Total edge irregularity strength of triple 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-labeling is a map f : V ⋃ E → {1, 2, …, k} such that for any two different edges xy and x′y′ in E, ω(xy) ≠ ω(x′y′) where ω(xy) = f(x) + f(y) + f(xy). The minimum k for which the graph G admits an edge irregular total k-labeling 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 book graphs and double book graphs. In this paper, we construct an edge irregular total k-labeling that can be used for book graphs, double book graphs, and triple book graphs. We also show the exact value of the total edge irregularity strength of triple book graphs.


Introduction
Let G(V, E) be a finite, simple and undirected graph with a vertex set V and an edge set E. Labelling of graph G is a function that assigns elements on the graph (vertices, edges or both) to numbers (usually positive integer and called labels) that satisfy certain conditions [1]. There are various labelling of graphs that pay attention to the number of labels of elements on the graph. In [2] 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 is 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). The smallest k value is called irregular strength of G and is denoted by s(G) if the graph G admits an irregular edge k -labelling. Furthermore in [3] an edge irregular total k-labelling on graph G(V, E) was introduced by Bača et al. as a function f from the set of vertex union the set of edge to the set of positive integer from 1 up to k such that for any two different edges xy and x y in G have distinct weights. Let xy is an edge in G, the weight of xy denoted by ω f (xy) is defined 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. 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.
For tes of trees, Ivanco and Jendrol [4] have determined it. Meanwhile, research on the tes cyclic graphs for various graph classes is still being done. Some results of the investigation of 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. Thus the vertex set of .., n, j = 1, ..., m − 2}. A triple 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 ≤ 2. Thus the vertex set of The construction of an edge irregular total k-labelling for the first book graph, the second book graph and the third book graphs is shown in Theorem 3.3 as below.  Proof: A triple book graph 3B n (C m ) is obtained from three book graphs B 1 n (C m ), B 2 n (C m ) and hence any triple book graph 3B n (C m ) has maximum degree ∆(3B n (C m )) = 2n + 2. From the Definition 2.1 we know that a triple book graph has m sides and 3n sheets so that it is obtained |E(3B n (C m ))| = 3(m − 1)n + 3. By using the lower bound given by Bača i.e. tes(G) ≥ max{ |E|+2 . Meanwhile, the upper bound is shown by constructing an edge irregular total k 3 -labeling with k 3 = 3((m−1)n+1)+2 3 as below.
Based on Definition 2.2, it is clear that We label the vertices of triple book graph as below:

3
, r q = k q − m−3 3 n + k q−1 and q = 1 for the first book, q = 2 for the second book, q = 3 for the third book, and For the edge labeling f the proof is devided into into 3 cases.
The edge labeling f for the first book B 1 n (C m ) is defined as below: By using labelling f , we obtained the edge weights as below: The edge labeling f for the second book B 2 n (C m ) is defined as below: By using labelling f , we obtained the edge weights as below: The edge labeling f for the third book B 3 n (C m ) is defined as below: By using labelling f , we obtained the edge weights as below: We found that the weight of the edges of 3B n (C m ) for m = 1(mod 3) by using the labeling f The edge labeling f for the first book B 1 n (C m ) is defined as below: By using labelling f , we obtained the weight of edges as follows: The edge labeling f for the second book B 2 n (C m ) is defined as below: By using labelling f , we obtained the edge weights as below: The edge labeling f for the third book B 3 n (C m ) is defined in the following way: By using labelling f , we obtained the weight of edges as follows: We found that the weight of the edges of 3B n (C m ) for m = 2(mod 3) by using the labeling f form the set {3, 4, ..., 3(m − 1)n + 5}. .
By using labelling f , we obtained the weight of edges as follows: .