On Tades of Disjoint Union of Some Graphs

Lourdusamy,  A.; Beaula, F.; Patrick, F.

doi:/10.61091/ars157-09

Abstract

References

Ars Combinatoria

Volume 157
Pages: 89-94

Research article

On Tades of Disjoint Union of Some Graphs

^¹, ^², ^¹

¹Department of Mathematics, St. Xavier’s College (Autonomous), Palayamkottai-627002, India.

²Center: PG and Research Department of Mathematics, St. Xavier’s College (Autonomous), Palayamkottai-627002, Manonmaniam Sundaranar University, Abisekapatti-627012, Tamilnadu, India.

Received: 03/12/2023
Accepted: 21/12/2023
Published: 25/12/2023

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Abstract

Consider a total labeling $ξ$ of a graph $G$ . For every two different edges $e$ and $f$ of $G$ , let $w t (e) \neq w t (f)$ where weight of $e = x y$ is defined as $w t (e) = | ξ (e) - ξ (x) - ξ (y) |$ . Then $ξ$ is called edge irregular total absolute difference $k$ -labeling of $G$ . Let $k$ be the minimum integer for which there is a graph $G$ with edge irregular total absolute difference labeling. This $k$ is called the total absolute difference edge irregularity strength of the graph $G$ , denoted $t a d e s (G)$ . We compute $t a d e s$ of $S C_{n}$ , disjoint union of grid and zigzag graph.

Keywords: Total absolute difference edge irregularity strength, Edge irregularity strength, Zigzag graph, Grid graph, Staircase graph

1. Introduction and Motivation

Chartrand et al. [1] introduced irregularity strength and irregular assignments of a graph in 1988. The other kind of total labeling the total edge irregularity strength of a graph was studied by Baca et al. [2]: Consider a graph $G = (V, E)$ . A labeling $ξ : V ⋃ E \to {1, 2, \dots, k}$ is called an edge irregular total $k$ -labeling if for every pair of distinct edges $u v$ and $x y$ , $ξ (u) + ξ (v) + ξ (u v) \neq ξ (x) + ξ (y) + ξ (x y)$ . If a graph $G$ admits an edge irregular total $k$ -labeling and $k$ is minimum then $G$ is said to have a total edge irrgularity strength denoted by $t e s (G)$ . The results about the $t e s (G)$ can be found in [3,4,5,6,7,8,9,10].

Ramalakshmi and Kathiresan introduced the total absolute difference edge irregularity strength of graphs to lower edge weights, using $t e s (G)$ and graceful labeling. Consider a total labeling $ξ$ of a graph $G$ . For every two different edges $e$ and $f$ of $G$ , let $w t (e) \neq w t (f)$ where weight of $e = x y$ is defined as $w t (e) = | ξ (e) - ξ (x) - ξ (y) |$ . Then $ξ$ is called edge irregular total absolute difference $k$ -labeling of $G$ . Let $k$ be the minimum integer for which there is a graph $G$ with edge irregular total absolute difference labeling. This $k$ is called the total absolute difference edge irregularity strength of the graph $G$ , denoted $t a d e s (G)$ .

Lourdusamy et al. [11] determined the total absolute difference edge irregular strength for snake related graphs, wheel related graphs, lotus inside the circle and double fan graph. Also, they obtained the $t a d e s$ of $T_{p}$ -tree related graphs [12]. Lourdusamy et al. [13] discussed the $t a d e s$ of super subdivision of certain families of graphs and corona graphs. Also, they obtained the $t a d e s$ of transformed tree and path related graphs [14]. Here, we discuss the $t a d e s$ of staircase graph, disjoint union of zigzag and grid graphs.

Theorem 1. [15] For a graph $G = (V, E)$ , we have $⌈ \frac{| E |}{2} ⌉ \leq t a d e s (G) \leq | E | + 1$ .

2. Main Results

In this section, we compute the exact value of total absolute difference edge irregularity strength of staircase graph.

Theorem 2. For $S C_{n}$ , the total absolute difference edge irregularity strength is $t a d e s (S C_{n}) = ⌈ \frac{n (n + 3)}{2} ⌉$ .

Proof. Let $k = ⌈ \frac{n (n + 3)}{2} ⌉$ . Let $V (S C_{n}) = {a_{r, s} : r = 0, 1, 0 \leq s \leq n} \cup {a_{r, s} : 2 \leq r \leq n, r - 1 \leq s \leq n}$ and $E (S C_{n}) = {a_{r, s} a_{r + 1, s} : r = 0, 0 \leq s \leq n} \cup {a_{r, s} a_{r + 1, s} : 1 \leq r \leq n - 1, r \leq s \leq n} \cup {a_{r, s} a_{r, s + 1} : r = 0, 1, 0 \leq s \leq n - 1} \cup {a_{r, s} a_{r, s + 1} : 2 \leq r \leq n, r - 1 \leq s \leq n - 1}$ . Note that $| V (S C_{n}) | = \frac{1}{2} (n + 1) (n + 2) + n$ and $| E (S C_{n}) | = n (n + 3)$ .

From Theorem 1, $t a d e s (S C_{n}) \geq k$ . To complete the proof we show that $t a d e s (S C_{n}) \leq k$ . We define a, $k$ -labeling $ξ : V (S C_{n}) \cup E (S C_{n}) \to {1, 2, \dots k}$ as follows:
$ξ (a_{r, 0}) = 1, r = 0, 1;$
For $1 \leq s \leq n$
$ξ (a_{0, s}) = ⌈ \frac{s^{2} + 3 s}{2} ⌉ - ⌊ \frac{s}{2} ⌋;$
Case 1. $s$ is odd
Let $1 \leq s \leq n$ and $s$ is odd .
Fix $ξ (a_{1, s}) = ⌈ \frac{s^{2} + 3 s}{2} ⌉ - ⌊ \frac{s}{2} ⌋ .$
Let $2 \leq r \leq n, r - 1 \leq s \leq n$ and $s$ is odd.
Fix $ξ (a_{r, s}) = {\begin{cases} ⌈ \frac{s^{2} + 3 s}{2} ⌉ - ⌊ \frac{s}{2} ⌋ + \frac{r - 1}{2} & if r is odd \\ ⌈ \frac{s^{2} + 3 s}{2} ⌉ - ⌊ \frac{s}{2} ⌋ + \frac{r}{2} & if r is even; \end{cases}$
Case 2. $s$ is even
Let $1 \leq s \leq n$ and $s$ is even.
Fix $ξ (a_{1, s}) = ⌈ \frac{s^{2} + 3 s}{2} ⌉ - ⌊ \frac{s}{2} ⌋ + 1.$
Let $2 \leq r \leq n, r - 1 \leq s \leq n$ and $s$ is even.
Fix $ξ (a_{r, s}) = {\begin{cases} ⌈ \frac{s^{2} + 3 s}{2} ⌉ - ⌊ \frac{s}{2} ⌋ + \frac{r + 1}{2} & if r is odd \\ ⌈ \frac{s^{2} + 3 s}{2} ⌉ - ⌊ \frac{s}{2} ⌋ + \frac{r}{2} & if r is even . \end{cases}$
We fix the edge labels as follows:
$ξ (a_{0, 0} a_{1, 0}) = 2$ ;
$ξ (a_{0, 0} a_{0, 1}) = 2$ ;
$ξ (a_{1, 0} a_{1, 1}) = 1$ ;
$ξ (a_{0, s} a_{1, s}) = 1$ , for $1 \leq s \leq n$ ;
$ξ (a_{r, s} a_{r + 1, s}) = 1$ , for $1 \leq r \leq n - 1$ and $r \leq s \leq n$ ;
$ξ (a_{r, s} a_{r, s + 1}) = 1$ , for $r = 0, 1$ and $1 \leq s \leq n - 1$ ;
$ξ (a_{r, s} a_{r, s + 1}) = 1$ , for $2 \leq r \leq n$ and $r - 1 \leq s \leq n - 1$ .
We then have the weight of the edges as follows:
$w t (a_{0, 0} a_{1, 0}) = 0$ ;
$w t (a_{0, 0} a_{0, 1}) = 1$ ;
$w t (a_{1, 0} a_{1, 1}) = 2$ ;
$w t (a_{0, s} a_{1, s}) = s^{2} + 2 s$ for $1 \leq s \leq n$ ;
$w t (a_{r, s} a_{r + 1, s}) = s^{2} + 2 s + r$ , for $1 \leq r \leq n - 1$ and $r \leq s \leq n$ ;
$w t (a_{r, s} a_{r, s + 1}) = s^{2} + 3 s + 1$ , for $r = 0, 1$ and $1 \leq s \leq n - 1$ ;
$w t (a_{r, s} a_{r, s + 1}) = s^{2} + 3 s + r + 1$ , for $2 \leq r \leq n$ and $r - 1 \leq s \leq n - 1$ .
Hence $ξ$ is total absolute difference edge irregular $k$ -labeling with $k = ⌈ \frac{n (n + 3)}{2} ⌉$ as the weights for the edges are different.

3. Disjoint Union of Zigzag Graph

In this section, we compute the exact value of total absolute difference edge irregularity strength of disjoint union of zigzag graphs $⋃_{j = 1}^{p} Z_{n_{j}}^{m_{j}}$ with $n_{j} \geq 2$ and $m_{j} \geq 2$ .

Theorem 3. For any integer $n_{j} \geq 2$ , $m_{j} \geq 2$ , $t a d e s (⋃_{j = 1}^{p} Z_{n_{j}}^{m_{j}}) = ⌈ \frac{\sum_{j = 1}^{p} (n_{j} - 1) (2 m_{j} - 1)}{2} ⌉$ .

Proof. Let $k = ⌈ \frac{\sum_{j = 1}^{p} (n_{j} - 1) (2 m_{j} - 1)}{2} ⌉$ . The disjoint union $⋃_{j = 1}^{p} Z_{n_{j}}^{m_{j}}$ of zigzag graphs $Z_{n}^{m}$ is defined to be a graph with vertex set $V (⋃_{j = 1}^{p} Z_{n_{j}}^{m_{j}}) = {a_{i, s}^{j} : 1 \leq i \leq n_{j}, 1 \leq s \leq m_{j}, 1 \leq j \leq p}$ and the edge set $E (⋃_{j = 1}^{p} Z_{n_{j}}^{m_{j}}) = {a_{i, s}^{j} a_{i + 1, s}^{j} : 1 \leq i \leq n_{j} - 1, 1 \leq s \leq m_{j}, 1 \leq j \leq p} ⋃ {a_{i, s}^{j} a_{i - 1, s + 1}^{j} : 1 \leq i \leq n_{j}, 1 \leq s \leq m_{j} - 1, 1 \leq j \leq p}$ . The disjoint union of zigzag graphs $⋃_{j = 1}^{p} Z_{n_{j}}^{m_{j}}$ has $\sum_{j = 1}^{p} n_{j} m_{j}$ vertices and $\sum_{j = 1}^{p} (n_{j} - 1) (2 m_{j} - 1)$ edges. Based on Theorem 1, we have $t a d e s (⋃_{j = 1}^{p} Z_{n_{j}}^{m_{j}}) \geq ⌈ \frac{\sum_{j = 1}^{p} (n_{j} - 1) (2 m_{j} - 1)}{2} ⌉$ .

Figure 2. $Z_{5}^{3} ⋃ Z_{4}^{3} ⋃ \dots ⋃ Z_{3}^{4}$

We define $ξ$ as follows:
for $1 \leq i \leq n_{j}$ , $1 \leq s \leq m_{j}$ and $1 \leq j \leq p - 1$ ,
$ξ (a_{i, s}^{j}) = ⌈ \frac{r}{2} ⌉ + (s - 1) (n_{j} - 1) + ⌊ \frac{i + \frac{1}{2} ((- 1)^{r} + 1)}{2} ⌋$ where $r = \sum_{q = 1}^{j - 1} (n_{q} - 1) (2 m_{q} - 1)$ ;
for $1 \leq i \leq n_{p}$ , $1 \leq s \leq m_{p} - 1$ and $r = \sum_{q = 1}^{p - 1} (n_{q} - 1) (2 m_{q} - 1)$ ,
$ξ (a_{i, s}^{p}) = ⌈ \frac{r}{2} ⌉ + (s - 1) (n_{p} - 1) + ⌊ \frac{i + \frac{1}{2} ((- 1)^{r} + 1)}{2} ⌋$ ;
$ξ (a_{i, m_{p}}^{p}) = {\begin{cases} ⌈ \frac{r}{2} ⌉ + (m_{p} - 1) (n_{p} - 1) + ⌊ \frac{i + \frac{1}{2} ((- 1)^{r} + 1)}{2} ⌋ & if 1 \leq i \leq n_{p} - 1 \\ k & if i = n_{p}; \end{cases}$
$ξ (a_{i, s}^{j} a_{i + 1, s}^{j}) = 2$ , for $1 \leq i \leq n_{j} - 1$ , $1 \leq s \leq m_{j}$ and $1 \leq j \leq p - 1$ ;
$ξ (a_{i, s}^{p} a_{i + 1, s}^{p} = 2$ , for $1 \leq i \leq n_{p} - 1$ , $1 \leq s \leq m_{p} - 1$ ;
$ξ (a_{i, m_{p}}^{p} a_{i + 1, m_{p}}^{p}) = 2$ , for $1 \leq i \leq n_{p} - 2$ ;
$ξ (a_{n_{p - 1}, m_{p}}^{p} a_{n_{p}, m_{p}}^{p}) = {\begin{cases} 1 & if | E (⋃_{j = 1}^{p} Z_{n_{j}}^{m_{j}}) | iseven \\ 2 & if | E (⋃_{j = 1}^{p} Z_{n_{j}}^{m_{j}}) | is odd \end{cases};$
$ξ (a_{i - 1, s + 1}^{j} a_{i, s}^{j}) = 2$ , for $2 \leq i \leq n_{j}$ , $1 \leq s \leq m_{j} - 1$ and $1 \leq j \leq p .$
We now arrive at the weight of the edges:
for $1 \leq i \leq n_{j} - 1$ , $1 \leq s \leq m_{j}$ , $1 \leq j \leq p$ and $r = \sum_{q = 1}^{j - 1} (n_{q} - 1) (2 m_{q} - 1)$ ,
$w t (a_{i, s}^{j} a_{i + 1, s}^{j}) = 2 ⌈ \frac{r}{2} ⌉ + 2 (s - 1) (n_{j} - 1) + i + \frac{1}{2} ((- 1)^{r} + 1) - 2;$
for $1 \leq i \leq n_{j}$ , $1 \leq s \leq m_{j} - 1$ , $1 \leq j \leq p$ and $r = \sum_{q = 1}^{j - 1} (n_{q} - 1) (2 m_{q} - 1)$ ,
$w t (a_{i, s}^{j} a_{i - 1, s + 1}^{j}) = 2 ⌈ \frac{r}{2} ⌉ + (2 s - 1) (n_{j} - 1) + i + \frac{1}{2} ((- 1)^{r} - 1) - 3.$
It is clear that, the labels for vertices and edges receive values are not more than $k$ . Also we see that the weights for the edges are all distinct. Hence $t a d e s (⋃_{j = 1}^{p} Z_{n_{j}}^{m_{j}}) = ⌈ \frac{\sum_{j = 1}^{p} (n_{j} - 1) (2 m_{j} - 1)}{2} ⌉$ .

Illustration for $t a d e s$ of $Z_{5}^{4} ⋃ Z_{6}^{5} ⋃ Z_{4}^{6} ⋃ Z_{3}^{7}$ is shown in Figure 3.

Figure 3. $Z_{5}^{4} ⋃ Z_{6}^{5} ⋃ Z_{4}^{6} ⋃ Z_{3}^{7}$

4. Disjoint Union of Grid Graph

In this section, we compute the exact value of total absolute difference edge irregularity strength of disjoint union of grid graphs $⋃_{j = 1}^{p} Z_{n_{j}}^{m_{j}}$ with $n_{j}, m_{j} \geq 2$ .

Theorem 4. For any integer $n_{j}, m_{j} \geq 2$ and $1 \leq j \leq p$ , $t a d e s (⋃_{j = 1}^{p} G_{n_{j}, m_{j}}) = ⌈ \frac{\sum_{j = 1}^{p} (2 n_{j} m_{j} - m_{j} - n_{j})}{2} ⌉$ .

Proof. Let $k = ⌈ \frac{\sum_{j = 1}^{p} (2 n_{j} m_{j} - m_{j} - n_{j})}{2} ⌉$ . We define disjoint union $⋃_{j = 1}^{p} G_{n_{j}, m_{j}}$ of grid graphs $G_{n, m}$ as follows:
Let $V (⋃_{j = 1}^{p} G_{n_{j}, m_{j}}) = {a_{i, s}^{j} : 1 \leq i \leq n_{j}, 1 \leq s \leq m_{j}, 1 \leq j \leq p}$ . Let $E (⋃_{j = 1}^{p} G_{n_{j}, m_{j}}) = {a_{i, s}^{j}, a_{i + 1, s}^{j} : 1 \leq i \leq n_{j} - 1, 1 \leq s \leq m_{j}, 1 \leq j \leq p} ⋃ {a_{i, s}^{j}, a_{i, s + 1}^{j} : 1 \leq i \leq n_{j}, 1 \leq s \leq m_{j - 1}, 1 \leq j \leq p}$ . From Theorem 1, $t a d e s (⋃_{j = 1}^{p} G_{n_{j}, m_{j}}) \geq ⌈ \sum_{j = 1}^{p} \frac{(2 n_{j} m_{j} - m_{j} - n_{j})}{2} ⌉$ . Now we prove the converse part.

Figure 4. $G_{5, 3} ⋃ G_{4, 3} ⋃ \dots ⋃ G_{3, 4}$

Let us define $ξ : V (⋃_{j = 1}^{p} G_{n_{j}, m_{j}}) \cup E (⋃_{j = 1}^{p} G_{n_{j}, m_{j}}) \to {1, 2, \dots, ⌈ \frac{\sum_{j = 1}^{p} (2 n_{j} m_{j} - m_{j} - n_{j})}{2} ⌉}$ as follows:
for $1 \leq i \leq n_{j}$ , $1 \leq s \leq m_{j}$ and $1 \leq j \leq p - 1$ ,
$ξ (a_{i, s}^{j}) = {\begin{cases} ⌈ \frac{t}{2} ⌉ + \frac{s - 1}{2} (2 n_{j} - 1) + ⌊ \frac{i + \frac{1}{2} ((- 1)^{t} + 1)}{2} ⌋ & if s is odd \\ ⌈ \frac{t}{2} ⌉ + n_{j} (s - 1) - \frac{s}{2} + ⌈ \frac{i + \frac{1}{2} ((- 1)^{t} + 1)}{2} ⌉ & if s is even; \end{cases}$
where $t = \sum_{q = 1}^{j - 1} 2 n_{q} m_{q} - m_{q} - n_{q}$
for $1 \leq i \leq n_{p}, 1 \leq s \leq m_{p} - 1$ and $t = \sum_{q = 1}^{p - 1} 2 n_{q} m_{q} - m_{q} - n_{q}$
$ξ (a_{i, s}^{p}) = {\begin{cases} ⌈ \frac{t}{2} ⌉ + \frac{s - 1}{2} (2 n_{p} - 1) + ⌊ \frac{i + \frac{1}{2} ((- 1)^{t} + 1)}{2} ⌋ & if s is odd \\ ⌈ \frac{t}{2} ⌉ + n_{p} (s - 1) - \frac{s}{2} + ⌈ \frac{i + \frac{1}{2} ((- 1)^{t} + 1)}{2} ⌉ & if s is even; \end{cases}$
for $1 \leq i \leq n_{p} - 1$ ,
$ξ (a_{i, m_{p}}^{p}) = {\begin{cases} ⌈ \frac{t}{2} ⌉ + \frac{m_{p} - 1}{2} (2 n_{p} - 1) + ⌊ \frac{i + \frac{1}{2} ((- 1)^{t} + 1)}{2} ⌋ & if s is odd \\ ⌈ \frac{t}{2} ⌉ + n_{p} (m_{p} - 1) - \frac{m_{p}}{2} + ⌈ \frac{i + \frac{1}{2} ((- 1)^{t} + 1)}{2} ⌉ & if s is even; \end{cases}$
$ξ (a_{n_{p}, m_{p}}^{p}) = k$ ;
$ξ (a_{i, s}^{j} a_{i + 1, s}^{j}) = 2$ , for $1 \leq i \leq n_{j} - 1$ , $1 \leq s \leq m_{j}$ and $1 \leq j \leq p - 1$ ;
$ξ (a_{i, s}^{p} a_{i + 1, s}^{p}) = 2$ , for $1 \leq i \leq n_{p} - 1$ , $1 \leq s \leq m_{p} - 1$ ;
$ξ (a_{i, m_{p}}^{p} a_{i + 1, m_{p}}^{p}) = 2$ , for $1 \leq i \leq n_{p} - 2$ ;
$ξ (a_{n_{p - 1}, a_{p}}^{p} a_{n_{p}, m_{p}}^{p}) = {\begin{cases} 1 & if | E (⋃_{j = 1}^{p} G_{n_{j}, m_{j}}) | iseven \\ 2 & if | E (⋃_{j = 1}^{p} G_{n_{j}, m_{j}}) | is odd \end{cases};$
$ξ (a_{i, s}^{j} a_{i, s + 1}^{j}) = 2$ , for $2 \leq i \leq n_{j}$ , $1 \leq s \leq m_{j} - 1$ and $1 \leq j \leq p .$
Below we arrive at the weight of the edges. for $1 \leq i \leq n_{j} - 1$ , $1 \leq s \leq m_{j}$ , $1 \leq j \leq p$ and $t = \sum_{q = 1}^{j - 1} 2 n_{q} m_{q} - m_{q} - n_{q}$ ,
$w t (a_{i, s}^{j} a_{i + 1, s}^{j}) = {\begin{cases} 2 ⌈ \frac{t}{2} ⌉ + (s - 1) (2 n_{j} - 1) + i + \frac{1}{2} ((- 1)^{t} + 1) - 2 & s is odd \\ 2 ⌈ \frac{t}{2} ⌉ + (s - 1) 2 n_{j} - s + i + 1 + \frac{1}{2} ((- 1)^{t} + 1) - 2 & s is even; \end{cases}$
for $1 \leq i \leq n_{j}$ , $1 \leq s \leq m_{j} - 1$ , $1 \leq j \leq p$ and $t = \sum_{k = 1}^{j - 1} 2 n_{j} m_{j} - m_{j} - n_{j}$ ,
$w t (a_{i, s}^{j} a_{i, s + 1}^{j}) = 2 ⌈ \frac{t}{2} ⌉ + (2 s - 1) n_{j} - s + i + \frac{1}{2} ((- 1)^{t} + 1) - 2.$
It is clear that, the labels for vertices and edges receive values are not more than $k$ . Also we see that the weights for the edges are all distinct. Hence $t a d e s (⋃_{j = 1}^{p} G_{n_{j}, m_{j}}) = ⌈ \frac{\sum_{j = 1}^{p} (2 n_{j} m_{j} - m_{j} - n_{j})}{2} ⌉ .$

Illustration for $t a d e s$ of $G_{6, 7} ⋃ G_{5, 6} ⋃ G_{4, 5}$ is shown in Figure 5.

Figure 5. $G_{6, 7} ⋃ G_{5, 6} ⋃ G_{4, 5}$

Conflict of Interest

The authors declare no conflict of interests.

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[1] Chartrand, G., Jacobson, M.S., Lehel, J., Oellermann, O.R., Ruiz, S. and Saba, F., 1988. Irregular networks. Congressus Numerantium, 64, pp.187-192.[Google Scholor]

[2] Baca, M., Miller, M. and Ryan, J., 2007. On irregular total labellings. Discrete Mathematics, 307(11-12), pp.1378-1388.[Google Scholor]

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

[4] Indriati, D., Widodo, Wijayanti, I.E. and Sugeng, K.A., 2013. On the total edge irregularity strength of generalized helm. AKCE International Journal of Graphs and Combinatorics, 10(2), pp.147-155.[Google Scholor]

[5] Indriati, D., Widodo, Wijayanti, I.E., Sugeng, K.A. and Baca, M., 2015. On total edge irregularity strength of generalized web graphs and related graphs. Mathematics in Computer Science, 9, pp.161-167.[Google Scholor]

[6] Ivanco, J. and Jendrol’, S., 2006. Total edge irregularity strength of trees. Discussiones mathematicae graph theory, 26(3), pp.449-456.[Google Scholor]

[7] Miškuf, J. and Soták, R., 2010. Total edge irregularity strength of complete graphs and complete bipartite graphs. Discrete mathematics, 310(3), pp.400-407.[Google Scholor]

[8] Miškuf, J. and Jendrol, S., 2007. On total edge irregularity strength of the grids. Tatra Mountains Mathematical Publications, 36, pp.147-151.[Google Scholor]

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[10] Siddiqui, M.K., 2012. On edge irregularity strength of subdivision of star Sn. International Journal of Mathematics and Soft Computing, 2(1), pp.75-82.[Google Scholor]

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Contents

Ars Combinatoria

On Tades of Disjoint Union of Some Graphs

Abstract

1. Introduction and Motivation

2. Main Results

3. Disjoint Union of Zigzag Graph

4. Disjoint Union of Grid Graph

Conflict of Interest

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