   even

f * ( x j y j ) = ( 2 n + m − 5 j + 2 ) N + 1 for 1 ≤ j ≤ m ,

f * ( u i x i + 1 2 ) = [ 2 n + n − 5 i + 1 2 ] N + 1 if i is odd and 1 ≤ i ≤ n − 1 ,

f * ( u i y i 2 ) = [ 2 n + n − 5 i 2 ] N + 1 if i is even and 2 ≤ i ≤ n − 2 .

Therefore, f is a one modulo N-difference mean labeling and hence A ( Q n ) is a one modulo N-difference mean graph.

Figure 6 shows a one modulo N-difference mean labeling of A ( Q 8 ) .

Theorem 2.8. The graph J l n ( P 3 ) ( n ≥ 1 ) is a one modulo N-difference mean graph.

Proof. Let v 0 i , v 1 i , v 2 i ( 1 ≤ i ≤ n ) be the vertices of the n copies of the path P 3 .

Then the graph J l n ( P 3 ) is obtained by identifying v 0 i = u and v 2 i = v .

Define f : V ( J l n ( P 3 ) ) → { 0,1, N , N + 1,2 N ,2 N + 1, ⋯ ,2 N ( 2 n − 1 ) + 1 } as follows:

f ( u ) = 0 ,

f ( v ) = 2 N ,

f ( v i 1 ) = ( 4 i − 2 ) N + 1 for 1 ≤ i ≤ n .

Then the induced edge labels are

f * ( x 1 u ) = 0 ,

f * ( x i v ) = 2 ( i − 1 ) N + 1 for 2 ≤ i ≤ n ,

f * ( u x i ) = ( 2 i − 1 ) N + 1 for 1 ≤ i ≤ n .

Therefore, f is a one modulo N-difference mean labeling and hence J l n ( P 3 ) is a one modulo N-difference mean graph.

Figure 7 shows a one modulo N-difference mean labeling of J l n ( P 3 ) .

Theorem 2.9. The corona graph C 4 ⊙ K 1, n ( n ≥ 1 ) is a one modulo N-difference mean graph.

Proof. Let v 1 , v 2 , v 3 , v 4 be the vertices of cycle C 4 and { v i j : 1 ≤ i ≤ n ,1 ≤ j ≤ 4 } be the vertices of the four stars K 1, n .

Define f : V ( C n ⊙ K 1, n ) → { 0,1, N , N + 1,2 N ,2 N + 1, ⋯ ,2 N ( 4 n + 3 ) + 1 } as follows:

We label the vertices of C 4 as follows:

f ( v 1 ) = 0 ,

f ( v 2 ) = 2 ( 4 n + 3 ) N + 1 ,

f ( v 3 ) = 4 N ,

f ( v 4 ) = 4 ( 2 n + 1 ) N + 1 .

Now, we label the vertices of K 1, n as follows:

f ( v i 1 ) = 2 ( i − 1 ) N + 1 for 1 ≤ i ≤ n ,

f ( v i 2 ) = 2 N ( 3 n − i + 4 ) for 1 ≤ i ≤ n ,

f ( v i 3 ) = 2 ( 2 n + i + 1 ) N + 1 for 1 ≤ i ≤ n ,

f ( v i 4 ) = 2 N ( n − i + 3 ) for 1 ≤ i ≤ n .

Let e i = { v i v i + 1 : 1 ≤ i ≤ 3 } and e i j = { v j v i j : 1 ≤ i ≤ n ,1 ≤ j ≤ 4 } .

Then the induced edge labels are

f * ( e 1 ) = ( 4 n + 3 ) N + 1 ,

f * ( e 2 ) = ( 4 n + 1 ) N + 1 ,

f * ( e 3 ) = 4 n N + 1 ,

f * ( v 4 v 1 ) = ( 4 n + 2 ) N + 1 ,

f * ( e i j ) = [ ( j − 1 ) n + i − 1 ] N + 1 for 1 ≤ i ≤ n , 1 ≤ j ≤ 4 .

Therefore, f is a one modulo N-difference mean labeling. Hence, C 4 ⊙ K 1, n is a one modulo N-difference mean graph.

Figure 8 shows a one modulo N-difference mean labeling of C 4 ⊙ K 1,3 .

Theorem 2.10. The graph D U P 2 ( K 1, n ) , n ≥ 2 is a one modulo N-difference mean graph.

Proof. Let { v , v i ( 1 ≤ i ≤ n ) , u , u i ( 1 ≤ i ≤ n ) } be the vertices and { v v i , u u i , v i u : 1 ≤ i ≤ n } be the edges of D U P 2 ( K 1, n ) .

Now, the vertex labels are defined as follows:

Define f : V ( D U P 2 ( K 1, n ) ) → { 0,1, N , N + 1,2 N ,2 N + 1, ⋯ ,2 N ( 3 n − 1 ) + 1 } by

f ( v ) = 2 N ( 2 n − 1 ) ,

f ( u ) = 0 ,

f ( v i ) = 2 N ( 2 n − i ) + 1 for 1 ≤ i ≤ n ,

f ( u i ) = 2 N ( 3 n − i ) + 1 for 1 ≤ i ≤ n .

Then the induced edge labels are

f * ( v v i ) = N ( i − 1 ) + 1 for 1 ≤ i ≤ n ,

f * ( u u i ) = N ( 3 n − i ) + 1 for 1 ≤ i ≤ n ,

f * ( v i u ) = N ( 2 n − i ) + 1 for 1 ≤ i ≤ n .

Therefore, f is a one modulo N-difference mean labeling. Hence, D U P 2 ( K 1, n ) is a one modulo N-difference mean graph. Figure 9 shows a one modulo N-difference mean labeling of D U P 2 ( K 1,7 ) .

Theorem 2.11. The graph D U P 2 ( B n , n ) , n ≥ 2 is a one modulo N-difference mean graph.

Proof. Let { v , v ′ , v i , v ′ i , u , u ′ , u i , u ′ i : 1 ≤ i ≤ n } be the vertices and

{ v v i , u u i , v i u , v ′ v ′ i , u ′ u ′ i , v ′ i u ′ , v u ′ , u u ′ , u v ′ : 1 ≤ i ≤ n } be the edges of D U P 2 ( B n , n ) .

Now, the vertex labels are defined as follows:

Define f : V ( D U P 2 ( B n , n ) ) → { 0,1, N , N + 1,2 N ,2 N + 1, ⋯ ,2 N ( 6 n + 2 ) + 1 } by

f ( v ) = 2 N ,

f ( u ) = 0 ,

f ( v i ) = 4 N ( 3 n − i + 2 ) + 1 for 1 ≤ i ≤ n ,

f ( u i ) = 2 N ( 4 n − i + 3 ) + 1 for 1 ≤ i ≤ n ,

f ( v ′ ) = 4 N + 1 ,

f ( u ′ ) = 2 N + 1 ,

f ( v ′ i ) = 2 N ( 2 n − i + 4 ) for 1 ≤ i ≤ n ,

f ( u ′ i ) = 2 N ( 3 n − i + 4 ) for 1 ≤ i ≤ n .

Then the induced edge labels are

f * ( v v i ) = ( 6 n − 2 i + 3 ) N + 1 for 1 ≤ i ≤ n ,

f * ( u u i ) = ( 4 n − i + 3 ) N + 1 for 1 ≤ i ≤ n ,

f * ( v i u ) = ( 6 n − 2 i + 4 ) N + 1 for 1 ≤ i ≤ n ,

f * ( v ′ v ′ i ) = [ 2 ( n − i ) + 3 ] N + 1 for 1 ≤ i ≤ n ,

f * ( u ′ u ′ i ) = ( 3 n − i + 3 ) N + 1 for 1 ≤ i ≤ n ,

f * ( v ′ i u ′ ) = 2 ( n − i + 2 ) N + 1 for 1 ≤ i ≤ n ,

f * ( u u ′ ) = N + 1 , f * ( v u ′ ) = 1 , f * ( u v ′ ) = 2 N + 1 .

Therefore, f is a one modulo N-difference mean labeling. Hence, D U P 2 ( B n , n ) is a one modulo N-difference mean graph. Figure 10 shows a one modulo N-difference mean labeling of D U P 2 ( B 5,5 ) .

Theorem 2.12. The friendship graph C 4 ( n ) , n ≥ 1 is a one modulo N-difference mean graph.

Proof. Let v 1 j , v 2 j , v 3 j , v 4 j ( 1 ≤ j ≤ n ) be the vertices of the cycle C 4 . Then the graph C 4 ( n ) is obtained by identifying the vertices v 1 j = v 1 for ( 1 ≤ j ≤ n ) .

Then E ( C 4 ( n ) ) = { v i j v i + 1 j , v 4 j v 1 j : 1 ≤ i ≤ 3,1 ≤ j ≤ n } .

We label the vertices as follows:

Define f : V ( C 4 ( n ) ) → { 0,1, N , N + 1,2 N ,2 N + 1, ⋯ ,2 N ( 4 n − 1 ) + 1 } by

f ( v 1 ) = 0 ,

f ( v 2 j ) = 2 N ( 4 n − 2 j + 1 ) + 1 for 1 ≤ j ≤ n ,

f ( v 3 1 ) = 2 N ( 4 n − 1 ) ,

f ( v 3 j ) = 4 N [ 2 ( n − j ) + 1 ] for 1 ≤ j ≤ n ,

f ( v 4 j ) = 2 N ( 4 n − 2 j ) + 1 for 1 ≤ j ≤ n .

Then the induced edge labels are

f * ( v 1 v 2 j ) = N ( 4 n − 2 j + 1 ) + 1 for 1 ≤ j ≤ n ,

f * ( v 2 1 v 3 1 ) = 1 , f * ( v 3 1 v 4 1 ) = N + 1 ,

f * ( v 2 j v 3 j ) = N ( 2 j − 1 ) + 1 for 2 ≤ j ≤ n ,

f * ( v 3 j v 4 j ) = 2 N ( j − 1 ) + 1 for 2 ≤ j ≤ n ,

f * ( v 4 j v 1 j ) = N ( 4 n − 2 j ) + 1 for 1 ≤ j ≤ n .

Therefore, f is a one modulo N-difference mean labeling. Hence, the graph C 4 ( n ) is a one modulo N-difference mean graph. Figure 11 shows a one modulo N-difference mean labeling of C 4 ( 5 ) .

Theorem 2.13. The graph n C 4 , n ≥ 1 is a one modulo N-difference mean graph.

Proof. Let v 1 j , v 2 j , v 3 j , v 4 j ( 1 ≤ j ≤ n ) be the vertices of n copies of the cycle C 4 .

Then E ( n C 4 ) = { v i j v i + 1 j , v 4 j v 1 j : 1 ≤ i ≤ 3,1 ≤ j ≤ n } .

We label the vertices as follows:

Define f : V ( n C 4 ) → { 0,1, N , N + 1,2 N ,2 N + 1, ⋯ ,2 N ( 4 n − 1 ) + 1 } by

f ( v 1 j ) = ( i − 1 ) N for 1 ≤ j ≤ n ,

f ( v 2 j ) = N ( 8 n − 3 j + 1 ) + 1 for 1 ≤ j ≤ n ,

f ( v 3 j ) = N ( 8 n − 7 j + 3 ) for 1 ≤ j ≤ n ,

f ( v 4 j ) = N ( 8 n − 3 j − 1 ) + 1 for 1 ≤ j ≤ n .

Then the induced edge labels are

f * ( v 1 j v 2 j ) = N ( 4 n − 2 j + 1 ) + 1 for 1 ≤ j ≤ n ,

f * ( v 2 j v 3 j ) = N ( 2 j − 1 ) + 1 for 2 ≤ j ≤ n ,

f * ( v 3 j v 4 j ) = 2 N ( j − 1 ) + 1 for 2 ≤ j ≤ n ,

f * ( v 4 j v 1 j ) = N ( 4 n − 2 j ) + 1 for 1 ≤ j ≤ n .

Therefore, f is a one modulo N-difference mean labeling. Hence, the graph n C 4 is a one modulo N-difference mean graph. Figure 12 shows a one modulo N-difference mean labeling of 6 C 4 .

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

Jeyanthi, P., Selvi, M. and Ramya, D. (2023) New Results on One Modulo N-Difference Mean Graphs. Open Journal of Discrete Mathematics, 13, 100-112. https://doi.org/10.4236/ojdm.2023.134010