EDGE IRREGULAR REFLEXIVE LABELING ON ALTERNATE TRIANGULAR SNAKE AND DOUBLE ALTERNATE QUADRILATERAL SNAKE

ABSTRACT

Reflexive Edge Strength; Alternate Triangular Snake; Double Alternate Quadrilateral Snake.

INTRODUCTION
Graphs discussed in this paper are connected, undirected, and simple graph.Here using vertex set denoted as () also edge set as ().Graph labeling is a mapping with component graph as domain and the co-domain is integers number [1].There is different types of labeling, which categorized as being vertex labeling, edge labeling, and total labeling.There are many categories of labeling on Gallian research.One type of labeling is irregular total -labeling [2].
Baca et al. [3] the total k-labeling is divided into two types namely edge irregular total k-labeling and vertex irregular total k-labeling.If all edges have different weights, an edge irregular total k-labeling similar with total k-labeling.In 2017, Ryan et al. in Baca et al. [4] introduced concept of irregular total k-labeling.Namely edge irregular reflexive total k-labeling and vertex irregular reflexive total k-labeling.In graph ,  = , denoted by () as the sum of labels vertex x, vertex y and the label of edge e.
An edge irregular reflexive k-labeling is determine as the function  ∶ () → { 1,2, . . .,   } and  ∶ () → {0,2, . . .,2  }, where  = {  , 2  }.In a graph, the minimum value of k that can be labeled with an edge irregular reflexive labeling is called the strength of the reflexive edge strength and is denoted by () [5].For this article, we will discuss about reflexive edge strength of , denoted as  ().When we prove the result, we will often use the lemma proven by Ryan et al. in Baca et al. [4].
The lower bound for  () appeared from the fact that the minimal edge weight under an edge irregular reflexive labeling is one, and the basis of the maximal edge weights, that is |()| can be achieved only as the sum of three numbers from whose at least two are even.
In the same paper, prove the res of prisms graph   and wheels .This article, we determine the  () of alternate triangular snake   and double alternate quadrilateral snake (  ).

RESULTS AND DISCUSSION
In this section describes the results about reflexive edge strength of alternate triangular snake (  ) and double alternate quadrilateral snake (  ).
Theorem 1.For every positive integer  ≥ 3, the reflexive edge strength of alternate triangular snake (  ) is Proof.a) First, we prove the lower bound of  ((  )).Because alternate triangular snake (  ) with n even has 2 − 1 edges, then by Lemma 1 we get.

𝑟𝑒𝑠 (𝐴(𝑇
It can be seen from the weight of an alternate triangular snake (  ) weight is different .Therefore, f requires that the component on an edge irregular reflexive k-labeling.The trial in the theorem is complete.▄ Example 1.The example of an edge irregular reflexive-4 labeling for alternate triangular snake graph ( 5 ) is shown in Figure 1.The red color shows edge weight, the blue color shows edge label, and the black color show vertex label.

Double Alternate Quadrilateral Snake 𝑫𝑨(𝑸 𝒏 )
A double alternate quadrilateral snake  (  ) consists of two alternate quadrilateral snakes that have a common path.Which is obtained from a path  1 ,  2 , . . .  by joining   and  +1 (alternatively) to new vertices   ,   and   ,   respectively and adding the edges     and     [15].
Proof.a) First, we prove the lower bound of  ( (  )).Because double alternate quadrilateral snake  (  ) with n odd has 4 − 4 edges, then by Lemma 1 we get.

𝑓(𝑣 𝑖
By proving the lower and upper bounds, found that the maximum labels for the vertices and edges ⌈

CONCLUSIONS
Based on these descriptions, conclusions are obtained

2 ,
1 for  odd, 4 − 4 ≡ 2, 3 (mod 6).In this case, the edge weight is expressed as,  (    ) = 2 (4 − 3), 1 ≤  ≤ ⌈  − 1 2 ⌉ .  (    ) = 8 − 7, 1 ≤  ≤ ⌈  −It can be seen from the weight of double alternate quadrilateral snake  (  ) have different weight.Therefore, f requires that the item on an edge irregular reflexive k-labeling.The trial in theorem is completed.▄Example The example of an edge irregular reflexive 6-labeling of double alternate quadrilateral snake  ( 5 ) is shown in Figure2.The red color shows edge weight, the blue color shows edge label, and the black color shows vertex label.

explained in this article uses literature studies related to reflexive edge strength for various types about graphs. The materials used in this research are books
, journals, papers, and articles.Related to reflexive edge strength, alternate triangular snake, and double alternate quadrilateral snake.The procedure used in this research are as follows, 1. Determine the lower bound () of alternate triangular snake (  ) and double alternate quadrilateral snake ( ) based on lemma proven by Ryan et al. in Baca et al. [4], 2. Labeling alternate triangular snake (  ) and double alternate quadrilateral snake (  ) that satisfies the lower bound, 3. Calculating the weight of each side of the alternate triangular snake (  ) and double alternate quadrilateral snake (  ) so that all edges have different weight, 4. Looking for general pattern  alternate triangular snake graph (  ) and double alternate quadrilateral (  ).