New Results on One Modulo N-Difference Mean Graphs

Pon Jeyanthi¹, Meganathan Selvi², Damodaran Ramya³

¹Research Centre, Department of Mathematics, Govindammal Aditanar College for Women, Tiruchendur, Tamilnadu, India.
²Department of Mathematics, Dr. Sivanthi Aditanar College of Engineering, Tiruchendur, Tamilnadu, India.
³Department of Mathematics, Government Arts College (Autonomous), Salem-7, Tamilnadu, India.

DOI: 10.4236/ojdm.2023.134010                

Abstract

A graph G is said to be one modulo N-difference mean graph if there is an injective function f from the vertex set of G to the set
, where N is the natural number and q is the number of edges of G and f induces a bijection  from the edge set of G to given by and the function f is called a one modulo N-difference mean labeling of G. In this paper, we show that the graphs such as arbitrary union of paths, , ladder, slanting ladder, diamond snake, quadrilateral snake, alternately quadrilateral snake, , , , , friendship graph and
admit one modulo N-difference mean labeling.

Keywords

Skolem Difference Mean Labeling, One Modulo N-Graceful Labeling, One Modulo N-Difference Mean Labeling and One Modulo N-Difference Mean Graph

1. Introduction and Preliminaries

Here we consider only finite and simple graphs. The vertex set and the edge set of a graph G are denoted by $V\left(G\right)$ and $E\left(G\right)$ 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 $G=\left(V\mathrm{,}E\right)$ with p vertices and q edges is said to have skolem difference mean labeling if it is possible to label the vertices $x\in V$ with distinct elements $f\left(x\right)$ from $\left\{\mathrm{1,2,3,}\cdots \mathrm{,}p+q\right\}$ in such a way that for each edge $e=uv$ ,

let $f^{\mathrm{<mo>\; *\; </mo>}}\left(e\right)=\lceil \frac{\left|f\left(u\right)-f\left(v\right)\right|}{2}\rceil$ and the resulting labels of the edges are distinct

and are $\mathrm{1,2,3,}\cdots \mathrm{,}q$ . 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 $\left\{\mathrm{0,1,2,}\cdots \mathrm{,}q\right\}$ such that, when each edge xy is assigned with the label $\left|f\left(x\right)-f\left(y\right)\right|$ , 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 $\phi$ from the vertex set of G to $\left\{\mathrm{0,1,}N\mathrm{,}\left(N+1\right)\mathrm{,2}N\mathrm{,}\left(2N+1\right)\mathrm{,}\cdots \mathrm{,}N\left(q-1\right)\mathrm{,\; <mi>\; </mi>}N\left(q-1\right)+1\right\}$ in such a way that 1) $\phi$ is 1-1; 2) $\phi$ induces a bijection ${\phi }^{\mathrm{<mo>\; *\; </mo>}}$ from the edge set of G to $\left\{\mathrm{1,}N+\mathrm{1,2}N+\mathrm{1,}\cdots \mathrm{,}N\left(q-1\right)+1\right\}$ where ${\phi }^{\mathrm{<mo>\; *\; </mo>}}\left(uv\right)=\left|\phi \left(u\right)-\phi \left(v\right)\right|$ .

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 $B_{m\mathrm{,}n}$ , $S_{m\mathrm{,}n}$ , $P_n@P_m$ , $B\left(l\mathrm{,}m\mathrm{,}n\right)$ , $T\left(n\mathrm{,}m\right)$ , shrub, caterpillar and $K_{\mathrm{1,}n}$ 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 $G=\left(V\mathrm{,}E\right)$ be a graph and $G^{\prime }=\left(V^{\prime }\mathrm{,}{E}^{\prime }\right)$ be the copy of G. Then the graph $M_2\left(G\right)$ of G is obtained from G and $G^{\prime }$ by joining each vertices in V to its corresponding vertices in $V^{\prime }$ by an edge.

Definition 1.2. The slanting ladder graph $S{L}_n$ is obtained from two paths $u_1\mathrm{,}{u}_2\mathrm{,}{u}_3\mathrm{,}\cdots \mathrm{,}{u}_n$ and $v_1\mathrm{,}{v}_2\mathrm{,}{v}_3\mathrm{,}\cdots \mathrm{,}{v}_n$ by joining $u_i$ with $v_{i+1}$ for $1\le i\le n-1$ .

Definition 1.3. Let $G=\left(V\mathrm{,}E\right)$ be a bipartite graph with $V=V_1\cup V_2$ . Let $G^{\prime }=\left(V^{\prime }\mathrm{,}{E}^{\prime }\right)$ be the copy of G with $V^{\prime }={V^{\prime }}_1\cup {V^{\prime }}_2$ such that ${V^{\prime }}_1$ and ${V^{\prime }}_2$ be the copies of $V_1$ and $V_2$ . Then the graph $DU{P}_2\left(G\right)$ is obtained from G and $G^{\prime }$ such that $V\left(DU{P}_2\left(G\right)\right)=V\cup V^{\prime }$ and $E\left(DU{P}_2\left(G\right)\right)=E\left(G\right)\cup E\left(G^{\prime }\right)\cup \left\{{v^{\prime }}_i{v}_j\mathrm{<mi>\; </mi>}/\mathrm{<mi>\; </mi>}{v}_i{v}_j\in E\left(G\right)\text{where}{v^{\prime }}_i\in V^{\prime },v_j\in V\right\}$ . That is, $DU{P}_2\left(G\right)$ is obtained from G and $G^{\prime }$ by joining each ${v^{\prime }}_i\in V^{\prime }$ to $v_j\in V$ if $v_i$ is adjacent to $v_j$ in G.

Definition 1.4. A quadrilateral snake graph $Q_n$ is obtained from a path $u_1\mathrm{,}{u}_2\mathrm{,}\cdots ,u_n$ by joining $u_i$ and $u_{i+1}$ to two new vertices $x_i\mathrm{,}{y}_i$ respectively and then joining $x_i$ and $y_i$ .

Definition 1.5. An alternate quadrilateral snake is obtained from a path $u_1\mathrm{,}{u}_2\mathrm{,}\cdots ,u_n$ 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 $i\equiv 1\left(\mathrm{mod}2\right)$ and $1\le i\le n-1$ . 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 $v_0\mathrm{,}{v}_1\mathrm{,}{v}_2$ . The graph $J{l}_n\left(P_3\right)$ is obtained by taking n copies of $P_3$ and then identifying the left end vertices $v_0^i\left(1\le i\le n\right)$ with u and the right end vertices $v_2^i\left(1\le i\le n\right)$ with v.

Definition 1.7. Two graphs G and H are isomorphic (written $G\simeq H$ ) 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 $G_1\cup G_2$ with $V\left(G_1\cup G_2\right)=V\left(G_1\right)\cup V\left(G_2\right)$ and $E\left(G_1\cup G_2\right)=E\left(G_1\right)\cup E\left(G_2\right)$ .

Definition 1.9. The corona $G_1\odot G_2$ 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 $v_1\mathrm{,}{v}_2\mathrm{,}\cdots \mathrm{,}{v}_n$ . The graph $C_n^{\left(t\right)}$ is obtained by taking t copies of $C_n$ and then identifying the vertices $v_1^{\left(i\right)}$ for $1\le i\le t$ .

2. Main Results

Theorem 2.1. The disjoint union of paths $\cup P_{n_i}$ ( $n_i\ge 2$ , 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 $1\le i\le m$ and $n=n_1+n_2+\cdots +n_m$ .

Define $f:\mathrm{<mi>\; </mi>}V\left(\cup P_{n_i}\right)\to \left\{0,1,N,N+1,2N,2N+1,\cdots ,2N\left(n-m-1\right)+1\right\}$ as follows:

$f\left(u_{i,j}\right)=N\left[{\displaystyle {\sum }_{k=1\left(n_k\text{is}\text{odd}\right)}^{i-1}}\left(n_k-1\right)+{\displaystyle {\sum }_{k=1\left(n_k\text{is}\text{even}\right)}^{i-1}}\left(n_k-2\right)+i+j-2\right]$ if j is odd,

$f\left(u_{i,j}\right)=\left[2\left(n-m-1\right)+i-j-1-{\displaystyle {\sum }_{k=1\left(n_k\text{is}\text{odd}\right)}^{i-1}}\left(n_k-1\right)-{\displaystyle {\sum }_{k=1\left(n_k\text{is}\text{even}\right)}^{i-1}}{n}_k\right]N+1$ if j is even.

Let $e_{i\mathrm{,}j}=u_{i\mathrm{,}j}{u}_{i\mathrm{,}j+1}$ for $1\le i\le m$ and $1\le j\le n_i-1$ .

The corresponding edge label $f^{\mathrm{<mo>\; *\; </mo>}}$ is

$f^{*}\left(e_{i,j}\right)=N\left(n-m-{\displaystyle {\sum }_{k=1}^{i-1}}{n}_k+i-j-1\right)+1$ for $1\le i\le m$ and $1\le j\le n_i-1$ .

Therefore, f is a one modulo N-difference mean labeling. Hence, $\cup P_{n_i}$ is a one modulo N-difference mean graph.

Figure 1 shows a one modulo N-difference mean labeling of $P_4\cup P_5\cup P_6\cup P_2\cup P_3\cup P_8$ .

Theorem 2.2. The graph $M_2\left(P_n\right)\left(n\ge 2\right)$ is a one modulo N-difference mean graph.

Proof. Let $\left\{v_i\mathrm{,}{v^{\prime }}_i\mathrm{<mo>\; :\; </mo>\; <mi>\; </mi>1}\le i\le n\right\}$ be the vertices and $\left\{e_i,{e^{\prime }}_i,a_i=v_i{v^{\prime }}_i:\mathrm{<mi>\; </mi>1}\le i\le n\right\}$ be the edges of the graph $M_2\left(P_n\right)$ . Then the graph has 2n vertices and $3n-2$ edges.

Define $f\mathrm{<mo>\; :\; </mo>\; <mi>\; </mi>}V\left(M_2\left(P_n\right)\right)\to \left\{\mathrm{0,1,}N\mathrm{,}N+\mathrm{1,2}N\mathrm{,2}N+\mathrm{1,}\cdots \mathrm{,2}N\left(3n-3\right)+1\right\}$ by

$f\left(v_1\right)=0$ ,

Figure 1. One modulo N-difference mean labeling of $P_4\cup P_5\cup P_6\cup P_2\cup P_3\cup P_8$ .

For $2\le i\le n$ ,

$f\left(v_i\right)=\{\begin{array}{ll}\left(3i-5\right)N\hfill & \text{if}i\text{is}\text{odd}\hfill \\ 3N\left(2n-i\right)+1\hfill & \text{if}i\text{is}\text{even}\hfill \end{array}$

For $1\le i\le n$ ,

$f\left({v^{\prime }}_i\right)=\{\begin{array}{ll}\left(6n-3i-2\right)N+1\hfill & \text{if}i\text{is}\text{odd}\hfill \\ 3iN\hfill & \text{if}i\text{is}\text{even}\hfill \end{array}$

Then the induced edge labels are

$f^{*}\left(e_1\right)=3\left(n-1\right)N+1$ ,

$f^{\mathrm{<mo>\; *\; </mo>}}\left(e_i\right)=\left[3\left(n-i\right)+1\right]N+1$ for $2\le i\le n$ ,

$f^{\mathrm{<mo>\; *\; </mo>}}\left({e^{\prime }}_i\right)=\left[3\left(n-i\right)-4\right]N+1$ for $1\le i\le n$ ,

$f^{\mathrm{<mo>\; *\; </mo>}}\left(v_1{v^{\prime }}_1\right)=\left[3\left(n-4\right)\right]N+1$ ,

$f^{\mathrm{<mo>\; *\; </mo>}}\left(a_i\right)=\left[3\left(n-i\right)\right]N+1$ for $2\le i\le n$ .

Therefore, f is a one modulo N-difference mean labeling and hence $M_2\left(P_n\right)$ is a one modulo N-difference mean graph.

Figure 2 shows a one modulo N-difference mean labeling of $M_2\left(P_5\right)$ .

Figure 2. One modulo N-difference mean labeling of $M_2\left(P_5\right)$ .

Corollary 2.3. The ladder graph $P_n\times P_2$ is a one modulo N-difference mean graph.

Theorem 2.4. The slanting ladder $S{L}_n\left(n\ge 2\right)$ is a one modulo N-difference mean graph.

Proof. Let $u_1\mathrm{,}{u}_2\mathrm{,}{u}_3\mathrm{,}\cdots \mathrm{,}{u}_n$ and $v_1\mathrm{,}{v}_2\mathrm{,}{v}_3\mathrm{,}\cdots \mathrm{,}{v}_n$ be the vertices of the path of length $n-1$ .

Then $E\left(S{L}_n\right)=\left\{u_i{u}_{i+1}\mathrm{,}{v}_i{v}_{i+1}\mathrm{,}{u}_i{v}_{i+1}\mathrm{<mo>\; :\; </mo>\; <mi>\; </mi>1}\le i\le n-1\right\}$ .

Define $f\mathrm{<mo>\; :\; </mo>\; <mi>\; </mi>}V\left(S{L}_n\right)\to \left\{\mathrm{0,1,}N\mathrm{,}N+\mathrm{1,2}N\mathrm{,2}N+\mathrm{1,}\cdots \mathrm{,2}N\left(3n-4\right)+1\right\}$ by

For $1\le i\le n$ ,

$f\left(u_i\right)=\{\begin{array}{ll}2\left(i-1\right)N\hfill & \text{if}i\text{is}\text{odd}\hfill \\ 2N\left[3n-2\left(i+1\right)\right]+1\hfill & \text{if}i\text{is}\text{even}\hfill \end{array}$

$f\left(v_1\right)=\{\begin{array}{ll}2\left(3n-4\right)N\hfill & \text{if}n\text{is}\text{odd}\hfill \\ 2\left(3n-5\right)N\hfill & \text{if}n\text{is}\text{even}\hfill \end{array}$

For $2\le i\le n$ ,

$f\left(v_i\right)=\{\begin{array}{ll}2\left(i-2\right)N\hfill & \text{if}i\text{is}\text{odd}\hfill \\ 2N\left[3n-2i\right]+1\hfill & \text{if}i\text{is}\text{even}\hfill \end{array}$

Then the induced edge labels are

For $1\le i\le n-1$ ,

$f^{\mathrm{<mo>\; *\; </mo>}}\left(u_i{u}_{i+1}\right)=\{\begin{array}{ll}3N\left(n-i-1\right)+1\hfill & \text{if}i\text{is}\text{odd}\hfill \\ N\left[3\left(n-i\right)-2\right]+1\hfill & \text{if}i\text{is}\text{even}\hfill \end{array}$

$f^{\mathrm{<mo>\; *\; </mo>}}\left(v_1{v}_2\right)=\{\begin{array}{ll}1\hfill & \text{if}n\text{is}\text{odd}\hfill \\ N+1\hfill & \text{if}n\text{is}\text{even}\hfill \end{array}$

For $2\le i\le n-1$ ,

$f^{\mathrm{<mo>\; *\; </mo>}}\left(v_i{v}_{i+1}\right)=\{\begin{array}{ll}3N\left(n-i\right)+1\hfill & \text{if}i\text{is}\text{odd}\hfill \\ N\left[3\left(n-i\right)+1\right]+1\hfill & \text{if}i\text{is}\text{even}\hfill \end{array}$

$f^{\mathrm{<mo>\; *\; </mo>}}\left(u_i{v}_{i+1}\right)=N\left[3\left(n-i\right)-1\right]+1$ for $1\le i\le n-1$ .

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 $S{L}_{10}$ .

Figure 3. One modulo N-difference mean labeling of $S{L}_{10}$ .

Theorem 2.5. The diamond snake graph $DS\left(n\right)\left(n\ge 1\right)$ is a one modulo N-difference mean graph.

Proof. Let $\left\{v_0\mathrm{,}{v}_i\mathrm{,}{a}_i\mathrm{,}{b}_i\mathrm{<mo>\; :\; </mo>\; <mi>\; </mi>1}\le i\le n\right\}$ be the vertices and $\left\{v_0{a}_1\mathrm{,}{v}_0{b}_1\mathrm{,}{v}_i{a}_{i+1}\mathrm{,}{a}_i{v}_i\mathrm{,}{v}_i{b}_{i+1}\mathrm{,}{b}_i{v}_i\mathrm{<mo>\; :\; </mo>\; <mi>\; </mi>1}\le i\le n\right\}$ be the edges of the diamond snake graph which has $4n-4$ vertices and 4n edges.

Define $f\mathrm{<mo>\; :\; </mo>\; <mi>\; </mi>}V\left(DS\left(n\right)\right)\to \left\{\mathrm{0,1,}N\mathrm{,}N+\mathrm{1,2}N\mathrm{,2}N+\mathrm{1,}\cdots \mathrm{,2}N\left(4n-1\right)+1\right\}$ by

$f\left(v_0\right)=0$ ,

$f\left(v_i\right)=4iN$ for $1\le i\le n$ ,

$f\left(a_i\right)=2N\left(4n-2i+1\right)+1$ for $1\le i\le n$ ,

$f\left(b_i\right)=4N\left(2n-i\right)+1$ for $1\le i\le n$ .

Then the induced edge labels are

$f^{\mathrm{<mo>\; *\; </mo>}}\left(v_0{a}_1\right)=\left(4n-1\right)N+1$ ,

$f^{\mathrm{<mo>\; *\; </mo>}}\left(v_i{a}_{i+1}\right)=\left(4n-4i-1\right)N+1$ for $1\le i\le n-1$ ,

$f^{\mathrm{<mo>\; *\; </mo>}}\left(a_i{v}_i\right)=\left(4n-4i+1\right)N+1$ for $1\le i\le n$ ,

$f^{\mathrm{<mo>\; *\; </mo>}}\left(v_0{b}_1\right)=\left(4n-2\right)N+1$ ,

$f^{\mathrm{<mo>\; *\; </mo>}}\left(v_i{b}_{i+1}\right)=\left(4n-4i-2\right)N+1$ for $1\le i\le n-1$ ,

$f^{\mathrm{<mo>\; *\; </mo>}}\left(b_i{v}_i\right)=\left(4n-4i\right)N+1$ for $1\le i\le n$ .

Therefore, f is a one modulo N-difference mean labeling and hence $DS\left(n\right)$ is a one modulo N-difference mean graph. A one modulo N-difference mean labeling of $DS\left(5\right)$ is shown in Figure 4.

Figure 4. One modulo N-difference mean labeling of $DS\left(5\right)$ .

Theorem 2.6. The quadrilateral snake $Q_n\left(n>1\right)$ is a one modulo N-difference mean graph.

Proof. Let $u_1\mathrm{,}{u}_2\mathrm{,}\cdots \mathrm{,}{u}_n$ be the vertices of the path $P_n$ of length $n-1$ .

Then $\left\{u_i\mathrm{,}{x}_j\mathrm{,}{y}_j\mathrm{<mo>\; :\; </mo>\; <mi>\; </mi>1}\le i\le n\mathrm{,\; <mi>\; </mi>1}\le j\le n-1\right\}$ be the vertices of and $\left\{u_i{u}_{i+1}\mathrm{,}{u}_i{x}_i\mathrm{,}{u}_{i+1}{y}_i\mathrm{,}{x}_i{y}_i\mathrm{<mo>\; :\; </mo>\; <mi>\; </mi>1}\le i\le n-1\right\}$ be the edges of $Q_n$ .

Define $f\mathrm{<mo>\; :\; </mo>\; <mi>\; </mi>}V\left(Q_n\right)\to \left\{\mathrm{0,1,}N\mathrm{,}N+\mathrm{1,2}N\mathrm{,2}N+\mathrm{1,}\cdots \mathrm{,2}N\left(4n-5\right)+1\right\}$ by

$f\left(u_1\right)=0$ ,

For $2\le i\le n$ ,

$f\left(u_i\right)=\{\begin{array}{ll}\left(3i-1\right)N\hfill & \text{if}i\text{is}\text{odd}\hfill \\ \left(8n-5i-2\right)N+1\hfill & \text{if}i\text{is}\text{even}\hfill \end{array}$

$f\left(x_1\right)=2\left(4n-5\right)N+1$ ,

For $2\le i\le n-1$ ,

$f\left(x_i\right)=\{\begin{array}{ll}\left(8n-5i-3\right)N+1\hfill & \text{if}i\text{is}\text{odd}\hfill \\ 3iN\hfill & \text{if}i\text{is}\text{even}\hfill \end{array}$

For $1\le i\le n-1$ ,

$f\left(y_i\right)=\{\begin{array}{ll}\left(3i+1\right)N\hfill & \text{if}i\text{is}\text{odd}\hfill \\ \left(8n-5i-6\right)N+1\hfill & \text{if}i\text{is}\text{even}\hfill \end{array}$

Then the induced edge labels are

$f^{\mathrm{<mo>\; *\; </mo>}}\left(u_1{u}_2\right)=\left[4n-6\right]N+1$

For $2\le i\le n-1$ ,

$f^{\mathrm{<mo>\; *\; </mo>}}\left(u_i{u}_{i+1}\right)=\{\begin{array}{ll}\left[4\left(n-i\right)-3\right]N+1\hfill & \text{if}i\text{is}\text{odd}\hfill \\ \left[4\left(n-i\right)-2\right]N+1\hfill & \text{if}i\text{is}\text{even}\hfill \end{array}$

$f^{\mathrm{<mo>\; *\; </mo>}}\left(x_1{y}_1\right)=\left(4n-7\right)N+1$ ,

For $2\le i\le n-1$ ,

$f^{\mathrm{<mo>\; *\; </mo>}}\left(x_i{y}_i\right)=\{\begin{array}{ll}\left[4\left(n-i\right)-2\right]N+1\hfill & \text{if}i\text{is}\text{odd}\hfill \\ \left[4\left(n-i\right)-3\right]N+1\hfill & \text{if}i\text{is}\text{even}\hfill \end{array}$

$f^{\mathrm{<mo>\; *\; </mo>}}\left(u_i{x}_i\right)=\left[4\left(n-i\right)-1\right]N+1$ for $1\le i\le n-1$ ,

$f^{\mathrm{<mo>\; *\; </mo>}}\left(u_{i+1}{y}_i\right)=\left[4\left(n-i-1\right)\right]N+1$ for $1\le i\le n-1$ .

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 $Q_7$ .

Figure 5. One modulo N-difference mean labeling of $Q_7$ .

Theorem 2.7. The alternately quadrilateral snake $A\left(Q_n\right)\left(n>1\right)$ is a one modulo N-difference mean graph.

Proof. Let $u_1\mathrm{,}{u}_2\mathrm{,}\cdots \mathrm{,}{u}_n$ be the vertices of the path $P_n$ of length $n-1$ .

Let $n=2m$ .

Then $\left\{u_i\mathrm{,}{x}_j\mathrm{,}{y}_j\mathrm{<mo>\; :\; </mo>\; <mi>\; </mi>1}\le i\le n\mathrm{,1}\le j\le m\right\}$ be the vertices of and $\left\{u_i{u}_{i+1}\mathrm{,}{u}_i{x}_j\mathrm{,}{u}_i{y}_j\mathrm{,}{x}_j{y}_j\mathrm{<mo>\; :\; </mo>\; <mi>\; </mi>1}\le i\le n\mathrm{,1}\le j\le m\right\}$ be the edges of $A\left(Q_n\right)$ .

Define $f\mathrm{<mo>\; :\; </mo>\; <mi>\; </mi>}V\left(A\left(Q_n\right)\right)\to \left\{\mathrm{0,1,}N\mathrm{,}N+\mathrm{1,2}N\mathrm{,2}N+\mathrm{1,}\cdots \mathrm{,2}N\left(2n+\frac{n}{2}-2\right)+1\right\}$ by

$f\left(u_1\right)=0$ ,

For $2\le i\le n$ ,

$f\left(u_i\right)=\{\begin{array}{ll}2Ni\hfill & \text{if}i\text{is}\text{odd}\hfill \\ \left(5n-3i\right)N+1\hfill & \text{if}i\text{is}\text{even}\hfill \end{array}$

$f\left(x_1\right)=\left(5n-4\right)N+1$ ,

$f\left(x_j\right)=\left(5n-6j+4\right)N+1$ for $2\le j\le m$ ,

$f\left(y_j\right)=4Nj$ for $1\le j\le m$ .

Then the induced edge labels are

For $1\le i\le n-1$ ,

$f^{\mathrm{<mo>\; *\; </mo>}}\left(u_i{u}_{i+1}\right)=\{\begin{array}{ll}\left[2n+\frac{n-5i-3}{2}\right]N+1\hfill & \text{if}i\text{is}\text{odd}\hfill \\ \left[2n+\frac{n-5i-2}{2}\right]N+1\hfill & \text{if}i\text{is}\text{even}\hfill \end{array}$

$f^{\mathrm{<mo>\; *\; </mo>}}\left(x_j{y}_j\right)=\left(2n+m-5j+2\right)N+1$ for $1\le j\le m$ ,

$f^{\mathrm{<mo>\; *\; </mo>}}\left(u_i{x}_{\frac{i+1}{2}}\right)=\left[2n+\frac{n-5i+1}{2}\right]N+1$ if i is odd and $1\le i\le n-1$ ,

$f^{\mathrm{<mo>\; *\; </mo>}}\left(u_i{y}_{\frac{i}{2}}\right)=\left[2n+\frac{n-5i}{2}\right]N+1$ if i is even and $2\le i\le n-2$ .

Therefore, f is a one modulo N-difference mean labeling and hence $A\left(Q_n\right)$ is a one modulo N-difference mean graph.

Figure 6 shows a one modulo N-difference mean labeling of $A\left(Q_8\right)$ .

Figure 6. One modulo N-difference mean labeling of $A\left(Q_8\right)$ .

Theorem 2.8. The graph $J{l}_n\left(P_3\right)\left(n\ge 1\right)$ is a one modulo N-difference mean graph.

Proof. Let $v_0^i\mathrm{,}{v}_1^i\mathrm{,}{v}_2^i\left(1\le i\le n\right)$ be the vertices of the n copies of the path $P_3$ .

Then the graph $J{l}_n\left(P_3\right)$ is obtained by identifying $v_0^i=u$ and $v_2^i=v$ .

Define $f\mathrm{<mo>\; :\; </mo>\; <mi>\; </mi>}V\left(J{l}_n\left(P_3\right)\right)\to \left\{\mathrm{0,1,}N\mathrm{,}N+\mathrm{1,2}N\mathrm{,2}N+\mathrm{1,}\cdots \mathrm{,2}N\left(2n-1\right)+1\right\}$ as follows:

$f\left(u\right)=0$ ,

$f\left(v\right)=2N$ ,

$f\left(v_i^1\right)=\left(4i-2\right)N+1$ for $1\le i\le n$ .

Then the induced edge labels are

$f^{\mathrm{<mo>\; *\; </mo>}}\left(x_{1}u\right)=0$ ,

$f^{\mathrm{<mo>\; *\; </mo>}}\left(x_{i}v\right)=2\left(i-1\right)N+1$ for $2\le i\le n$ ,

$f^{\mathrm{<mo>\; *\; </mo>}}\left(u{x}_i\right)=\left(2i-1\right)N+1$ for $1\le i\le n$ .

Therefore, f is a one modulo N-difference mean labeling and hence $J{l}_n\left(P_3\right)$ is a one modulo N-difference mean graph.

Figure 7 shows a one modulo N-difference mean labeling of $J{l}_n\left(P_3\right)$ .

Figure 7. One modulo N-difference mean labeling of $J{l}_5\left(P_3\right)$ .

Theorem 2.9. The corona graph $C_4\odot K_{\mathrm{1,}n}\left(n\ge 1\right)$ is a one modulo N-difference mean graph.

Proof. Let $v_1\mathrm{,}{v}_2\mathrm{,}{v}_3\mathrm{,}{v}_4$ be the vertices of cycle $C_4$ and $\left\{v_i^j\mathrm{<mo>\; :\; </mo>\; <mi>\; </mi>1}\le i\le n\mathrm{,1}\le j\le 4\right\}$ be the vertices of the four stars $K_{\mathrm{1,}n}$ .

Define $f\mathrm{<mo>\; :\; </mo>\; <mi>\; </mi>}V\left(C_n\odot K_{\mathrm{1,}n}\right)\to \left\{\mathrm{0,1,}N\mathrm{,}N+\mathrm{1,2}N\mathrm{,2}N+\mathrm{1,}\cdots \mathrm{,2}N\left(4n+3\right)+1\right\}$ as follows:

We label the vertices of $C_4$ as follows:

$f\left(v_1\right)=0$ ,

$f\left(v_2\right)=2\left(4n+3\right)N+1$ ,

$f\left(v_3\right)=4N$ ,

$f\left(v_4\right)=4\left(2n+1\right)N+1$ .

Now, we label the vertices of $K_{\mathrm{1,}n}$ as follows:

$f\left(v_i^1\right)=2\left(i-1\right)N+1$ for $1\le i\le n$ ,

$f\left(v_i^2\right)=2N\left(3n-i+4\right)$ for $1\le i\le n$ ,

$f\left(v_i^3\right)=2\left(2n+i+1\right)N+1$ for $1\le i\le n$ ,

$f\left(v_i^4\right)=2N\left(n-i+3\right)$ for $1\le i\le n$ .

Let $e_i=\left\{v_i{v}_{i+1}\mathrm{<mo>\; :\; </mo>\; <mi>\; </mi>1}\le i\le 3\right\}$ and $e_i^j=\left\{v_j{v}_i^j\mathrm{<mo>\; :\; </mo>\; <mi>\; </mi>1}\le i\le n\mathrm{,1}\le j\le 4\right\}$ .

Then the induced edge labels are

$f^{\mathrm{<mo>\; *\; </mo>}}\left(e_1\right)=\left(4n+3\right)N+1$ ,

$f^{\mathrm{<mo>\; *\; </mo>}}\left(e_2\right)=\left(4n+1\right)N+1$ ,

$f^{\mathrm{<mo>\; *\; </mo>}}\left(e_3\right)=4nN+1$ ,

$f^{\mathrm{<mo>\; *\; </mo>}}\left(v_4{v}_1\right)=\left(4n+2\right)N+1$ ,

$f^{\mathrm{<mo>\; *\; </mo>}}\left(e_i^j\right)=\left[\left(j-1\right)n+i-1\right]N+1$ for $1\le i\le n$ , $1\le j\le 4$ .

Therefore, f is a one modulo N-difference mean labeling. Hence, $C_4\odot K_{\mathrm{1,}n}$ is a one modulo N-difference mean graph.

Figure 8 shows a one modulo N-difference mean labeling of $C_4\odot K_{\mathrm{1,3}}$ .

Figure 8. One modulo N-difference mean labeling of $C_4\odot K_{\mathrm{1,3}}$ .

Theorem 2.10. The graph $DU{P}_2\left(K_{\mathrm{1,}n}\right)\mathrm{,\; <mi>\; </mi>}n\ge 2$ is a one modulo N-difference mean graph.

Proof. Let $\left\{v\mathrm{,}{v}_i\left(1\le i\le n\right)\mathrm{,}u\mathrm{,}{u}_i\left(1\le i\le n\right)\right\}$ be the vertices and $\left\{v{v}_i\mathrm{,}u{u}_i,v_{i}u\mathrm{<mo>\; :\; </mo>\; <mi>\; </mi>1}\le i\le n\right\}$ be the edges of $DU{P}_2\left(K_{\mathrm{1,}n}\right)$ .

Now, the vertex labels are defined as follows:

Define $f\mathrm{<mo>\; :\; </mo>\; <mi>\; </mi>}V\left(DU{P}_2\left(K_{\mathrm{1,}n}\right)\right)\to \left\{\mathrm{0,1,}N\mathrm{,}N+\mathrm{1,2}N\mathrm{,2}N+\mathrm{1,}\cdots \mathrm{,2}N\left(3n-1\right)+1\right\}$ by

$f\left(v\right)=2N\left(2n-1\right)$ ,

$f\left(u\right)=0$ ,

$f\left(v_i\right)=2N\left(2n-i\right)+1$ for $1\le i\le n$ ,

$f\left(u_i\right)=2N\left(3n-i\right)+1$ for $1\le i\le n$ .

Then the induced edge labels are

$f^{\mathrm{<mo>\; *\; </mo>}}\left(v{v}_i\right)=N\left(i-1\right)+1$ for $1\le i\le n$ ,

$f^{\mathrm{<mo>\; *\; </mo>}}\left(u{u}_i\right)=N\left(3n-i\right)+1$ for $1\le i\le n$ ,

$f^{\mathrm{<mo>\; *\; </mo>}}\left(v_{i}u\right)=N\left(2n-i\right)+1$ for $1\le i\le n$ .

Therefore, f is a one modulo N-difference mean labeling. Hence, $DU{P}_2\left(K_{\mathrm{1,}n}\right)$ is a one modulo N-difference mean graph. Figure 9 shows a one modulo N-difference mean labeling of $DU{P}_2\left(K_{\mathrm{1,7}}\right)$ .

Figure 9. One modulo N-difference mean labeling of $DU{P}_2\left(K_{\mathrm{1,7}}\right)$ .

Theorem 2.11. The graph $DU{P}_2\left(B_{n\mathrm{,}n}\right)\mathrm{,}n\ge 2$ is a one modulo N-difference mean graph.

Proof. Let $\left\{v\mathrm{,}{v}^{\prime }\mathrm{,}{v}_i\mathrm{,}{v^{\prime }}_i\mathrm{,}u\mathrm{,}{u}^{\prime }\mathrm{,}{u}_i\mathrm{,}{u^{\prime }}_i\mathrm{<mo>\; :\; </mo>\; <mi>\; </mi>1}\le i\le n\right\}$ be the vertices and

$\left\{v{v}_i\mathrm{,}u{u}_i,v_{i}u\mathrm{,}{v}^{\prime }{v^{\prime }}_i\mathrm{,}{u}^{\prime }{u^{\prime }}_i,{v^{\prime }}_i{u}^{\prime }\mathrm{,}v{u}^{\prime }\mathrm{,}u{u}^{\prime }\mathrm{,}u{v}^{\prime }\mathrm{<mo>\; :\; </mo>\; <mi>\; </mi>1}\le i\le n\right\}$ be the edges of $DU{P}_2\left(B_{n\mathrm{,}n}\right)$ .

Now, the vertex labels are defined as follows:

Define $f\mathrm{<mo>\; :\; </mo>\; <mi>\; </mi>}V\left(DU{P}_2\left(B_{n\mathrm{,}n}\right)\right)\to \left\{\mathrm{0,1,}N\mathrm{,}N+\mathrm{1,2}N\mathrm{,2}N+\mathrm{1,}\cdots \mathrm{,2}N\left(6n+2\right)+1\right\}$ by

$f\left(v\right)=2N$ ,

$f\left(u\right)=0$ ,

$f\left(v_i\right)=4N\left(3n-i+2\right)+1$ for $1\le i\le n$ ,

$f\left(u_i\right)=2N\left(4n-i+3\right)+1$ for $1\le i\le n$ ,

$f\left(v^{\prime }\right)=4N+1$ ,

$f\left(u^{\prime }\right)=2N+1$ ,

$f\left({v^{\prime }}_i\right)=2N\left(2n-i+4\right)$ for $1\le i\le n$ ,

$f\left({u^{\prime }}_i\right)=2N\left(3n-i+4\right)$ for $1\le i\le n$ .

Then the induced edge labels are

$f^{\mathrm{<mo>\; *\; </mo>}}\left(v{v}_i\right)=\left(6n-2i+3\right)N+1$ for $1\le i\le n$ ,

$f^{\mathrm{<mo>\; *\; </mo>}}\left(u{u}_i\right)=\left(4n-i+3\right)N+1$ for $1\le i\le n$ ,

$f^{\mathrm{<mo>\; *\; </mo>}}\left(v_{i}u\right)=\left(6n-2i+4\right)N+1$ for $1\le i\le n$ ,

$f^{\mathrm{<mo>\; *\; </mo>}}\left(v^{\prime }{v^{\prime }}_i\right)=\left[2\left(n-i\right)+3\right]N+1$ for $1\le i\le n$ ,

$f^{\mathrm{<mo>\; *\; </mo>}}\left(u^{\prime }{u^{\prime }}_i\right)=\left(3n-i+3\right)N+1$ for $1\le i\le n$ ,

$f^{\mathrm{<mo>\; *\; </mo>}}\left({v^{\prime }}_i{u}^{\prime }\right)=2\left(n-i+2\right)N+1$ for $1\le i\le n$ ,

$f^{\mathrm{<mo>\; *\; </mo>}}\left(u{u}^{\prime }\right)=N+1$ , $f^{\mathrm{<mo>\; *\; </mo>}}\left(v{u}^{\prime }\right)=1$ , $f^{\mathrm{<mo>\; *\; </mo>}}\left(u{v}^{\prime }\right)=2N+1$ .

Therefore, f is a one modulo N-difference mean labeling. Hence, $DU{P}_2\left(B_{n\mathrm{,}n}\right)$ is a one modulo N-difference mean graph. Figure 10 shows a one modulo N-difference mean labeling of $DU{P}_2\left(B_{\mathrm{5,5}}\right)$ .

Figure 10. One modulo N-difference mean labeling of $DU{P}_2\left(B_{\mathrm{5,5}}\right)$ .

Theorem 2.12. The friendship graph $C_4^{\left(n\right)}\mathrm{,}n\ge 1$ is a one modulo N-difference mean graph.

Proof. Let $v_1^j\mathrm{,}{v}_2^j\mathrm{,}{v}_3^j\mathrm{,}{v}_4^j\left(1\le j\le n\right)$ be the vertices of the cycle $C_4$ . Then the graph $C_4^{\left(n\right)}$ is obtained by identifying the vertices $v_1^j=v_1$ for $\left(1\le j\le n\right)$ .

Then $E\left(C_4^{\left(n\right)}\right)=\left\{v_i^j{v}_{i+1}^j\mathrm{,}{v}_4^j{v}_1^j\mathrm{<mo>\; :\; </mo>\; <mi>\; </mi>1}\le i\le \mathrm{3,1}\le j\le n\right\}$ .

We label the vertices as follows:

Define $f\mathrm{<mo>\; :\; </mo>\; <mi>\; </mi>}V\left(C_4^{\left(n\right)}\right)\to \left\{\mathrm{0,1,}N\mathrm{,}N+\mathrm{1,2}N\mathrm{,2}N+\mathrm{1,}\cdots \mathrm{,2}N\left(4n-1\right)+1\right\}$ by

$f\left(v_1\right)=0$ ,

$f\left(v_2^j\right)=2N\left(4n-2j+1\right)+1$ for $1\le j\le n$ ,

$f\left(v_3^1\right)=2N\left(4n-1\right)$ ,

$f\left(v_3^j\right)=4N\left[2\left(n-j\right)+1\right]$ for $1\le j\le n$ ,

$f\left(v_4^j\right)=2N\left(4n-2j\right)+1$ for $1\le j\le n$ .

Then the induced edge labels are

$f^{\mathrm{<mo>\; *\; </mo>}}\left(v_1{v}_2^j\right)=N\left(4n-2j+1\right)+1$ for $1\le j\le n$ ,

$f^{\mathrm{<mo>\; *\; </mo>}}\left(v_2^1{v}_3^1\right)=1$ , $f^{\mathrm{<mo>\; *\; </mo>}}\left(v_3^1{v}_4^1\right)=N+1$ ,

$f^{\mathrm{<mo>\; *\; </mo>}}\left(v_2^j{v}_3^j\right)=N\left(2j-1\right)+1$ for $2\le j\le n$ ,

$f^{\mathrm{<mo>\; *\; </mo>}}\left(v_3^j{v}_4^j\right)=2N\left(j-1\right)+1$ for $2\le j\le n$ ,

$f^{\mathrm{<mo>\; *\; </mo>}}\left(v_4^j{v}_1^j\right)=N\left(4n-2j\right)+1$ for $1\le j\le n$ .

Therefore, f is a one modulo N-difference mean labeling. Hence, the graph $C_4^{\left(n\right)}$ is a one modulo N-difference mean graph. Figure 11 shows a one modulo N-difference mean labeling of $C_4^{\left(5\right)}$ .

Figure 11. One modulo N-difference mean labeling of $C_4^{\left(5\right)}$ .

Theorem 2.13. The graph $n{C}_4\mathrm{,}n\ge 1$ is a one modulo N-difference mean graph.

Proof. Let $v_1^j\mathrm{,}{v}_2^j\mathrm{,}{v}_3^j\mathrm{,}{v}_4^j\left(1\le j\le n\right)$ be the vertices of n copies of the cycle $C_4$ .

Then $E\left(n{C}_4\right)=\left\{v_i^j{v}_{i+1}^j\mathrm{,}{v}_4^j{v}_1^j\mathrm{<mo>\; :\; </mo>\; <mi>\; </mi>1}\le i\le \mathrm{3,1}\le j\le n\right\}$ .

We label the vertices as follows:

Define $f\mathrm{<mo>\; :\; </mo>\; <mi>\; </mi>}V\left(n{C}_4\right)\to \left\{\mathrm{0,1,}N\mathrm{,}N+\mathrm{1,2}N\mathrm{,2}N+\mathrm{1,}\cdots \mathrm{,2}N\left(4n-1\right)+1\right\}$ by

$f\left(v_1^j\right)=\left(i-1\right)N$ for $1\le j\le n$ ,

$f\left(v_2^j\right)=N\left(8n-3j+1\right)+1$ for $1\le j\le n$ ,

$f\left(v_3^j\right)=N\left(8n-7j+3\right)$ for $1\le j\le n$ ,

$f\left(v_4^j\right)=N\left(8n-3j-1\right)+1$ for $1\le j\le n$ .

Then the induced edge labels are

$f^{\mathrm{<mo>\; *\; </mo>}}\left(v_1^j{v}_2^j\right)=N\left(4n-2j+1\right)+1$ for $1\le j\le n$ ,

$f^{\mathrm{<mo>\; *\; </mo>}}\left(v_2^j{v}_3^j\right)=N\left(2j-1\right)+1$ for $2\le j\le n$ ,

$f^{\mathrm{<mo>\; *\; </mo>}}\left(v_3^j{v}_4^j\right)=2N\left(j-1\right)+1$ for $2\le j\le n$ ,

$f^{\mathrm{<mo>\; *\; </mo>}}\left(v_4^j{v}_1^j\right)=N\left(4n-2j\right)+1$ for $1\le j\le n$ .

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 $6C_4$ .

Figure 12. One modulo N-difference mean labeling of $6C_4$ .

Conflicts of Interest

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

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