On Topological Cordial Labeling of Some Graphs

Let G be the graph. A topological cordial labeling of a graph G with |V (G)| = n is an injective function f from the vertex set V(G) to the set 2X where X is a nonempty set such that |X| < n and it forms a topology on X, denoted by {f (V (G))}, that induced a function f ∗ from E (G) → {0, 1} defined by

f ∗ (uv) = {1 0 if f (u) ∩ f (v)is not an empty set and not a singleton set, 0 otherwise

for all uv ∈ E (G) such that  |ef (0) − ef (1) | ≤ 1 where ef (0) is the number of vertices labeled with 0 and where ef (1) is the number of vertices labeled with 1. The graph that satisfies the condition of a topological cordial labeling is called topological cordial graph.

Introduction

In this paper, the graphs considered were finite and undirected. Graph $G$ consists of a set $V$ of vertices and a collection $E$ of unordered pairs. For such notations and terminologies, we refer to Balakrishnan [1]. Graph labeling is an assignment of integers to vertices, or edges, or both, under certain conditions [2]. Graph Labeling has various applications such as x-ray, coding theory, radar, circuits, etc. Cordial Labeling is a graph labeling in which the possible choices to be labeled are either 0 or 1 of the edges and vertices of a certain graph. Cordial Labeling occurs in many ways depending on the condition that must be considered. Cordial Labeling was introduced by Cahit in 1987 as a weaker version of harmonious and graceful graphs [3].

Topological cordial labelling was introduced by Selistin et al. in 2020 [4] and 2021 [5] and Prijith et al. in 2023 [6]. A graph is called a topological cordial graph subject to certain condition, that is, the function $f$ is an injective that maps from the vertex set of $G$ to $2^X$ where $X$ is the nonempty set such that the cardinality is less than the vertex set of $G$ and it forms a topology on $X$ that induces a function $f^{\ast }$ defined by

$$\begin{array}{r}{f}^{\ast }\left(uv\right)=\{\begin{array}{ll}1& iff\left(u\right)\cap f\left(v\right)isnotanemptysetandnotasingletonset\\ 0& otherwise\end{array},\end{array}$$

for all $uv\in E\left(G\right)$ such that $|e_f\left(0\right)-e_f\left(1\right)|\le 1$ where $e_f\left(0\right)$ is the number of vertices labeled with 0 and where $e_f\left(1\right)$ is the number of vertices labeled with 1. The graph that satisfies the condition of a topological cordial labeling is called topological cordial graph. In this paper, we discuss the Topological cordial labeling.

Basic Concepts

Definition 1. [7] Let $X$ be the set. A topology (or topological structure) in $X$ is a family $\mathrm{????}$ of subsets of $X$ that satisfies:

i) $\varnothing$ and $\mathrm{????}$ are members of $\mathrm{????}$.

ii) Each union of members of $\mathrm{????}$ is also a member of $\mathrm{????}$.

iii) Each finite intersection of members of $\mathrm{????}$ is also a member of $\mathrm{????}$.

A set $X$ for which a topology $\mathrm{????}$ has been specified is called topological space.

Definition 2. [6] A topological cordial labeling of a graph $G$ with $\left|V\left(G\right)\right|=n$ is an injective function $f$ from the vertex set $V(G)$ to the set $2^X$ where $X$ is a nonempty set such that $|X|<n$ and it forms a topology on $X$, denoted by $\left\{f\left(V\left(G\right)\right)\right\}$, that induced a function $f^{\ast }$ from $E\left(G\right)\to \{0,1\}$ defined by

$$\begin{array}{r}{f}^{\ast }\left(uv\right)=\{\begin{array}{ll}1& iff\left(u\right)\cap f\left(v\right)isnotanemptysetandnotasingletonset\\ 0& otherwise\end{array}\end{array}$$

Definition 3. [8] A Bi-star graph $B_{n,n}$ is a graph obtained by joining graph obtained by joining the center vertices of two copies of $K_{1,n}$ by an edge. The vertex set of $B_{n,n}$ is $V\left(B_{n,n}\right)=\{u,v,u_i,v_i|1\le i\le n\}$, where $u,v$ are the center vertices and $u_i,v_i$ are pendant vertices. The edge set of $B_{n,n}$ is $E\left(B_{(n,n)}\right)=\{uv,u{u}_i,v{v}_i|1\le i\le n\}$ so, $\left|V\left(B_{n,n}\right)\right|=2n+2$ and $\left|E\left(B_{n,n}\right)\right|=2n+1$.

Definition 4. [8] The Splitting Graph $S^{\prime }(G)$ of a graph $G$ is constructed by adding to each vertex $v$, a new vertex $v^{\prime }$ such that $v^{\prime }$ is adjacent to every vertex that is adjacent to $v$, that is, $N(v)=N(v^{\prime })$.

Definition 5. [9] Let $G=(V\left(G\right),E\left(G\right))$ be a graph with $V=S_1\cup S_2\cup \cdots \cup S_i\cup T$ where each $S_i$ is the set of vertices having at least two vertices of the same degree and $T=V\mathrm{\setminus }\bigcup S_i$. The degree splitting graph of $G$ denoted by $DS(G)$ is obtained from $G$ by adding vertices $w_1,w_2,\dots ,w_t$ and joining to each vertex of $S_i$ for $1\le i\le t$.

Definition 6. [10] A Firecracker Graph $F_{m,n}$ is the graph obtained by the concatenation of mn-stars by linking on leaf from each. Firecracker graph $F_{m,n}$ is the graph with (n + 1) m order and (n + 1) m − 1 size, consisting of vertex set $V(F_{m,n})=\{u_i|1\le i\le m\}\cup \{v_{i,j}|1\le i\le m,1\le j\le n\}$ and edge set $E(F_{m,n})=\{u_i{v}_{i,j}|1\le i\le m,1\le j\le n\}\cup \{v_{i,1}{v}_{(i+1),1}|1\le i\le m-1\}$.

Definition 7. [11] The Banana Tree Graph $B{N}_{n,k}$ is the graph obtained by connecting one leaf of each of $n$ copies of a $k$-star graph with a single root vertex that is distinct for all the stars. The $B{N}_{n,k}$ has order $nk+1$ and size $nk$.

Definition 8. [12] The Jellyfish Graph $J_{m,n}$ is obtained from 4-cycle $u,v,w,x$ by joining $u$ and $v$ with an edge and appending $m$ pendant edges $\{v_1,v_2,\dots ,v_m\}$ to $v$ and $n$ pendant edges $\{x_1,x_2,\dots ,x_n\}$ to $x.$

Results and Discussions

Theorem 1. Let $G$ be a graph with $n$ vertices and let $Y_{n-1}=\{1,2,3,\cdots ,n-1\}$. Then

$$\begin{array}{r}\mathrm{????}=\{\varnothing ,\left\{1\right\},\{1,2\},\dots ,\{1,2,3,\dots ,n-1\}\}\end{array}$$

is a topology on $Y_{n-1}$.

Proof: Let $G$ be the graph of order $n$. Suppose that $Y_{n-1}=\{1,2,3,\dots ,n-1\}$ and let the set $\mathrm{????}=\{Y_0,Y_1,Y_2,\dots ,Y_{n-1}\}$ where

$$\begin{array}{c}{Y}_0=\mathrm{\varnothing }\\ Y_1=\left\{1\right\}\\ Y_2=\{1,2\}\\ ⋮\\ Y_{n-2}=\left\{1,2,3,\dots ,n-2\right\}\\ {Y_{n-}}_1=\left\{1,2,3,\dots ,n-1\right\}.\end{array}$$

Note that $Y_i\subset Y_j$ if $i<j$. By Definition 1,

i) $\varnothing$ and $Y_{n-1}$ is in $\mathrm{????}$. Hence, the first condition is satisfied.

ii) Let $I=\{0,1,2,\dots ,n-1\}$ and suppose $\{Y_{\alpha }:\alpha \in I\}$ is a collection of elements of $\mathrm{????}$ and let $J$ be a subset of $I$. We want to show that

$$\begin{array}{r}\bigcup _{\alpha \in J}{Y}_{\alpha }\in \mathrm{????}\end{array}$$

Let $m\in J$ such that $\alpha \le m$ for every $\alpha \in J$. Then

$$\begin{array}{r}\bigcup _{\alpha \in J}{Y}_{\alpha }=Y_m\in \mathrm{????}\end{array}$$

Hence, the arbitrary union of the elements of any subcollection of $\mathrm{????}$ is in $\mathrm{????}$.

iii) Suppose $J\subseteq I$. Let $s\in J$ such that $s\le \alpha$ for every $\alpha \in J$. Then

$$\begin{array}{r}\bigcap _{\alpha \in J}{Y}_{\alpha }=Y_s\in \mathrm{????}\end{array}$$

Hence, the intersection of any finite subcollection of $\mathrm{????}$ is in $\mathrm{????}$.

Thus, three conditions are satisfied, therefore, $\mathrm{????}$ is a topology on $Y_{n-1}$, that is,

$$\begin{array}{r}\mathrm{????}=\{\varnothing ,\left\{1\right\},\{1,2\},\dots ,\{1,2,3,\dots ,n-1\}\}\end{array}$$

is a topology on $Y_{n-1}$.

Theorem 2. A Splitting of a Bi-star Graph $B_{n,n}$ is a topological cordial graph for all $n\in \mathbb{N}$.

Proof: Suppose $S^{\prime }\left(B_{n,n}\right)$ is the splitting of a bi-star graph $B_{n,n}.$ Let $V\left(S^{\prime }\left(B_{n,n}\right)\right)=V\left(B_{n,n}\right)\cup \{u^{\prime },v^{\prime },u_i^{\prime },v_i^{\prime }|1\le i\le n\}$ be a vertex set of $S^{\prime }\left(B_{n,n}\right)$. The order of $S^{\prime }\left(B_{n,n}\right)$ is $4n+4$. Also, let $E\left(S^{\prime }\left(B_{n,n}\right)\right)=E\left(B_{n,n}\right)\cup \{u{v}^{\prime },v{u}^{\prime },u{u}_i^{\prime }v{v}_i^{\prime }{u}^{\prime }{v}_i{v}^{\prime }{v}_i|1\le i\le n\}$be the edge set of $E\left(S^{\prime }\left(B_{n,n}\right)\right).$ The size of $S^{\prime }\left(B_{n,n}\right)$ is $6n+3$. Now, let $X=\{1,2,3,\dots ,4n+3\}$ and suppose that

$$\begin{array}{r}\mathrm{????}=\{\varnothing ,\left\{1\right\},\{1,2\},\dots ,\{1,2,\dots ,4n+2\}\}.\end{array}$$

According to Theorem 1, $\mathrm{????}$ is a topology on $X$. Now, the function $f:V\left(DS\left(B_{n,n}\right)\right)\to \mathrm{????}$ is defined by

$$f\left(v\right)=\{1,2\},f\left(u\right)=\varnothing$$

$$f\left(u^{\prime }\right)=\{1,2\cdots ,4n+3\},f\left(v^{\prime }\right)=\{1\}$$

$$f\left(u_i^{\prime }\right)=\{1,2,\dots ,i+2\}\text{for}1\le i\le n,$$

$$f\left(v_i^{\prime }\right)=\{1,2,\dots ,n+i+2\}\text{for}1\le i\le n,$$

$$f\left(u_i\right)=\{1,2,\dots ,2n+i+2\}\text{for}1\le i\le n,$$

$$f\left(v_i\right)=\{1,2,\dots ,3n+i+2\}\text{for}1\le i\le n.$$

Thus, the induced edge labels are

$$f^{\ast }(uv)=0,f^{\ast }(u{v}^{\prime })=0,$$

$$f^{\ast }(u{v}^{\prime })=1,$$

$$f^{\ast }(u{u}_i)=0\text{for}1\le i\le n,$$

$$f^{\ast }(u{u}_i^{\prime })=0\text{for}1\le i\le n,$$

$$f^{\ast }(v{v}_i)=1\text{for}1\le i\le n,$$

$$f^{\ast }(u^{\prime }{u}_i)=1\text{for}1\le i\le n,$$

$$f^{\ast }(v{v}_i)=1\text{for}1\le i\le n,$$

$$f^{\ast }(v^{\prime }{v}_i)=1\text{for}1\le i\le n,$$

It can be observed that $e_f\left(0\right)=3n+2$ and $e_f\left(1\right)=3n+1$. Thus $|e_f\left(0\right)-e_f\left(1\right)|=|3n+2-(3n+1)|=1$. Therefore, the Splitting of a Bi-star Graph $S^{\prime }\left(B_{n,n}\right)$ is a topological cordial graph for $n\in \mathbb{N}$.

Theorem 3. A Degree Splitting of a Bistar Graph $DS(B_{n,n})$ is a topological cordial graph for all $n\in \mathbb{N}$.

Proof: Suppose $DS\left(B_{n,n}\right)$ is the degree splitting of a bistar graph $B_{n,n}$. Let the vertex set $V\left(DS\left(B_{n,n}\right)\right)=V\left(B_{n,n}\right)\cup \{w_1,w_2,u_i,v_i|1\le i\le n\}$. The order of $DS\left(B_{n,n}\right)$ is $2n+4$. In addition, let the edge set $E\left(DS\left(B_{n,n}\right)\right)=E\left(B_{n,n}\right)\cup \{w_1{u}_i,w_1{v}_i,u{w}_2,v{w}_2|1\le i\le n\}$. The size of $DS\left(B_{n,n}\right)$ is $4n+3$. Now, let $X=\{1,2,3,\dots ,4n+3\}$ and suppose $\mathrm{????}=\{\varnothing ,\left\{1\right\},\{1,2\},\dots ,\{1,2,\dots ,4n+2\}\}.$ By Theorem 1, $\mathrm{????}$ is a topology on $X$. Now, the function $f:V\left(DS\left(B_{n,n}\right)\right)\to \mathrm{????}$ is defined by:

$$\begin{array}{r}f\left(w_1\right)=\varnothing f\left(w_2\right)=\left\{1\right\}\end{array}$$

$$\begin{array}{r}f\left(u\right)=\{1,2\cdots ,2n+2\},\end{array}$$

$$\begin{array}{r}f\left(v\right)=\{1,2,\dots ,2n+3\},\end{array}$$

$$\begin{array}{r}f\left(u_i\right)=\{1,2,\dots ,i+1\}\text{for}1\le i\le n,\end{array}$$

$$\begin{array}{r}f\left(v_i\right)=\{1,2,\dots ,n+i+1\}\text{for}1\le i\le n,\end{array}$$

Thus, the induced edge labels are:

$$\begin{array}{r}{f}^{\ast }\left(w_{2}u\right)=0,f^{\ast }\left(w_{2}v\right)=0,f^{\ast }\left(uv\right)=1,\end{array}$$

$$\begin{array}{r}{f}^{\ast }\left(w_1{u}_i\right)=0\text{for}1\le i\le n,f^{\ast }\left(w_1{v}_i\right)=0\text{for}1\le i\le n,\end{array}$$

$$\begin{array}{r}{f}^{\ast }\left(v{v}_i\right)=1\text{for}1\le i\le n,f^{\ast }\left(u{u}_i\right)=1\text{for}1\le i\le n.\end{array}$$

It can be observed that e_f (0) = 3n + 2 and e_f (1) = 3n + 1. Thus $|e_f(0)-e_f(1)|=|3n+2-(3n+1)|=1$. Therefore, the Splitting of a Bi-star Graph $S^{\prime }(B_{n,n})$ is a topological cordial graph for $n\in \mathbb{N}$.

Theorem 4. A Firecracker Graph $F_{m,n}$ where $n=2$ and $n\in \mathbb{N}$ is a topological cordial graph.

Proof: Suppose $F_{m,n}$ is a firecracker graph where $n=2$ and $n\in \mathbb{N}$ and assume $V\left(F_{2,n}\right)=\{u_1,u_2,u_{1,j},u_{2,j}|1\le j\le n\}$ be the vertex set of $F_{2,n}$. The order of $F_{2,n}$ is $2(n+1)-1$. Also, let $E\left(F_{2,n}\right)=\{u_i{u}_{i,j}|1\le i\le 2,1\le j\le n\}\cup \{u_{1,1}{u}_{2,1}\}$ be an edge set of $F_{2,n}$. The size of $F_{2,n}$ is $2(n+1)-1$. Moreover, assume that $X=\{1,2,3,\dots ,2n+1\}$ and suppose $\mathrm{????}=\{\varnothing ,\left\{1\right\},\{1,2\},\dots ,\{1,2,\dots ,2n\}\}$. By Theorem 1, $\mathrm{????}$ is a topology on $X$. Now, the function $f:V\left(F_{2,n}\right)\to \mathrm{????}$ is defined by:

$$\begin{array}{r}f\left(x\right)=\{1,2,\dots ,2k\},f\left(x_1\right)=\varnothing ,\end{array}$$

$$\begin{array}{r}f\left(x_2\right)=\{1,2\cdots ,k\},\end{array}$$

$$\begin{array}{r}f(x_{1,j})=\{1,2,\cdots ,j\}\text{for}1\le j\le k-1,\end{array}$$

$$\begin{array}{r}f(x_2,j)=\{1,2,\cdots ,k+j\}\text{for}1\le j\le k-1.\end{array}$$

Thus, the induced edge labels are:

$$\begin{array}{r}{f}^{\ast }(x{x}_{2,1})=1,f^{\ast }(x{x}_{1,1})=0,\end{array}$$

$$\begin{array}{r}{f}^{\ast }\left(x_1{x}_{1,j}\right)=0\text{for}1\le j\le k-1,\end{array}$$

$$\begin{array}{r}{f}^{\ast }(x_2{x}_{2,j})=1\text{for}1\le j\le k-1.\end{array}$$

It can be observed that $e_f\left(0\right)=n+1$ and $e_f\left(1\right)=n$. Thus $|e_f\left(0\right)-e_f\left(1\right)|=|n+1-n|=1$. Therefore, the Firecracker Graph $F_{m,n}$ where $m=2$ and $n\in \mathbb{N}$ is a topological cordial graph.

Theorem 5. A Banana Tree Graph $B{N}_{n,k}$ where $n=2$ and is a topological cordial graph for all $n\in \mathbb{N}$.

Proof: Suppose $B{N}_{n,k}$ is a banana tree graph where $n=2$ and $k\in \mathbb{N}$ and let the vertex set $V\left(B{N}_{2,k}\right)=\{x,x_1,x_2,x_{1,j},x_{2,j}|1\le j\le k-1\}.$ The order of $B{N}_{2,k}$ is $2k+1$. In addition, suppose that edge set $E\left(B{N}_{2,k}\right)=\{x_1{x}_{1,j},x_2{x}_{2,j}|1\le j\le k-1\}\cup \{x{x}_{1,1},x{x}_{2,1}\}$. The size of $B{N}_{2,k}$ is $2k$. Now, assume $X=\{1,2,\dots ,2k\}$ and suppose $\mathrm{????}=\{\varnothing ,X,\left\{1\right\},\{1,2\},\dots ,\{1,2,\dots ,2k-1\}\}.$ According to Theorem 1, $\mathrm{????}$ is a topology on $X$. Now, the function $f:V\left(B{N}_{2,k}\right)\to \mathrm{????}$ is defined by:

$$\begin{array}{r}f(x)=\{1,2,\dots ,2k\},f(x_1)=\varnothing ,\end{array}$$

$$\begin{array}{r}f(x_2)=\{1,2\cdots ,k\},\end{array}$$

$$\begin{array}{r}f(x_{1,j})=\{1,2,\cdots ,j\}\text{for}1\le j\le k-1,\end{array}$$

$$\begin{array}{r}f(x_{2,j})=\{1,2,\cdots ,k+j\}\text{for}1\le j\le k-1.\end{array}$$

Thus, the induced edge labels are:

$$\begin{array}{r}{f}^{\ast }\left(u_{1,1}{u}_{2,1}\right)=0,\end{array}$$

$$\begin{array}{r}{f}^{\ast }\left(u_1{u}_{1,j}\right)=0\text{for}1\le j\le n,\end{array}$$

$$\begin{array}{r}{f}^{\ast }\left(u_2{u}_{2,j}\right)=1\text{for}1\le j\le n.\end{array}$$

It can be observed that $e_f\left(0\right)=k$ and $e_f\left(1\right)=k$. Thus $|e_f\left(0\right)-e_f\left(1\right)|=|k-k|=0<1$. Therefore, the Banana Tree Graph $B{N}_{n,k}$ where $n=2$ and $k\in \mathbb{N}$ is a topological cordial graph.

Theorem 6. A Jellyfish Graph $J_{m,n}$ is a topological cordial graph for $m-n=i$ where $i=0,1,2$ and for $m,n\in \mathbb{N}$.

Proof: Suppose $J_{m,n}$ is a Jellyfish graph. To prove this theorem, we consider the following cases:

Case 1: $m-n=0$.

Let $V\left(J_{m,m}\right)=\{u,v,w,x\}\cup \{v_i,x_i|1\le i\le m\}$. The order of $J_{m,m}$ is $2m+4$. Furthermore, let $E\left(J_{m,m}\right)=\{uv,vw,uw,wx\}\cup \{v{v}_i,x{x}_i|1\le i\le m\}.$ The size of $J_{m,m}$ is $2m+5$. Now, let $X=\{1,2,\dots ,2m+3\}$ and assume that $\mathrm{????}=\{\varnothing ,X,\left\{1\right\},\{1,2\},\dots ,\{1,2,\dots ,2m+2\}\}$. By Theorem 1, $\mathrm{????}$ is a topology on $X$. Define the function $f:V\left(J_{m,m}\right)\to \mathrm{????}$ by:

$$\begin{array}{r}f(u)=\{1,2,\dots ,m+2\},f(v)=\varnothing ,\end{array}$$

$$\begin{array}{r}f(x)=\{1,2,\dots ,m+3\},f(w)=\{1,2,\dots ,m+1\},\end{array}$$

$$\begin{array}{r}f(v_i)=\{1,2,\dots ,i\}\text{for}1\le i\le m,\end{array}$$

$$\begin{array}{r}f(x_j)=\{1,2,\dots ,m+i+3\}\text{for}1\le i\le n.\end{array}$$

Thus, the induced edge labels are:

$$\begin{array}{r}{f}^{\ast }(ux)=1,f^{\ast }(uv)=0,\end{array}$$

$$\begin{array}{r}{f}^{\ast }(vw)=0,f^{\ast }(uw)=1,\\ f^{\ast }(wx)=1,\end{array}$$

$$\begin{array}{r}{f}^{\ast }(v{v}_i)=0,\text{for}1\le i\le n,\end{array}$$

$$\begin{array}{r}{f}^{\ast }(x{x}_j)=1,\text{for}1\le i\le n.\end{array}$$

It can be observed that $e_f\left(0\right)=m+2$ and $e_f\left(1\right)=m+3$. Thus $|e_f\left(0\right)-e_f\left(1\right)|=|m+2-(m+3)|=1$. Hence, if $m-n=0,$ the Jellyfish Graph $J_{m,m}$ is a topological cordial graph.

Case 2: $m-n=1$.

Let $V\left(J_{m,n}\right)=\{u,v,w,x\}\cup \{v_i,x_j|1\le i\le n+1,1\le j\le n\}$. The order of $J_{m,n}$ is $2n+5$. In addition, let $E\left(J_{m,n}\right)=\{uv,vw,uw,wx\}\cup \{v{v}_i,x{x}_j|1\le i\le n+1,1\le j\le n\}.$ The size of $J_{m,n}$ is $2m+6$. Now, let $X=\{1,2,\dots ,2m+4\}$ and assume that $\mathrm{????}=\{\varnothing ,X,\left\{1\right\},\{1,2\},\dots ,\{1,2,\dots ,2m+3\}\}$. By Theorem 1, $\mathrm{????}$ is a topology on $X$. Define the function $f:V\left(J_{m,m}\right)\to \mathrm{????}$ by:

$$\begin{array}{r}f(u)=\{1,2,\dots ,n+3\},f(v)=\varnothing ,\end{array}$$

$$\begin{array}{r}f(x)=\{1,2,\dots ,n+4\},f(w)=\{1,2,\dots ,n+2\},\end{array}$$

$$\begin{array}{r}f(v_i)=\{1,2,\dots ,i\},\text{for}1\le i\le n+1,\end{array}$$

$$\begin{array}{r}f(x_j)=\{1,2,\dots ,m+i+4\},\text{for}1\le j\le n.\end{array}$$

Thus, the induced edge labels are:

$$\begin{array}{r}{f}^{\ast }(ux)=1,f^{\ast }(uv)=0,\end{array}$$

$$\begin{array}{r}{f}^{\ast }(vw)=0,f^{\ast }(uw)=1,\end{array}$$

$$\begin{array}{r}{f}^{\ast }(v{v}_i)=0\text{for}1\le i\le n+1f^{\ast }(wx)=1\end{array}$$

$$\begin{array}{r}{f}^{\ast }(x{x}_j)=1\text{for}1\le j\le n.\end{array}$$

Observe that $e_f\left(0\right)=n+3$ and $e_f\left(1\right)=n+3$. Thus $|e_f\left(0\right)-e_f\left(1\right)|=|n+3-(n+3)|=0$. Hence, if $m-n=1,$ the Jellyfish Graph $J_{m,m}$ is a topological cordial graph.

Case 3: $m-n=2$.

Let $V\left(J_{m,n}\right)=\{u,v,w,x\}\cup \{v_i,x_j|1\le i\le n+2,1\le j\le n\}$. The order of $J_{m,n}$ is $2n+6$. In addition, let $E\left(J_{m,n}\right)=\{uv,vw,uw,wx\}\cup \{v{v}_i,x{x}_j|1\le i\le n+2,1\le j\le n\}.$ The size of $J_{m,n}$ is $2m+7$. Now, let $X=\{1,2,\dots ,2m+5\}$ and assume that $\mathrm{????}=\{\varnothing ,X,\left\{1\right\},\{1,2\},\dots ,\{1,2,\dots ,2m+4\}\}$. By Theorem 1, $\mathrm{????}$ is a topology on $X$. Define the function $f:V\left(J_{m,m}\right)\to \mathrm{????}$ by:

$$\begin{array}{r}f(u)=\{1,2,\dots ,n+4\},f(v)=\varnothing ,\end{array}$$

$$\begin{array}{r}f(x)=\{1,2,\dots ,n+5\},f(w)=\{1,2,\dots ,n+3\},\end{array}$$

$$\begin{array}{r}f(v_i)=\{1,2,\dots ,i\},\text{for}1\le i\le n+2,\end{array}$$

$$\begin{array}{r}f(x_j)=\{1,2,\dots ,n+i+5\},\text{for}1\le j\le n.\end{array}$$

Thus, the induced edge labels are:

$$\begin{array}{r}{f}^{\ast }(ux)=1f^{\ast }(uv)=0,\end{array}$$

$$\begin{array}{r}{f}^{\ast }(vw)=0f^{\ast }(uw)=1,\end{array}$$

$$\begin{array}{r}{f}^{\ast }(v{v}_i)=0\text{for}1\le i\le n+2f^{\ast }(wx)=1\end{array}$$

$$\begin{array}{r}{f}^{\ast }(x{x}_j)=1\text{for}1\le j\le n\end{array}$$

It can be observed that $e_f\left(0\right)=n+4$ and $e_f\left(1\right)=n+3$. Thus $|e_f\left(0\right)-e_f\left(1\right)|=|n+4-(n+3)|=1$. Hence, if $m-n=2,$ the Jellyfish Graph $J_{m,n}$ is a topological cordial graph.

Considering the cases above, we can say that Jellyfish Graph $J_{m,n}$ for $m-n=i$ where $i=0,1,2$ is a topological cordial graph.