The Total Edge Irregularity Strength of Web Graph
Jusmawati Massalesse, Nurdin Hinding*
Department of Mathematics, Faculty of Mathematics and Natural Sciences, Hasanuddin
University,
Jl. Perintis Kemerdekaan Km.10 Tamalanrea, Makassar, Indonesia
*Corresponding Author: nurdin1701@unhas.ac.id
ABSTRACT
The total edge irregularity strength of any graph 𝐺 is the minimum integer number 𝑘 such that 𝐺
have a total edge irregular 𝑘 − labeling. In this paper we find that the total edge irregularity of
(2𝑚+1)𝑛+2
web graph is ⌈
⌉ for 𝑚 ≥ 2 and 𝑛 ≥ 3.
3
Keyword: Total edge irregularity strength, total edge irregular labeling, web graph.
1. Introduction
Graph labeling defined that mapping carry elements graph to a set of integer non negative number
[3]. One of some graph labeling is irregular labeling. Concept of irregular labeling introduced by
Chantrand, et al. [2]. Ba𝑐̌ a, et al. in 2007 extend to total labeling [1]. The total edge irregular 𝑘 −
labeling on any graph 𝐺 is defined as a mapping 𝑓: 𝑉 ∪ 𝐸 → {1,2, ⋯ , 𝑘} thus all edges in 𝐺 have
different weight which is
𝑓(𝑥) + 𝑓(𝑥𝑦) + 𝑓(𝑦) ≠ 𝑓(𝑢) + 𝑓(𝑢𝑣) + 𝑓(𝑣)
for each two edges 𝑥𝑦 and 𝑢𝑣 are distinct in 𝐸. The total edge irregularity strength of 𝐺, denoted
by 𝑡𝑒𝑠(𝐺), is the minimum positive integer number 𝑘 such that 𝐺 have a total edge irregular 𝑘 −
𝑙𝑎𝑏𝑒𝑙𝑖𝑛𝑔.
2. Total Edge Irregular Labeling
In 2007, Ba𝑐̌ a, et al. [1] introducet the total edge irregular labeling and the total edge irregularity
strength of a graph. Formally, the total edge irregular labeling define as follows.
Definition 1. Suppose 𝐺 = (𝑉, 𝐸) is a graph and 𝑓: 𝑉 ∪ 𝐸 → { 1, 2, ⋯ , 𝑘} is a total labeling on 𝐺.
The weight edge 𝑥𝑦 ∈ 𝐸 under 𝑓 defined by the number of label 𝑥𝑦 and label both end vertices of
𝑥𝑦, i.e.
𝑤𝑡(𝑥𝑦) = 𝑓(𝑥) + 𝑓(𝑥𝑦) + 𝑓(𝑦).
Definition 2. Suppose 𝐺 = (𝑉, 𝐸) is a graph. The function 𝑓: 𝑉 ∪ 𝐸 → { 1, 2, ⋯ , 𝑘} is called total
edge irregular 𝑘-labeling on 𝐺 if every two distinct edges in 𝐺, the weight of them are distinct, i.e.
𝑓(𝑥) + 𝑓(𝑥𝑦) + 𝑓(𝑦) ≠ 𝑓(𝑢) + 𝑓(𝑢𝑣) + 𝑓(𝑣),
for every two edges 𝑥𝑦 and 𝑢𝑣 in 𝐸.
Definition 3. The total edge irregularity strength of 𝐺, denoted by 𝑡𝑒𝑠(𝐺), is the smallest positive
integer number 𝑘 for which 𝐺 have a total edge irregular 𝑘-labeling.
3. Result and Discuss
In this section we give the total edge irregularity strength of web graph and describe in Theorem
1. as below.
Lemma 1. Suppose 𝑗, 𝑗 ′ ≤ 𝑛, for some 𝑛 ∈ 𝑍 + , then 2𝑖𝑛 + 𝑗 ≠ 2𝑖 ′ 𝑛 + 𝑗 ′ ∀ 𝑖, 𝑖 ′ ∈ 𝑍 + with 𝑗 ≠ 𝑗′
and 𝑖 ≠ 𝑖′.
Proof. Since 𝑗, 𝑗′ ≤ 𝑛, and 𝑗 ≠ 𝑗′, |𝑗 − 𝑗′| < 𝑛.
It is easy to proof that, 2𝑖𝑛 + 𝑗 ≠ 2𝑖 ′ 𝑛 +
𝑗 ′ ∀ 𝑖, 𝑖 ′ ∈ 𝑍 + with 𝑗 ≠ 𝑗′ and 𝑖 ≠ 𝑖′.
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4. Conclusion
By discussion we proof that the total edge irregularity strength of web graph 𝑊𝑚,𝑛 is ⌈
(2𝑚+1)𝑛+2
3
⌉
for 𝑚 ≥ 2 and 𝑛 ≥ 3.
REFERENCES
[1] M. Ba𝑐̌ a, S. Jendrol’, M. Miller, M., J. Ryan, On Irregular Total Labelings. Discrete
Mathematics, 307: 1378-1388, 2007
[2] G. Chartrand, Introductory Graph Theory. Dover Publications Inc, New York, 1986
[3] W.D. Wallis, Magic Graph. Birkh𝑎̈ user, Boston – Basel – Berlin, 2001.