Total Edge Irregularity Strength of q Tuple Book Graphs

Let G(V, E) be a simple, undirected, and finite graph with a vertex set V and an edge set E. An edge irregular total k-labelling is a function f from the set V ∪ E to the set of non-negative integer set {1, 2, . . . , k} such that any two different edges in E have distinct weights. The weight of edge xy is defined as the sum of the label of vertex x, the label of vertex y and the label of edge xy. The minimum k for which the graph G can be labelled by an edge irregular total k-labelling is called the total edge irregularity strength of G, denoted by tes(G). We have constructed the formula of an edge irregular total k-labelling and determined the total edge irregularity strength of triple book graphs, quadruplet book graphs and quintuplet book graphs. In this paper, we construct an edge irregular total of k-labelling and show the exact value of the total edge irregularity strength of q tuple book graphs.


Introduction
Labelling of a graph is a function that assigns elements on the graph (vertices, edges or both) to numbers (usually positive integer and called labels) that satisfies certain conditions (Wallis, 2001). There are various labelling of graphs that attention the sum of labels of elements on the graph. In 1988, Chartrand et al. introduced an irregular edge k-labelling as a function f from the set of edge to the set of number from 1 until k such that all vertices in G have different weights. Let v be a vertex in G, the weight of vertex v is the sum of all labels of edges that are incident to vertex v and denote by ω f (v). If the graph G admits an irregular edge k -labelling, the smallest value of k is called irregular strength of G and is denoted by s(G) (Chartrand et al., 1988). Furthermore, in 2007, an edge irregular total k-labelling on graph G was introduced by Bača et al. as a function f from the union of vertex set and edge set to the set of positive integer from 1 up to k such that any two different edges xy and x y in G have distinct weights. Let xy be an edge in G, the weight of xy denoted by ω f (xy) is defined as ω f (xy) = f (x) + f (y) + f (xy). If the graph G can be labelled with an edge irregular total k-labelling then the smallest k is called the total edge irregularity strength of G and is denoted by tes(G). Bača et al. (2007) also give a lower bound of tes(G) which is tes(G) ≥ max{ |E|+2 3 , ∆(G)+1 2 } where ∆(G) is the maximum vertex degree of G, the degree of vertex x in a graph G is the number of edges of G incident with x. Ivanco and Jendrol (2006) have determined the tes for trees. Meanwhile, research on the tes cyclic graphs for various graph classes is still being done. Some results of the investigation of tes for some cyclic graphs, including some book graphs, have been determined and can be seen in , (Nurdin et al., 2008), (Chunling et al., 2009), (Ahmad et al., 2014), (Bača and Siddiqui, 2014), (Indriati et al., 2015), (Jayanthi and Sudha, 2015), (Siddiqui et al., 2017), (Putra and Susanti, 2018), (Ratnasari and Susanti, 2018), (Ratnasari et al., 2019), .
In the previous research, an irregular total edge of k-labelling has been constructed for triple book graphs, quadruplet book graphs and quintuplet book graphs and we obtained the tes of triple three graphs, quadruplet book graphs and quintuplet book graphs respectively as follows tes(3B n (C m )) = 3((m−1)n+1)+2 3 , tes(4B n (C m )) = 4((m−1)n+1)+2 3 , and tes(5B n (C m )) = 5((m−1)n+1)+2 3 . Based on our observations, we see some similarities in the pattern of labelling for the first with the fourth book graphs and the second with the fifth book graphs. Here, we prove this for arbitrary q tuple book graphs.
In this paper, the first part is an introduction that explains the background of the problem. The second part contains the proof of the theorem, which is shown by constructing an edge irregular total k-labeling of the q tuple book graph and obtained the exact value of tes of q tuple book graph. The third part is a discussion that contains the conclusions obtained.

Results
We investigate the tes of q tuple book graphs by first giving the definitions of book graphs and q tuple book graphs.
A book graph with m sides and n sheets denoted by B n (C m ) is the graph obtained from cycle graphs C i m , i = 1, ..., n by merging edge uv from each cycle. The vertex set of Definition 2.2 Let B q n (C m ), 1 ≤ q ≤ s with s a positive integer, be the q th copy of book graph B n (C m ) as defined at the Definition 2.1. Let the vertices of B q n (C m ) be V(B q n (C m )) = {u q , v q } ∪ {x q i, j : i = 1, ..., n, j = 1, ..., m − 2}. A q tuple book graph is a graph obtained from three copies of book graphs B q n (C m ) by identifying vertex v q from book graph B q n (C m ) with vertex u q+1 from book graph B q+1 n (C m ) and renaming this vertex by w q , 1 ≤ q ≤ s − 1. The vertex set of qB n (C m ) is From the construction of an edge irregular total k-labelling of triple book graphs, quadruplet book graphs and quintuplet book graphs, it is found that there are similar patterns in the labelling of the first book graph with the fourth book graph and the second book graph with the fifth book graph. Therefore, we construct the edge irregular total k-labelling and determine the tes of q tuple book graphs by dividing it into 3 Lemmas as below: Lemma 2.3 Let qB n (C m ) be q tuple book graph and m ≡ 0 mod 3. Then tes(qB n (C m )) = q((m−1)n+1)+2

3
. Meanwhile, the upper bound is shown by constructing an edge irregular total k q -labelling with k q = q((m−1)n+1)+2 3 as below.
Based on Definition 2.2, it is clear that u q+1 = v q = w q , 1 ≤ q ≤ s − 1, with s a positive integer. The function f is a mapping from V(qB n (C m )) ∪ E(qB n (C m )) to {1, 2, ..., k q }.
We construct the vertex labelling f of q tuple book graph as below:

3
. We define the function g as a mapping from V(qB n (C m )) ∪ E(qB n (C m )) to {1, 2, ..., k q }. For the upper bound, we constuct labelling g as follows. For the vertex labelling g restricted on V(G) is equal to f in Lemma 2.3 restricted on V(G). Meanwhile, the edge labelling g restricted on E(G) is defined as follows: The edge labelling g for the q th book B q n (C m ) with q ≡ 1 mod 3, n ≥ 2 is defined as below: g(u q v q ) = k q−1 + n, g(u q x q i,1 ) = k q−1 , By using labelling g, we obtained the edge weights as below: The edge labelling g for the q th book B q n (C m ) with q ≡ 2 mod 3, n ≥ 2 is defined as below: g(u q v q ) = k q−1 + n, g(u q x q i,1 ) = k q−1 + 1, By using labelling g, we obtained the edge weights as below: The edge labelling g for the q th book B q n (C m ) with q ≡ 0 mod 3, n ≥ 2 is defined as below: By using labelling g, we obtained the edge weights as below: We found that the weight of the edges of qB n (C m ) for m = 1(mod 3) by using the labelling g form the set {3, ..., q((m − 1)n + 1) + 2}.

3
. Similar to the proofs of Lemma 2.3 and Lemma 2.4, we define the function h as a mapping from V(qB n (C m )) ∪ E(qB n (C m )) to {1, 2, ..., k q }. For the upper bound, the vertex labelling h restricted on V(G) is equal to f in Lemma 2.3 restricted on V(G) and we construct the edge labelling h restricted on E(G) as follows: The edge labelling h for the q th book B q n (C m ) with q ≡ 1 mod 3, n ≥ 2 is defined as below: By using labelling h, we obtained the weight of edges as follows: The edge labelling h for the q th book B q n (C m ) with q ≡ 2 mod 3, n ≥ 2 is defined as below: with 1 ≤ i ≤ n.
From the vertex labelling and the edge labelling of f , g, and h, which are defined in Lemma 2.3, Lemma 2.4, and Lemma 2.5, respectively, it is obtained that the weights of edges form the set of integer from 3 up to q((m − 1)n + 1) + 2. It shows that the weights of all edges in q tuple book graph qB n (C m ) are all different. Therefore, f , g and h are all edge irregular total k q -labellings with k q = q((m−1)n+1)+2 3 for q tuple book graphs qB n (C m ) with m ≡ 0 mod 3, m ≡ 1 mod 3, m ≡ 2 mod 3, respectively. We obtain tes(qB n (C m )) = q((m−1)n+1)+2 3 . Hence, the following Theorem is proven.

Discussion
We studied the construction of edge irregular total k-labelling of q tuple book graphs qB n (C m ) and we found that the total edge irregularity strength of q tuple book graphs B n (C m ) is equal to q((m−1)n+1)+2