Sum Divisor Cordial Labeling of $T_{p}$-Tree Related Graphs

Lourdusamy, A.; Wency, S.; Patrick, F.

doi:10.61091/ars157-01

Abstract

1. Introduction
2. Tp-Tree related graphs
Conflict of Interest

References

Ars Combinatoria

Volume 157
Pages: 3-22

Research article

Sum Divisor Cordial Labeling of $T_{p}$ -Tree Related Graphs

^¹, ^², ^¹

¹Department of Mathematics, St. Xavier’s College (Autonomous),Palayamkottai – 627 002, Tamilnadu, India.

²Research Scholar, Department of Mathematics, Manonmaniam Sundaranar University, Tirunelveli, Tamilnadu, India.

Received: 22/04/2020
Accepted: 20/01/2021
Published: 22/11/2023

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Abstract

A sum divisor cordial labeling of a graph $G$ with vertex set $V (G)$ is a bijection $f$ from $V (G)$ to ${1, 2, \dots, | V (G) |}$ such that an edge $u v$ is assigned the label $1$ if $2$ divides $f (u) + f (v)$ and $0$ otherwise; and the number of edges labeled with $1$ and the number of edges labeled with $0$ differ by at most $1$ . A graph with a sum divisor cordial labeling is called a sum divisor cordial graph. In this paper, we discuss the sum divisor cordial labeling of transformed tree related graphs.

Keywords: Sum divisor cordial labeling, Corona, Transformed tree

1. Introduction

All graphs considered here are simple, finite, connected and undirected. The vertex set and the edge set of a graph are denoted by $V (G)$ and $E (G)$ respectively. We follow the basic notations and terminology of graph theory as in [1]. A labeling of a graph is a map that carries the graph elements to the set of numbers, usually to the set of non-negative or positive integers. If the domain is the set of vertices then the labeling is called vertex labeling. If the domain is the set of edges then the labeling is called edge labeling. If the labels are assigned to both vertices and edges then the labeling is called total labeling. A detailed survey of graph labeling is available in [2]. The concept of cordial labeling was introduced by Cahit in [3].
Lourdusamy et al. introduced the concept of sum divisor cordial labeling in [4]. They prove that paths, combs, stars, complete bipartite, $K_{2} + m K_{1}$ , bistars, jewels, crowns, flowers, gears, subdivisions of stars, the graph obtained from $K_{1, 3}$ by attaching the root of $K_{1, n}$ at each pendent vertex of $K_{1, 3}$ , and the square $B_{n, n}$ are sum divisor cordial graphs. Also they discussed the sum divisor labeling of star related graphs, path related graphs and cycle related graphs in [5,6, 7].
In [8, 9, 10, 11], Sugumaran et al. investigated the behaviour of sum divisor cordial labeling of swastiks, path unions of finite number of copies of swastiks, cycles of $k$ copies of swastiks, when $k$ is odd, jelly fish, Petersen graphs, theta graphs, the fusion of any two vertices in the cycle of swastiks, duplication of any vertex in the cycle of swastiks, the switchings of a central vertex in swastiks, the path unions of two copies of a swastik, the star graph of the theta graphs, the Herschel graph, the fusion of any two adjacent vertices of degree 3 in Herschel graphs, the duplication of any vertex of degree 3 in the Herschel graph, the switching of central vertex in Herschel graph, the path union of two copies of the Herschel graph, $H$ -graph $H_{n}$ , when $n$ is odd, $C_{3} @ K_{1, n}$ , $< F_{n}^{1} Δ F_{n}^{2} >$ and open star of swastik graphs $S (t . S w_{n})$ , when $t$ is odd.
In [12, 13, 14, 15] Sugumaran et al. proved that the following graphs are sum divisor cordial graphs: $H$ -graph $H_{n}$ , when $n$ is even, duplication of all edges of the $H$ -graph $H_{n}$ , when $n$ is even, $H_{n} ⊙ K_{1}$ , $P (r . H_{n})$ , $C (r . H_{n})$ , plus graphs, umbrella graphs, path unions of odd cycles, kites, complete binary trees, drums graph, twigs, fire crackers of the form $P_{n} ⊙ S_{n}$ , where $n$ is even, and the double arrow graph $D A_{m}^{n}$ , where $| m - n | \leq 1$ and $n$ is even. Further results on sum divisor cordial labeling are given in [16, 17].
In this paper, we discuss the sum divisor cordial labeling of transformed tree related graphs like $T \hat{O} P_{n}$ , $T \hat{O} C_{n} (n \equiv 1, 3, 0 (m o d 4))$ , $T \hat{O} K_{1, n}$ , $T \hat{O} \bar{K_{n}}$ , $T \hat{O} Q_{n}$ , $T \tilde{O} C_{n} (n \equiv 1, 3, 0 (m o d 4))$ and $T \tilde{O} Q_{n}$ . We use the following definitions in the subsequent sections.

Definition 1. Let $G = (V (G), E (G))$ be a simple graph and $f : V (G) \to {1, 2, \dots, | V (G) |}$ be a bijection. For each edge $u v$ , assign the label $1$ if $2 | (f (u) + f (v))$ and the label $0$ otherwise. The function $f$ is called a sum divisor cordial labeling if $| e_{f} (1) - e_{f} (0) | \leq 1$ . A graph which admits a sum divisor cordial labeling is called a sum divisor cordial graph.

Definition 2. [18] Let $T$ be a tree and $u_{0}$ and $v_{0}$ be two adjacent vertices in $T$ . Let there be two pendant vertices $u$ and $v$ in $T$ such that the length of $u_{0} - u$ path is equal to the length of $v_{0} - v$ path. If the edge $u_{0} v_{0}$ is deleted from $T$ and $u, v$ are joined by an edge $u v$ , then such a transformation of $T$ is called an elementary parallel transformation (or an ept) and the edge $u_{0} v_{0}$ is called transformable edge.

If by the sequence of ept’s, $T$ can be reduced to a path, then $T$ is called a $T_{p}$ -tree (transformed tree) and such a sequence regarded as a composition of mappings (ept’s) denoted by $P$ , is called a parallel transformation of $T$ . The path, the image of $T$ under $P$ is denoted as $P (T)$ .

Figure 1. A $T_{P}$ -tree and a sequence of two ept’s reducing it to a path

Definition 3. The corona $G_{1} ⊙ G_{2}$ of two graphs $G_{1} (p_{1}, q_{1})$ and $G_{2} (p_{2}, q_{2})$ is defined as the graph obtained by taking one copy of $G_{1}$ and $p_{1}$ copies of $G_{2}$ and joining the $i^{t h}$ vertex of $G_{1}$ with an edge to every vertex in the $i^{t h}$ copy of $G_{2}$ .

Definition 4. [19] Let $G_{1}$ be a graph with $p$ vertices and $G_{2}$ be any graph. A graph $G_{1} \hat{O} G_{2}$ is obtained from $G_{1}$ and $p$ copies of $G_{2}$ by identifying one vertex of $i^{t h}$ copy of $G_{2}$ with $i^{t h}$ vertex of $G_{1}$ .

Definition 5. [19] Let $G_{1}$ be a graph with $p$ vertices and $G_{2}$ be any graph. A graph $G_{1} \tilde{O} G_{2}$ is obtained from $G_{1}$ and $p$ copies of $G_{2}$ by joining one vertex of $i^{t h}$ copy of $G_{2}$ with $i^{t h}$ vertex of $G_{1}$ by an edge.

Theorem 1. [7] Every $T_{p}$ -tree is sum divisor cordial graph.

2. $T_{p}$ -Tree related graphs

Theorem 2. If $T$ be a $T_{p}$ -tree on $m$ vertices, then the graph $T \hat{O} P_{n}$ is sum divisor cordial graph.

Proof. Let $T$ be a $T_{p}$ -tree with $m$ vertices. By the definition of a transformed tree there exists a parallel transformation $P$ of $T$ such that for the path $P (T)$ , we have $(i) V (P (T)) = V (T)$ and $(i i) E (P (T)) = (E (T) - E_{d}) ⋃ E_{p}$ , where $E_{d}$ is the set of edges deleted from $T$ and $E_{p}$ is the set of edges newly added through the sequence $P = (P_{1}, P_{2}, \dots, P_{k})$ of the epts $P$ used to arrive at the path $P (T)$ . Clearly, $E_{d}$ and $E_{p}$ have the same number of edges. Denote the vertices of $P (T)$ successively as $v_{1}, v_{2}, \dots, v_{m}$ starting from one pendant vertex of $P (T)$ right up to the other. Let $u_{1}^{j}, u_{2}^{j}, \dots, u_{n}^{j} (1 \leq j \leq m)$ be the vertices of $j^{t h}$ copy of $P_{n}$ with $u_{1}^{j} = v_{j}$ . Then $V (T \hat{O} P_{n}) = {v_{j}, u_{i}^{j} : 1 \leq i \leq n, 1 \leq j \leq m with u_{1}^{j} = v_{j}}$ and $E (T \hat{O} P_{n}) = E (T) ⋃ {u_{i}^{j} u_{i + 1}^{j} : 1 \leq i \leq n - 1, 1 \leq j \leq m}$ .
Define $f : V (T \hat{O} P_{n}) \to {1, 2, \dots, m n}$ as follows:
Case 1.1] $n$ is even.
For $1 \leq j \leq m$ and $1 \leq i \leq n$ ,
$f (u_{i}^{j}) = {\begin{cases} n (j - 1) + i + 1 & if i \equiv 1 (m o d 4) \\ n (j - 1) + i - 1 & if i \equiv 2 (m o d 4) \\ n (j - 1) + i & if i \equiv 3, 0 (m o d 4) . \end{cases}$

Since $v_{i + t} v_{j - t}$ is an edge in the path $P (T)$ , it follows that $i + t + 1 = j - t$ which implies $j = i + 2 t + 1$ . Therefore, $i$ and $j$ are of opposite parity.
The induced edge label of $v_{i} v_{j}$ is given by $\begin{aligned} f^{*} (v_{i} v_{j}) & = f^{*} (v_{i} v_{i + 2 t + 1}) \\ = 2 | (f (v_{i}) + f (v_{i + 2 t + 1})) \\ = 1. \end{aligned}$ The induced edge label of $v_{i + t} v_{j - t}$ is given by $\begin{aligned} f^{*} (v_{i + t} v_{j - t}) & = f^{*} (v_{i + t} v_{i + t + 1}) \\ = 2 | (f (v_{i + t}) + f (v_{i + t + 1})) \\ = 1. \end{aligned}$ Therefore, $f^{*} (v_{i} v_{j}) = f^{*} (v_{i + t} v_{j - t})$ .
The induced edge labels are as follows:
$f^{*} (v_{j} v_{j + 1}) = 1, 1 \leq j \leq m - 1;$
for $1 \leq i \leq n - 1$ and $1 \leq j \leq m$ ,
$f^{*} (u_{i}^{j} u_{i + 1}^{j}) = {\begin{cases} 0 & if i is odd \\ 1 & if i is even . \end{cases}$
Case 2.1] $n$ is odd.
For $1 \leq j \leq m$ ,
choose ‘if $j \equiv 1, 2 (m o d 4)$ ’,
$f (u_{i}^{j}) = {\begin{cases} n (j - 1) + i & if i \equiv 1, 0 (m o d 4) and 1 \leq i \leq n \\ n (j - 1) + i + 1 & if i \equiv 2 (m o d 4) and 1 \leq i \leq n \\ n (j - 1) + i - 1 & if i \equiv 3 (m o d 4) and 1 \leq i \leq n; \end{cases}$
choose ‘if $j \equiv 3, 0 (m o d 4)$ ’ and $n \equiv 3 (m o d 4)$ ,
$f (u_{i}^{j}) = {\begin{cases} n (j - 1) + i + 1 & if i \equiv 1 (m o d 4) and 1 \leq i \leq n \\ n (j - 1) + i - 1 & if i \equiv 2 (m o d 4) and 1 \leq i \leq n \\ n (j - 1) + i & if i \equiv 3, 0 (m o d 4) and 1 \leq i \leq n; \end{cases}$
choose ‘if $j \equiv 3, 0 (m o d 4)$ ’ and $n \equiv 1 (m o d 4)$ ,
$f (u_{i}^{j}) = {\begin{cases} n (j - 1) + i + 1 & if i \equiv 1 (m o d 4) and 1 \leq i \leq n - 2 \\ n (j - 1) + i - 1 & if i \equiv 2 (m o d 4) and 1 \leq i \leq n - 2 \\ n (j - 1) + i & if i \equiv 3, 0 (m o d 4) and 1 \leq i \leq n - 2 \\ n (j - 1) + i + 1 & if i = n - 1 \\ n (j - 1) + i - 1 & if i = n . \end{cases}$

Since $v_{i + t} v_{j - t}$ is an edge in the path $P (T)$ , it follows that $i + t + 1 = j - t$ which implies $j = i + 2 t + 1$ . Therefore, $i$ and $j$ are of opposite parity.
The induced edge label of $v_{i} v_{j}$ is given by $\begin{aligned} f^{*} (v_{i} v_{j}) & = f^{*} (v_{i} v_{i + 2 t + 1}) \\ = {\begin{cases} 0 & if i is odd \\ 1 & if i is even . \end{cases} \end{aligned}$ The induced edge label of $v_{i + t} v_{j - t}$ is given by $\begin{aligned} f^{*} (v_{i + t} v_{j - t}) & = f^{*} (v_{i + t} v_{i + t + 1}) \\ = {\begin{cases} 0 & if i is odd \\ 1 & if i is even . \end{cases} \end{aligned}$ Therefore, $f^{*} (v_{i} v_{j}) = f^{*} (v_{i + t} v_{j - t})$ .
The induced edge labels are as follows:
$f^{*} (v_{j} v_{j + 1}) = {\begin{cases} 0 & if i is odd and 1 \leq j \leq m - 1 \\ 1 & if i is even and 1 \leq j \leq m - 1; \end{cases}$
for $1 \leq j \leq m$ ,
choose ‘if $j \equiv 1, 2 (m o d 4)$ ’,
$f^{*} (u_{i}^{j} u_{i + 1}^{j}) = {\begin{cases} 1 & if i is odd and 1 \leq i \leq n - 1 \\ 0 & if i is even and 1 \leq i \leq n - 1; \end{cases}$
choose ‘if $j \equiv 3, 0 (m o d 4)$ ’ and $n \equiv 3 (m o d 4)$ ,
$f^{*} (u_{i}^{j} u_{i + 1}^{j}) = {\begin{cases} 0 & if i is odd and 1 \leq i \leq n - 1 \\ 1 & if i is even and 1 \leq i \leq n - 1; \end{cases}$
choose ‘if $j \equiv 3, 0 (m o d 4)$ ’ and $n \equiv 1 (m o d 4)$ ,
$f^{*} (u_{i}^{j} u_{i + 1}^{j}) = {\begin{cases} 0 & if i is odd and 1 \leq i \leq n - 3 \\ 1 & if i is even and 1 \leq i \leq n - 3 \\ 1 & if i = n - 2 \\ 1 & if i = n - 1 . \end{cases}$
In the above two cases,
when $m$ is odd and $n$ is odd,
$e_{f} (0) = e_{f} (1) = \frac{m n - 1}{2}$ ;
when $m$ is odd and $n$ is even,
$e_{f} (0) = ⌈ \frac{m n - 1}{2} ⌉$ and $e_{f} (1) = ⌊ \frac{m n - 1}{2} ⌋$ ;
when $m$ is even and $n$ is odd or even,
$e_{f} (0) = ⌈ \frac{m n - 1}{2} ⌉$ and $e_{f} (1) = ⌊ \frac{m n - 1}{2} ⌋$ .
Clearly $| e_{f} (0) - e_{f} (1) | \leq 1$ . Hence $T \hat{O} P_{n}$ is sum divisor cordial graph.

Example 1. Sum divisor cordial labeling of $T \hat{O} P_{5}$ where $T$ is a $T_{p}$ -tree with 11 vertices is shown in Figure 2.

Figure 2. Sum divisor cordial labeling of $T \hat{O} P_{5}$ where $T$ is a $T_{p}$ -tree with 11 vertices

Theorem 3. If $T$ be a $T_{p}$ -tree on $m$ vertices, then the graph $T \hat{O} C_{n}$ is sum divisor cordial graph if $n \equiv 0, 3, 1 (m o d 4)$ .

Proof. Let $T$ be a $T_{p}$ -tree with $m$ vertices. By the definition of a transformed tree there exists a parallel transformation $P$ of $T$ such that for the path $P (T)$ , we have $(i) V (P (T)) = V (T)$ and $(i i) E (P (T)) = (E (T) - E_{d}) ⋃ E_{p}$ , where $E_{d}$ is the set of edges deleted from $T$ and $E_{p}$ is the set of edges newly added through the sequence $P = (P_{1}, P_{2}, \dots, P_{k})$ of the epts $P$ used to arrive at the path $P (T)$ . Clearly, $E_{d}$ and $E_{p}$ have the same number of edges. Denote the vertices of $P (T)$ successively as $v_{1}, v_{2}, \dots, v_{m}$ starting from one pendant vertex of $P (T)$ right up to the other. Let $u_{1}^{j}, u_{2}^{j}, \dots, u_{n}^{j} (1 \leq j \leq m)$ be the vertices of $j^{t h}$ copy of $C_{n}$ with $u_{1}^{j} = v_{j}$ . Then $V (T \hat{O} C_{n}) = {u_{i}^{j} : 1 \leq i \leq n, 1 \leq j \leq m}$ and $E (T \hat{O} C_{n}) = E (T) ⋃ E (C_{n})$ . Define
$f : V (T \hat{O} C_{n}) \to {1, 2, 3, \dots, m n}$ as follows:
Case 1. $n \equiv 0 (m o d 4)$ .
Choose ‘if $j \equiv 1, 2 (m o d 4)$ ’ and $1 \leq j \leq m$ ,
$f (u_{i}^{j}) = {\begin{cases} n (j - 1) + i & if i \equiv 1, 0 (m o d 4) and 1 \leq i \leq n \\ n (j - 1) + i + 1 & if i \equiv 2 (m o d 4) and 1 \leq i \leq n \\ n (j - 1) + i - 1 & if i \equiv 3 (m o d 4) and 1 \leq i \leq n; \end{cases}$
choose ‘if $j \equiv 3, 0 (m o d 4)$ ’ and $1 \leq j \leq m$ ,
$f (u_{i}^{j}) = {\begin{cases} n (j - 1) + i + 1 & if i \equiv 1 (m o d 4) and 1 \leq i \leq n \\ n (j - 1) + i - 1 & if i \equiv 2 (m o d 4) and 1 \leq i \leq n \\ n (j - 1) + i & if i \equiv 3, 0 (m o d 4) and 1 \leq i \leq n . \end{cases}$

Since $v_{i + t} v_{j - t}$ is an edge in the path $P (T)$ , it follows that $i + t + 1 = j - t$ which implies $j = i + 2 t + 1$ . Therefore, $i$ and $j$ are of opposite parity.
The induced edge label of $v_{i} v_{j}$ is given by $\begin{aligned} f^{*} (v_{i} v_{j}) & = f^{*} (v_{i} v_{i + 2 t + 1}) \\ = {\begin{cases} 1 & if i is odd \\ 0 & if i is even . \end{cases} \end{aligned}$ The induced edge label of $v_{i + t} v_{j - t}$ is given by $\begin{aligned} f^{*} (v_{i + t} v_{j - t}) & = f^{*} (v_{i + t} v_{i + t + 1}) \\ = {\begin{cases} 1 & if i is odd \\ 0 & if i is even . \end{cases} \end{aligned}$ Therefore, $f^{*} (v_{i} v_{j}) = f^{*} (v_{i + t} v_{j - t})$ .
The induced edge labels are as follows:
$f^{*} (v_{j} v_{j + 1}) = {\begin{cases} 1 & if j is odd and 1 \leq j \leq m - 1 \\ 0 & if j is even and 1 \leq j \leq m - 1; \end{cases}$
for $1 \leq j \leq m$ and $1 \leq i \leq n - 1$ ,
$f^{*} (u_{n}^{j} u_{1}^{j}) = {\begin{cases} 0 & if j \equiv 1, 2 (m o d 4) \\ 1 & if j \equiv 3, 0 (m o d 4); \end{cases}$
$f^{*} (u_{i}^{j} u_{i + 1}^{j}) = {\begin{cases} 1 & if i is odd and j \equiv 1, 2 (m o d 4) \\ 0 & if i is even and j \equiv 1, 2 (m o d 4) \\ 0 & if i is odd and j \equiv 3, 0 (m o d 4) \\ 1 & if i is even and j \equiv 3, 0 (m o d 4) . \end{cases}$
Case 2. $n \equiv 3 (m o d 4)$ .
Choose ‘if $j \equiv 1, 3 (m o d 4)$ ’ and $1 \leq j \leq m$ ,
$f (u_{i}^{j}) = {\begin{cases} n (j - 1) + i & if i \equiv 1, 0 (m o d 4) and 1 \leq i \leq n \\ n (j - 1) + i + 1 & if i \equiv 2 (m o d 4) and 1 \leq i \leq n \\ n (j - 1) + i - 1 & if i \equiv 3 (m o d 4) and 1 \leq i \leq n; \end{cases}$
choose ‘if $j \equiv 2, 0 (m o d 4)$ ’ and $1 \leq j \leq m$ ,
$f (u_{i}^{j}) = {\begin{cases} n (j - 1) + i + 1 & if i \equiv 1 (m o d 4) and 1 \leq i \leq n \\ n (j - 1) + i - 1 & if i \equiv 2 (m o d 4) and 1 \leq i \leq n \\ n (j - 1) + i & if i \equiv 3, 0 (m o d 4) and 1 \leq i \leq n . \end{cases}$

Example 2. Sum divisor cordial labeling of $T \hat{O} C_{7}$ where $T$ is a $T_{p}$ -tree with 8 vertices is shown in Figure 3.

Figure 3. Sum divisor cordial labeling of $T \hat{O} C_{7}$ where $T$ is a $T_{p}$ -tree with 8 vertices

Theorem 4. If $T$ be a $T_{p}$ -tree on $m$ vertices, then the graph $T \hat{O} K_{1, n}$ is sum divisor cordial graph.

Let $v_{i} v_{j}$ be a transformed edge in $T$ , $1 \leq i < j \leq m$ and let $P_{1}$ be the ept obtained by deleting the edge $v_{i} v_{j}$ and adding the edge $v_{i + t} v_{j - t}$ where $t$ is the distance of $v_{i}$ from $v_{i + t}$ and the distance of $v_{j}$ from $v_{j - t}$ . Let $P$ be a parallel transformation of $T$ that contains $P_{1}$ as one of the constituent epts. Since $v_{i + t} v_{j - t}$ is an edge in the path $P (T)$ , it follows that $i + t + 1 = j - t$ which implies $j = i + 2 t + 1$ . Therefore, $i$ and $j$ are of opposite parity.
The induced edge label of $v_{i} v_{j}$ is given by $\begin{aligned} f^{*} (v_{i} v_{j}) & = f^{*} (v_{i} v_{i + 2 t + 1}) \\ = 2 | (f (v_{i}) + f (v_{i + 2 t + 1})) \\ = 1. \end{aligned}$ The induced edge label of $v_{i + t} v_{j - t}$ is given by $\begin{aligned} f^{*} (v_{i + t} v_{j - t}) & = f^{*} (v_{i + t} v_{i + t + 1}) \\ = 2 | (f (v_{i + t}) + f (v_{i + t + 1})) \\ = 1. \end{aligned}$ Therefore, $f^{*} (v_{i} v_{j}) = f^{*} (v_{i + t} v_{j - t})$ .
The induced edge labels are as follows:
$f^{*} (v_{j} v_{j + 1}) = 1, 1 \leq j \leq m - 1;$
for $1 \leq j \leq m$ ,
$f^{*} (u_{0}^{j} u_{n}^{j}) = 0$ ;
when $n$ is odd and $1 \leq i \leq n - 1$ ,
$f^{*} (u_{0}^{j} u_{i}^{j}) = {\begin{cases} 1 & if i is odd \\ 0 & if i is even; \end{cases}$
when $n$ is even and $1 \leq i \leq n - 1$ ,
$f^{*} (u_{0}^{j} u_{i}^{j}) = {\begin{cases} 1 & if i is odd and j is odd \\ 0 & if i is even and j is odd \\ 0 & if i is odd and j is even \\ 1 & if i is even and j is even . \end{cases}$
In view of above labeling we get,
when $m$ is odd and $n$ is even,
$e_{f} (0) = e_{f} (1) = \frac{m n + m - 1}{2}$ ;
when $m$ is odd and $n$ is odd,
$e_{f} (0) = ⌈ \frac{m n + m - 1}{2} ⌉$ and $e_{f} (1) = ⌊ \frac{m n + m - 1}{2} ⌋$ ;
when $m$ is even and $n$ is odd or even,
$e_{f} (0) = ⌈ \frac{m n + m - 1}{2} ⌉$ and $e_{f} (1) = ⌊ \frac{m n + m - 1}{2} ⌋$ .
Clearly $| e_{f} (0) - e_{f} (1) | \leq 1$ . Hence $T \hat{O} K_{1, n}$ is sum divisor cordial graph.

Example 3. Sum divisor cordial labeling of $T \hat{O} K_{1, 3}$ where $T$ is a $T_{p}$ -tree with 12 vertices is shown in Figure 4.

Figure 4. Sum divisor cordial labeling of $T \hat{O} K_{1, 3}$ where $T$ is a $T_{p}$ -tree with 12 vertices

Theorem 5. If $T$ be a $T_{p}$ -tree on $m$ vertices, then the graph $T ⊙ \bar{K_{n}}$ is sum divisor cordial graph.

Proof. Let $T$ be a $T_{p}$ -tree with $m$ vertices. By the definition of $T_{p}$ -tree there exists a parallel transformation $P$ of $T$ such that for the path $P (T)$ , we have $(i) V (P (T)) = V (T)$ and $(i i) E (P (T)) = (E (T) - E_{d}) ⋃ E_{p}$ , where $E_{d}$ is the set of edges deleted from $T$ and $E_{p}$ is the set of edges newly added through the sequence $P = (P_{1}, P_{2}, \dots, P_{k})$ of the epts $P$ used to arrive at the path $P (T)$ . Clearly, $E_{d}$ and $E_{p}$ have the same number of edges. Denote the vertices of $P (T)$ successively as $v_{1}, v_{2}, \dots, v_{m}$ starting from one pendant vertex of $P (T)$ right up to the other. Let $u_{1}^{j}, u_{2}^{j}, \dots, u_{n}^{j} (1 \leq j \leq m)$ be the pendant vertices joined with $v_{j} (1 \leq j \leq m)$ by an edge. Then $V (T ⊙ \bar{K_{n}}) = {v_{j}, u_{i}^{j} : 1 \leq i \leq n, 1 \leq j \leq m}$ and $E (T ⊙ \bar{K_{n}}) = E (T) ⋃ {v_{j} u_{i}^{j} : 1 \leq j \leq m, 1 \leq i \leq n}$ .
Define $f : V (T ⊙ \bar{K_{n}}) \to {1, 2, \dots, m n + m}$ as follows:
For $1 \leq j \leq m$ ,
$f (v_{j}) = 2 j - 1;$
$f (u_{n}^{j}) = 2 j;$
$f (u_{i}^{j}) = 2 m + (n - 1) (j - 1) + i, 1 \leq i \leq n - 1.$

Let $v_{i} v_{j}$ be a transformed edge in $T$ , $1 \leq i < j \leq m$ and let $P_{1}$ be the ept obtained by deleting the edge $v_{i} v_{j}$ and adding the edge $v_{i + t} v_{j - t}$ where $t$ is the distance of $v_{i}$ from $v_{i + t}$ and the distance of $v_{j}$ from $v_{j - t}$ . Let $P$ be a parallel transformation of $T$ that contains $P_{1}$ as one of the constituent epts. Since $v_{i + t} v_{j - t}$ is an edge in the path $P (T)$ , it follows that $i + t + 1 = j - t$ which implies $j = i + 2 t + 1$ . Therefore, $i$ and $j$ are of opposite parity.
The induced edge label of $v_{i} v_{j}$ is given by $\begin{aligned} f^{*} (v_{i} v_{j}) & = f^{*} (v_{i} v_{i + 2 t + 1}) \\ = 2 | (f (v_{i}) + f (v_{i + 2 t + 1})) \\ = 1. \end{aligned}$ The induced edge label of $v_{i + t} v_{j - t}$ is given by $\begin{aligned} f^{*} (v_{i + t} v_{j - t}) & = f^{*} (v_{i + t} v_{i + t + 1}) \\ = 2 | (f (v_{i + t}) + f (v_{i + t + 1})) \\ = 1. \end{aligned}$ Therefore, $f^{*} (v_{i} v_{j}) = f^{*} (v_{i + t} v_{j - t})$ .
The induced edge labels are as follows:
$f^{*} (v_{j} v_{j + 1}) = 1, 1 \leq j \leq m - 1;$
for $1 \leq j \leq m$ ,
$f^{*} (v_{j} u_{n}^{j}) = 0;$
when $n$ is odd and $1 \leq i \leq n - 1$ ,
$f^{*} (v_{j} u_{i}^{j}) = {\begin{cases} 1 & if i is odd \\ 0 & if i iseven; \end{cases}$
when $n$ is even and $1 \leq i \leq n - 1$ ,
$f^{*} (v_{j} u_{i}^{j}) = {\begin{cases} 1 & if i is odd and j is odd \\ 0 & if i is even and j is odd \\ 0 & if i is odd and j is even \\ 1 & if i is even and j is even . \end{cases}$
It can be verified that $| e_{f} (0) - e_{f} (1) | \leq 1$ . Hence $T ⊙ \bar{K_{n}}$ is sum divisor cordial graph.

Example 4. Sum divisor cordial labeling of $T ⊙ \bar{K_{4}}$ where $T$ is a $T_{p}$ -tree with 10 vertices is shown in Figure 5.

Sum divisor cordial labeling of T \odot \overline{K_{4}} where T is a T_{p}-tree with 10 vertices — Figure 5. Sum divisor cordial labeling of $T ⊙ \bar{K_{4}}$ where $T$ is a $T_{p}$ -tree with 10 vertices

Theorem 6. If $T$ be a $T_{p}$ -tree on $m$ vertices, then the graph $T \hat{O} Q_{n}$ is sum divisor cordial graph.

Proof. Let $T$ be a $T_{p}$ -tree with $m$ vertices. By the definition of a transformed tree there exists a parallel transformation $P$ of $T$ such that for the path $P (T)$ , we have $(i) V (P (T)) = V (T)$ and $(i i) E (P (T)) = (E (T) - E_{d}) ⋃ E_{p}$ , where $E_{d}$ is the set of edges deleted from $T$ and $E_{p}$ is the set of edges newly added through the sequence $P = (P_{1}, P_{2}, \dots, P_{k})$ of the epts $P$ used to arrive at the path $P (T)$ . Clearly, $E_{d}$ and $E_{p}$ have the same number of edges. Denote the vertices of $P (T)$ successively as $v_{1}, v_{2}, \dots, v_{m}$ starting from one pendant vertex of $P (T)$ right up to the other. Let $u_{1}^{j}, u_{2}^{j}, \dots, u_{n}^{j}, u_{n + 1}^{j} (1 \leq j \leq m)$ be the vertices of $j^{t h}$ copy of $Q_{n}$ with $u_{n + 1}^{j} = v_{j}$ . Then $V (T \hat{O} Q_{n}) = {u_{i}^{j} : 1 \leq i \leq n + 1, 1 \leq j \leq m} ⋃ {x_{i}^{j}, y_{i}^{j} : 1 \leq i \leq n, 1 \leq j \leq m}$ and $E (T \hat{O} Q_{n}) = E (T) ⋃ E (Q_{n})$ . We note that $| V (T \hat{O} Q_{n}) | = 3 n m + m$ and $| E (T \hat{O} Q_{n}) | = 4 m n + m - 1$ . Define $f : V (T \hat{O} Q_{n}) \to {1, 2, \dots, 3 m n + m}$ as follows:
Case 1.1] $m$ is odd.
For $1 \leq j \leq m$ and $1 \leq i \leq n$ ,
$f (u_{i}^{j}) = m + 3 n (j - 1) + 3 i - 2;$
$f (x_{i}^{j}) = m + 3 n (j - 1) + 3 i - 1;$
$f (y_{i}^{j}) = m + 3 n (j - 1) + 3 i;$
$f (v_{j}) = f (u_{n + 1}^{j}) = {\begin{cases} j & if j \equiv 1, 0 (m o d 4) \\ j + 1 & if j \equiv 2 (m o d 4) \\ j - 1 & if j \equiv 3 (m o d 4) . \end{cases}$

Since $v_{i + t} v_{j - t}$ is an edge in the path $P (T)$ , it follows that $i + t + 1 = j - t$ which implies $j = i + 2 t + 1$ . Therefore, $i$ and $j$ are of opposite parity.
The induced edge label of $v_{i} v_{j}$ is given by $\begin{aligned} f^{*} (v_{i} v_{j}) & = f^{*} (v_{i} v_{i + 2 t + 1}) \\ = {\begin{cases} 1 & if i is odd \\ 0 & if i is even . \end{cases} \end{aligned}$ The induced edge label of $v_{i + t} v_{j - t}$ is given by $\begin{aligned} f^{*} (v_{i + t} v_{j - t}) & = f^{*} (v_{i + t} v_{i + t + 1}) \\ = {\begin{cases} 1 & if i is odd \\ 0 & if i is even . \end{cases} \end{aligned}$ Therefore, $f^{*} (v_{i} v_{j}) = f^{*} (v_{i + t} v_{j - t})$ .
The induced edge labels are as follows:
$f^{*} (v_{j} v_{j + 1}) = {\begin{cases} 1 & if i is odd and 1 \leq j \leq m - 1 \\ 0 & if i is even and 1 \leq j \leq m - 1; \end{cases}$
for $1 \leq j \leq m$ ,
$f^{*} (u_{i}^{j} x_{i}^{j}) = 0, 1 \leq i \leq n;$
$f^{*} (u_{i}^{j} y_{i}^{j}) = 1, 1 \leq i \leq n;$
$f^{*} (x_{i}^{j} u_{i + 1}^{j}) = 1, 1 \leq i \leq n - 1;$
$f^{*} (y_{i}^{j} u_{i + 1}^{j}) = 0, 1 \leq i \leq n - 1;$
$f^{*} (x_{n}^{j} v_{j}) = {\begin{cases} 1 & if n is odd and j \equiv 1, 0 (m o d 4) \\ 0 & if n is odd and j \equiv 2, 3 (m o d 4) \\ 0 & if n is even and j \equiv 1, 2 (m o d 4) \\ 1 & if n is even and j \equiv 3, 0 (m o d 4); \end{cases}$
$f^{*} (y_{n}^{j} v_{j}) = {\begin{cases} 0 & if n is odd and j \equiv 1, 0 (m o d 4) \\ 1 & if n is odd and j \equiv 2, 3 (m o d 4) \\ 1 & if n is even and j \equiv 1, 2 (m o d 4) \\ 0 & if n is even and j \equiv 3, 0 (m o d 4) . \end{cases}$
Case 2.1] $m$ is even.
For $1 \leq j \leq m$ and $1 \leq i \leq n$ ,
$f (u_{i}^{j}) = m + 3 n (j - 1) + 3 i - 2;$
$f (x_{i}^{j}) = m + 3 n (j - 1) + 3 i - 1;$
$f (y_{i}^{j}) = m + 3 n (j - 1) + 3 i;$
$f (v_{j}) = f (u_{n + 1}^{j}) = {\begin{cases} j & if j \equiv 1, 2 (m o d 4) \\ j + 1 & if j \equiv 3 (m o d 4) \\ j - 1 & if j \equiv 0 (m o d 4) . \end{cases}$

Since $v_{i + t} v_{j - t}$ is an edge in the path $P (T)$ , it follows that $i + t + 1 = j - t$ which implies $j = i + 2 t + 1$ . Therefore, $i$ and $j$ are of opposite parity.
The induced edge label of $v_{i} v_{j}$ is given by $\begin{aligned} f^{*} (v_{i} v_{j}) & = f^{*} (v_{i} v_{i + 2 t + 1}) \\ = {\begin{cases} 0 & if i is odd \\ 1 & if i is even . \end{cases} \end{aligned}$ The induced edge label of $v_{i + t} v_{j - t}$ is given by $\begin{aligned} f^{*} (v_{i + t} v_{j - t}) & = f^{*} (v_{i + t} v_{i + t + 1}) \\ = {\begin{cases} 0 & if i is odd \\ 1 & if i is even . \end{cases} \end{aligned}$ Therefore, $f^{*} (v_{i} v_{j}) = f^{*} (v_{i + t} v_{j - t})$ .
The induced edge labels are as follows:
$f^{*} (v_{j} v_{j + 1}) = {\begin{cases} 0 & if i is odd and 1 \leq j \leq m - 1 \\ 1 & if i is even and 1 \leq j \leq m - 1; \end{cases}$
for $1 \leq j \leq m$ ,
$f^{*} (u_{i}^{j} x_{i}^{j}) = 0, 1 \leq i \leq n;$
$f^{*} (u_{i}^{j} y_{i}^{j}) = 1, 1 \leq i \leq n;$
$f^{*} (x_{i}^{j} u_{i + 1}^{j}) = 1, 1 \leq i \leq n - 1;$
$f^{*} (y_{i}^{j} u_{i + 1}^{j}) = 0, 1 \leq i \leq n - 1;$
$f^{*} (x_{n}^{j} v_{j}) = {\begin{cases} 0 & if n is odd and j \equiv 1, 2 (m o d 4) \\ 1 & if n is odd and j \equiv 3, 0 (m o d 4) \\ 1 & if n is even and j \equiv 1, 0 (m o d 4) \\ 0 & if n is even and j \equiv 2, 3 (m o d 4); \end{cases}$
$f^{*} (y_{n}^{j} v_{j}) = {\begin{cases} 1 & if n is odd and j \equiv 1, 2 (m o d 4) \\ 0 & if n is odd and j \equiv 3, 0 (m o d 4) \\ 1 & if n is even and j \equiv 2, 3 (m o d 4) \\ 0 & if n is even and j \equiv 1, 0 (m o d 4) . \end{cases}$
In the above two cases,
when $m$ is odd,
$e_{f} (1) = e_{f} (0) = \frac{4 m n + m - 1}{2}$ ;
when $m$ is even,
$e_{f} (1) = ⌊ \frac{4 m n + m - 1}{2} ⌋$ and $e_{f} (0) = ⌈ \frac{4 m n + m - 1}{2} ⌉$ .
It can be verified that $| e_{f} (0) - e_{f} (1) | \leq 1$ . Hence $T \hat{O} Q_{n}$ is sum divisor cordial graph.

Example 5. Sum divisor cordial labeling of $T \hat{O} Q_{2}$ where $T$ is a $T_{p}$ -tree with 8 vertices is shown in Figure $6$ .

Figure 6. Sum divisor cordial labeling of $T \hat{O} Q_{2}$ where $T$ is a $T_{p}$ -tree with 8 vertices

Theorem 7. If $T$ be a $T_{p}$ -tree on $m$ vertices, then the graph $T \tilde{O} C_{n}$ is sum divisor cordial graph if $n \equiv 0, 3, 1 (m o d 4)$ .

Proof. Let $T$ be a $T_{p}$ -tree with $m$ vertices. By the definition of a transformed tree there exists a parallel transformation $P$ of $T$ such that for the path $P (T)$ , we have $(i) V (P (T)) = V (T)$ and $(i i) E (P (T)) = (E (T) - E_{d}) ⋃ E_{p}$ , where $E_{d}$ is the set of edges deleted from $T$ and $E_{p}$ is the set of edges newly added through the sequence $P = (P_{1}, P_{2}, \dots, P_{k})$ of the epts $P$ used to arrive at the path $P (T)$ . Clearly, $E_{d}$ and $E_{p}$ have the same number of edges. Denote the vertices of $P (T)$ successively as $v_{1}, v_{2}, \dots, v_{m}$ starting from one pendant vertex of $P (T)$ right up to the other. Let $u_{1}^{j}, u_{2}^{j}, \dots, u_{n}^{j} (1 \leq j \leq m)$ be the vertices of $j^{t h}$ copy of $C_{n}$ . Then $V (T \tilde{O} C_{n}) = {v_{j}, u_{i}^{j} : 1 \leq i \leq n, 1 \leq j \leq m}$ and $E (T \tilde{O} C_{n}) = E (T) ⋃ E (C_{n}) ⋃ {v_{j} u_{1}^{j} : 1 \leq j \leq m}$ . Define $f : V (T \tilde{O} C_{n}) \to {1, 2, \dots, m n + m}$ as follows:
Case 1. $n \equiv 0 (m o d 4)$ .
$f (v_{j}) = (n + 1) j, 1 \leq j \leq m;$
for $1 \leq j \leq m$ and $1 \leq i \leq n$ ,
$f (u_{i}^{j}) = {\begin{cases} (n + 1) (j - 1) + i & if i \equiv 1, 0 (m o d 4) \\ (n + 1) (j - 1) + i + 1 & if i \equiv 2 (m o d 4) \\ (n + 1) (j - 1) + i - 1 & if i \equiv 3 (m o d 4) . \end{cases}$

Let $v_{i} v_{j}$ be a transformed edge in $T$ , $1 \leq i < j \leq m$ and let $P_{1}$ be the ept obtained by deleting the edge $v_{i} v_{j}$ and adding the edge $v_{i + t} v_{j - t}$ where $t$ is the distance of $v_{i}$ from $v_{i + t}$ and also the distance of $v_{j}$ from $v_{j - t}$ . Let $P$ be a parallel transformation of $T$ that contains $P_{1}$ as one of the constituent epts. Since $v_{i + t} v_{j - t}$ is an edge in the path $P (T)$ , it follows that $i + t + 1 = j - t$ which implies $j = i + 2 t + 1$ . Therefore, $i$ and $j$ are of opposite parity.
The induced edge label of $v_{i} v_{j}$ is given by $\begin{aligned} f^{*} (v_{i} v_{j}) & = f^{*} (v_{i} v_{i + 2 t + 1}) \\ = 0. \end{aligned}$ The induced edge label of $v_{i + t} v_{j - t}$ is given by $\begin{aligned} f^{*} (v_{i + t} v_{j - t}) & = f^{*} (v_{i + t} v_{i + t + 1}) \\ = 0. \end{aligned}$ Therefore, $f^{*} (v_{i} v_{j}) = f^{*} (v_{i + t} v_{j - t})$ .
The induced edge labels are as follows:
$f^{*} (v_{j} v_{j + 1}) = 0, 1 \leq j \leq m - 1;$
$f^{*} (u_{1}^{j} v_{j}) = 1, 1 \leq j \leq m$ ;
$f^{*} (u_{n}^{j} u_{1}^{j}) = 0, 1 \leq j \leq m$ ;
for $1 \leq j \leq m$ ,
$f^{*} (u_{i}^{j} u_{i + 1}^{j}) = {\begin{cases} 1 & if i is odd and 1 \leq i \leq n - 1 \\ 0 & if j is even and 1 \leq i \leq n - 1. \end{cases}$
Case 2. $n \equiv 3 (m o d 4)$ .
$f (v_{j}) = {\begin{cases} (n + 1) j & if j \equiv 1, 2 (m o d 4) and 1 \leq j \leq m \\ (n + 1) (j - 1) + 1 & if j \equiv 3, 0 (m o d 4) and 1 \leq j \leq m; \end{cases}$
choose ‘if $j \equiv 1, 2 (m o d 4)$ ’ and $1 \leq j \leq m$ ,
$f (u_{i}^{j}) = {\begin{cases} (n + 1) (j - 1) + i + 1 & if i \equiv 1 (m o d 4) and 1 \leq i \leq n \\ (n + 1) (j - 1) + i - 1 & if i \equiv 2 (m o d 4) and 1 \leq i \leq n \\ (n + 1) (j - 1) + i & if i \equiv 3, 0 (m o d 4) and 1 \leq i \leq n; \end{cases}$
choose ‘if $j \equiv 3, 0 (m o d 4)$ ’ and $1 \leq j \leq m$ ,
$f (u_{i}^{j}) = {\begin{cases} (n + 1) (j - 1) + i + 2 & if i \equiv 1 (m o d 4) and 1 \leq i \leq n \\ (n + 1) (j - 1) + i & if i \equiv 2 (m o d 4) and 1 \leq i \leq n \\ (n + 1) (j - 1) + i + 1 & if i \equiv 3, 0 (m o d 4) and 1 \leq i \leq n . \end{cases}$

Let $v_{i} v_{j}$ be a transformed edge in $T$ , $1 \leq i < j \leq m$ and let $P_{1}$ be the ept obtained by deleting the edge $v_{i} v_{j}$ and adding the edge $v_{i + t} v_{j - t}$ where $t$ is the distance of $v_{i}$ from $v_{i + t}$ and the distance of $v_{j}$ from $v_{j - t}$ . Let $P$ be a parallel transformation of $T$ that contains $P_{1}$ as one of the constituent epts. Since $v_{i + t} v_{j - t}$ is an edge in the path $P (T)$ , it follows that $i + t + 1 = j - t$ which implies $j = i + 2 t + 1$ . Therefore, $i$ and $j$ are of opposite parity.
The induced edge label of $v_{i} v_{j}$ is given by $\begin{aligned} f^{*} (v_{i} v_{j}) & = f^{*} (v_{i} v_{i + 2 t + 1}) \\ = {\begin{cases} 1 & if i is odd \\ 0 & if i is even . \end{cases} \end{aligned}$ The induced edge label of $v_{i + t} v_{j - t}$ is given by $\begin{aligned} f^{*} (v_{i + t} v_{j - t}) & = f^{*} (v_{i + t} v_{i + t + 1}) \\ = {\begin{cases} 1 & if i is odd \\ 0 & if i is even . \end{cases} \end{aligned}$ Therefore, $f^{*} (v_{i} v_{j}) = f^{*} (v_{i + t} v_{j - t})$ .
The induced edge labels are as follows:
$f^{*} (v_{j} v_{j + 1}) = {\begin{cases} 1 & if i is odd and 1 \leq j \leq m - 1 \\ 0 & if i is even and 1 \leq j \leq m - 1; \end{cases}$
for $1 \leq j \leq m$ ,
$f^{*} (u_{1}^{j} v_{j}) = 1;$
$f^{*} (u_{n}^{j} u_{1}^{j}) = 0;$
$f^{*} (u_{i}^{j} u_{i + 1}^{j}) = {\begin{cases} 0 & if i is odd and 1 \leq i \leq n - 1 \\ 1 & if i is even and 1 \leq i \leq n - 1 . \end{cases}$
Case 3. $n \equiv 1 (m o d 4)$ .
$f (v_{j}) = {\begin{cases} (n + 1) j & if j \equiv 1, 2 (m o d 4) and 1 \leq j \leq m \\ (n + 1) (j - 1) + 1 & if j \equiv 3, 0 (m o d 4) and 1 \leq j \leq m; \end{cases}$
choose ‘if $j \equiv 1, 2 (m o d 4)$ ’ and $1 \leq j \leq m$ ,
$f (u_{i}^{j}) = {\begin{cases} (n + 1) (j - 1) + i & if i \equiv 1, 0 (m o d 4) and 1 \leq i \leq n \\ (n + 1) (j - 1) + i + 1 & if i \equiv 2 (m o d 4) and 1 \leq i \leq n \\ (n + 1) (j - 1) + i - 1 & if i \equiv 3 (m o d 4) and 1 \leq i \leq n; \end{cases}$
choose ‘if $j \equiv 3, 0 (m o d 4)$ ’ and $1 \leq j \leq m$ ,
$f (u_{i}^{j}) = {\begin{cases} (n + 1) (j - 1) + i + 1 & if i \equiv 1, 0 (m o d 4) and 1 \leq i \leq n \\ (n + 1) (j - 1) + i + 2 & if i \equiv 2 (m o d 4) and 1 \leq i \leq n \\ (n + 1) (j - 1) + i & if i \equiv 3 (m o d 4) and 1 \leq i \leq n; \end{cases}$

Let $v_{i} v_{j}$ be a transformed edge in $T$ , $1 \leq i < j \leq m$ and let $P_{1}$ be the ept obtained by deleting the edge $v_{i} v_{j}$ and adding the edge $v_{i + t} v_{j - t}$ where $t$ is the distance of $v_{i}$ from $v_{i + t}$ and the distance of $v_{j}$ from $v_{j - t}$ . Let $P$ be a parallel transformation of $T$ that contains $P_{1}$ as one of the constituent epts. Since $v_{i + t} v_{j - t}$ is an edge in the path $P (T)$ , it follows that $i + t + 1 = j - t$ which implies $j = i + 2 t + 1$ . Therefore, $i$ and $j$ are of opposite parity.
The induced edge label of $v_{i} v_{j}$ is given by $\begin{aligned} f^{*} (v_{i} v_{j}) & = f^{*} (v_{i} v_{i + 2 t + 1}) \\ = {\begin{cases} 1 & if i is odd \\ 0 & if i is even . \end{cases} \end{aligned}$ The induced edge label of $v_{i + t} v_{j - t}$ is given by $\begin{aligned} f^{*} (v_{i + t} v_{j - t}) & = f^{*} (v_{i + t} v_{i + t + 1}) \\ = {\begin{cases} 1 & if i is odd \\ 0 & if i is even . \end{cases} \end{aligned}$ Therefore, $f^{*} (v_{i} v_{j}) = f^{*} (v_{i + t} v_{j - t})$ .
The induced edge labels are as follows:
$f^{*} (v_{j} v_{j + 1}) = {\begin{cases} 1 & if i is odd and 1 \leq j \leq m - 1 \\ 0 & if i is even and 1 \leq j \leq m - 1; \end{cases}$
for $1 \leq j \leq m$ ,
$f^{*} (u_{1}^{j} v_{j}) = 0;$
$f^{*} (u_{n}^{j} u_{1}^{j}) = 1;$
$f^{*} (u_{i}^{j} u_{i + 1}^{j}) = {\begin{cases} 1 & if i is odd and 1 \leq i \leq n - 1 \\ 0 & if i is even and 1 \leq i \leq n - 1 . \end{cases}$
In the above three cases,
when $m$ is odd,
$e_{f} (1) = {\begin{cases} \frac{m n + 2 m - 1}{2} & if n \equiv 1, 3 (m o d 4) \\ ⌈ \frac{m n + 2 m - 1}{2} ⌉ & if n \equiv 0 (m o d 4), \end{cases}$
$e_{f} (0) = {\begin{cases} \frac{m n + 2 m - 1}{2} & if n \equiv 1, 3 (m o d 4) \\ ⌊ \frac{m n + 2 m - 1}{2} ⌋ & if n \equiv 0 (m o d 4); \end{cases}$
when $m$ is even and $n \equiv 1, 3, 0 (m o d 4)$ ,
$e_{f} (1) = ⌈ \frac{m n + 2 m - 1}{2} ⌉$ and $e_{f} (0) = ⌊ \frac{m n + 2 m - 1}{2} ⌋$ .
It can be verified that $| e_{f} (0) - e_{f} (1) | \leq 1$ . Hence $T \tilde{O} C_{n}$ is sum divisor cordial graph.

Example 6. Sum divisor cordial labeling of $T \tilde{O} C_{5}$ where $T$ is a $T_{p}$ -tree with 8 vertices is shown in Figure $7$ .

Figure 7. Sum divisor cordial labeling of $T \tilde{O} C_{5}$ where $T$ is a $T_{p}$ -tree with 8 vertices

Theorem 8. If $T$ be a $T_{p}$ -tree on $m$ vertices, then the graph $T \tilde{O} Q_{n}$ is sum divisor cordial graph.

Proof. Let $T$ be a $T_{p}$ -tree with $m$ vertices. By the definition of a transformed tree there exists a parallel transformation $P$ of $T$ such that for the path $P (T)$ , we have $(i) V (P (T)) = V (T)$ and $(i i) E (P (T)) = (E (T) - E_{d}) ⋃ E_{p}$ , where $E_{d}$ is the set of edges deleted from $T$ and $E_{p}$ is the set of edges newly added through the sequence $P = (P_{1}, P_{2}, \dots, P_{k})$ of the epts $P$ used to arrive at the path $P (T)$ . Clearly, $E_{d}$ and $E_{p}$ have the same number of edges. Denote the vertices of $P (T)$ successively as $v_{1}, v_{2}, \dots, v_{m}$ starting from one pendant vertex of $P (T)$ right up to the other. Let $u_{1}^{j}, u_{2}^{j}, \dots, u_{n}^{j}, u_{n + 1}^{j} (1 \leq j \leq m)$ be the vertices of $j^{t h}$ copy of $Q_{n}$ . Then $V (T \tilde{O} Q_{n}) = {v_{j}, u_{i}^{j} : 1 \leq i \leq n + 1, 1 \leq j \leq m} ⋃ {x_{i}^{j}, y_{i}^{j} : 1 \leq i \leq n, 1 \leq j \leq m}$ and $E (T \tilde{O} Q_{n}) = E (T) ⋃ E (Q_{n}) ⋃ {v_{j} u_{n + 1}^{j} : 1 \leq j \leq m}$ . We note that $| V (T \tilde{O} Q_{n}) | = m (3 n + 2)$ and $| E (T \tilde{O} Q_{n}) | = 4 m n + 2 m - 1$ . Define $f : V (T \tilde{O} Q_{n}) \to {1, 2, \dots, m (3 n + 2)}$ as follows:
Case 1.1] $n$ is odd.
$f (v_{j}) = (3 n + 2) j, 1 \leq j \leq m;$
for $1 \leq j \leq m$ and $1 \leq i \leq n + 1$ ,
$f (u_{i}^{j}) = {\begin{cases} (3 n + 2) (j - 1) + 3 i - 1 & if i is odd \\ (3 n + 2) (j - 1) + 3 i - 3 & if i is even; \end{cases}$
for $1 \leq j \leq m$ and $1 \leq i \leq n$ ,
$f (x_{i}^{j}) = {\begin{cases} (3 n + 2) (j - 1) + 3 i - 2 & if i is odd \\ (3 n + 2) (j - 1) + 3 i - 1 & if i is even; \end{cases}$
$f (y_{i}^{j}) = {\begin{cases} (3 n + 2) (j - 1) + 3 i + 1 & if i is odd \\ (3 n + 2) (j - 1) + 3 i & if i is even . \end{cases}$

Example 7. Sum divisor cordial labeling of $T \tilde{O} Q_{2}$ where $T$ is a $T_{p}$ -tree with 8 vertices is shown in Figure 8.

Figure 8. Sum divisor cordial labeling of $T \tilde{O} Q_{2}$ where $T$ is a $T_{p}$ -tree with 8 vertices

Conflict of Interest

The authors declare no conflict of interests.

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Gallian, J.A., 2018. A dynamic survey of graph labeling. Electronic Journal of combinatorics, 1(DynamicSurveys), p.DS6.[Google Scholor]
Cahit, I., 1987. Cordial graphs-a weaker version of graceful and harmonious graphs. Ars combinatoria, 23, pp.201-207.[Google Scholor]
Lourdusamy, A. and Patrick, F., 2016. Sum divisor cordial graphs. Proyecciones Journal of Mathematics, 35(1), pp.119-136.[Google Scholor]
Lourdusamy, A. and Patrick, F., 2019. Sum divisor cordial labeling for path and cycle related graphs. Journal of Prime Research in Mathematics, 15, pp.101-114.[Google Scholor]
Lourdusamy, A. and Patrick, F., 2016. Sum divisor cordial labeling for star and ladder related graphs. Proyecciones Journal of Mathematics, 35(4), pp.437-455.[Google Scholor]
Lourdusamy, A. and Patrick, F., 2022. Several Results on Sum Divisor Cordial Graph. Jordan Journal of Mathematics and Statistics (JJMS), 15(3B), pp.643-660.[Google Scholor]
Sugumaran, A. and Rajesh, K., 2017. Some new results on sum divisor cordial graphs. Annals of pure and applied mathematics, 14(1), pp.45-52.[Google Scholor]
Sugumaran, A. and Rajesh, K., 2017. Sum divisor cordial labeling of Theta graph. Annals of pure and applied mathematics, 14(2), pp.313-320.[Google Scholor]
Sugumaran, A. and Rajesh, K., 2017. Sum divisor cordial labeling of Herschel graph. Annals of pure and applied mathematics, 14(3), pp.465-472.[Google Scholor]
Sugumaran, A. and Rajesh, K., 2017. Further results on sum divisor cordial labeling. Journal of computer and mathematical sciences, 8(12), pp.802-807.[Google Scholor]
Sugumaran, A. and Rajesh, K., 2018. Some graph operations on sum divisor cordial labeling related to $H$ -graph. International journal of statistics and applied mathematics, 3(1), pp.389-395.[Google Scholor]
Sugumaran, A. and Rajesh, K., 2018. Sum divisor cordial labeling on some special graphs. International Journal of Mathematics and its applications, 6(2-B), pp.347-353.[Google Scholor]
Sugumaran, A. and Rajesh, K., 2019. Sum divisor cordial labeling on some classes of graphs. Journal of Computer and Mathematical Sciences, 10(2), pp.316-325.[Google Scholor]
Sugumaran, A. and Rajesh, K., 2019. Extended results on sum divisor cordial labeling. Proyecciones Journal of Mathematics, 38(4), pp.653-663.[Google Scholor]
Raj, P.L.R. and Surya, S.H., 2016. Some new families of sum divisor cordial graphs. International Journal of Mathematics Trends and Technology, 40(2), pp.175-179.[Google Scholor]
Adalja, D.G. and Ghodasara, G.V., 2018. Sum divisor cordial labeling of snakes related graphs. Journal of Computer and Mathematical Sciences, 9(7), pp.754-768.[Google Scholor]
Hegde, S.M. and Shetty, S., 2002. On graceful trees. Applied Mathematics E-Notes, 2, 192-197.[Google Scholor]
Lourdusamy, A. and Patrick, F., 2019. Even vertex equitable even labeling for corona and $T_{p}$ -tree related graphs. Utilitas Mathematica, 110, 223-242.[Google Scholor]

[1] Harary, F., 1972. Graph Theory, Addison-wesley, Reading, Mass.

[2] Gallian, J.A., 2018. A dynamic survey of graph labeling. Electronic Journal of combinatorics, 1(DynamicSurveys), p.DS6.[Google Scholor]

[3] Cahit, I., 1987. Cordial graphs-a weaker version of graceful and harmonious graphs. Ars combinatoria, 23, pp.201-207.[Google Scholor]

[4] Lourdusamy, A. and Patrick, F., 2016. Sum divisor cordial graphs. Proyecciones Journal of Mathematics, 35(1), pp.119-136.[Google Scholor]

[5] Lourdusamy, A. and Patrick, F., 2019. Sum divisor cordial labeling for path and cycle related graphs. Journal of Prime Research in Mathematics, 15, pp.101-114.[Google Scholor]

[6] Lourdusamy, A. and Patrick, F., 2016. Sum divisor cordial labeling for star and ladder related graphs. Proyecciones Journal of Mathematics, 35(4), pp.437-455.[Google Scholor]

[7] Lourdusamy, A. and Patrick, F., 2022. Several Results on Sum Divisor Cordial Graph. Jordan Journal of Mathematics and Statistics (JJMS), 15(3B), pp.643-660.[Google Scholor]

[8] Sugumaran, A. and Rajesh, K., 2017. Some new results on sum divisor cordial graphs. Annals of pure and applied mathematics, 14(1), pp.45-52.[Google Scholor]

[9] Sugumaran, A. and Rajesh, K., 2017. Sum divisor cordial labeling of Theta graph. Annals of pure and applied mathematics, 14(2), pp.313-320.[Google Scholor]

[10] Sugumaran, A. and Rajesh, K., 2017. Sum divisor cordial labeling of Herschel graph. Annals of pure and applied mathematics, 14(3), pp.465-472.[Google Scholor]

[11] Sugumaran, A. and Rajesh, K., 2017. Further results on sum divisor cordial labeling. Journal of computer and mathematical sciences, 8(12), pp.802-807.[Google Scholor]

[12] Sugumaran, A. and Rajesh, K., 2018. Some graph operations on sum divisor cordial labeling related to $H$ -graph. International journal of statistics and applied mathematics, 3(1), pp.389-395.[Google Scholor]

[13] Sugumaran, A. and Rajesh, K., 2018. Sum divisor cordial labeling on some special graphs. International Journal of Mathematics and its applications, 6(2-B), pp.347-353.[Google Scholor]

[14] Sugumaran, A. and Rajesh, K., 2019. Sum divisor cordial labeling on some classes of graphs. Journal of Computer and Mathematical Sciences, 10(2), pp.316-325.[Google Scholor]

[15] Sugumaran, A. and Rajesh, K., 2019. Extended results on sum divisor cordial labeling. Proyecciones Journal of Mathematics, 38(4), pp.653-663.[Google Scholor]

[16] Raj, P.L.R. and Surya, S.H., 2016. Some new families of sum divisor cordial graphs. International Journal of Mathematics Trends and Technology, 40(2), pp.175-179.[Google Scholor]

[17] Adalja, D.G. and Ghodasara, G.V., 2018. Sum divisor cordial labeling of snakes related graphs. Journal of Computer and Mathematical Sciences, 9(7), pp.754-768.[Google Scholor]

[18] Hegde, S.M. and Shetty, S., 2002. On graceful trees. Applied Mathematics E-Notes, 2, 192-197.[Google Scholor]

[19] Lourdusamy, A. and Patrick, F., 2019. Even vertex equitable even labeling for corona and $T_{p}$ -tree related graphs. Utilitas Mathematica, 110, 223-242.[Google Scholor]

Contents

Ars Combinatoria