Total edge irregularity strength of quadruplet and quintuplet book graphs

Let G= (V, E) be a finite, simple and undirected graph with a vertex set V and an edge set E. An edge irregular total k-labelling is a function f : V  E → {1,2,...,k} such that for any two different edges xy and x'y' in E, their weights are distinct. The weight of edge xy is the sum of label of edge xy, labels of vertex x and of vertex y. The minimum k for which the graph G admits an edge irregular total k-labelling is called the total edge irregularity strength of G, denoted by tes(G). We have determined the total edge irregularity strength of book graphs, double book graphs and triple book graphs. In this paper, we show the exact value of the total edge irregularity strength of quadruplet book graphs and quintuplet book graphs.


Introduction
Graph labelling is a function of the set of integers to the set of elements on the graph (vertices, edges or both) with certain conditions [1]. Irregular edge k-labelling was introduced by Chartrand et al. [2] as a function of the set of edges to the set {1,2, … , } such that any two different vertices in graph have different weights. Let be a vertex ; the weight of is the sum of labels of edges that are incident to vertex . If graph can be labelled with an irregular edge -labelling, then the minimum is called irregularity strength of (denoted by ( )).
Bača et al. defined an edge irregular total -labelling of graph as a function from the union of the set of vertices and the set of edges to the set {1,2, … , } such that any two different edges of have different weights [3]. Let be an edge. The weight of the edge (denoted by ( )) is ( ) = ( ) + ( ) + ( ). If the graph can be labelled with a total irregular -labelling, then the minimum is called the total edge irregularity strength (denoted by ( )). In [3], Bača et al. have also given the lower bound of ( ). Ivanco and Jendrol in [4] have determined of trees. Research on the of cyclic graphs for various graph classes is still being done. Several studies on the exact value of in some cyclic graphs, including some book graphs, have been conducted by [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19].
In previous research [20], we have shown the of triple book graphs, and we have constructed the formula of an edge irregular total k-labelling of the first book, the second book and the third book. The results, the first book, the second book and the third book have *Corresponding author: ratnasari.lucia@gmail.com different edge labelling formulas. In this research, we investigate the formula of an edge irregular total k-labelling and determine the of quadruplet and quintuplet book graphs.

Main result
We define the book graph, quadruplet book graph and quintuplet book graph as below: We show the of quadruplet and of quintuplet book graphs in section 2.1. and section 2.2, respectively.

Total edge irregularity strength of quadruplet book graphs
We determine the tes quadruplet book graphs as described in Theorem 2.4.
We know from Definition 2.2. that 1 = 1 = 2 , 2 = 2 = 3 , 3 = 3 = 4 and we define the vertex labelling in the following way: We have defined the edge labelling for the first book, the second book and the third book in [20]. We prove the edge labelling of the fourth book for three different cases.

For ≡
. The edge labelling for the fourth book is defined as follows: Under the labelling , we have the weight of the edges of the fourth book graph as below: with 1 ≤ ≤ .

For
≡ . The edge labelling for the fourth book is defined as follows: ( 4 , 4 ) = 3 + , Under the labelling , we have the weight of the edges of the fourth book graph as below:

Total edge irregularity strength of quintuplet book graphs
In this section we discuss the tes of quintuplet book graphs.

Proof.
A quintuplet book graph 5 ( ) has sides and 5 sheets; therefore from Definition 2.3. is obtained | ( 5 ( ))| = 5( − 1) + 7. Bača et al. in [2] give the lower bound for tes(G); that is, tes(G) ≥ For the three cases, we define the edge labelling for the fifth book as follows: 1. For ≡ . The edge labelling for the fifth book is defined in the following way: Under the labelling , we have the weights of the edges of the fifth book graph as follows: Under the labelling , we have the weights of the edges of the fifth book graph as follows: 3. For ≡ . We define the edge labelling for the fifth book as below: Under the labelling , we have the weights of the edges of the fifth book graph as below:
Based on the labelling, we see that there is a similarity labelling for the first book with the fourth book and for the second book with the fifth book as well. By the results, it is reasonable to find further formulations of the labelling for general cases.