Abstract
LetG= (V, E)be a finite, simple and undirected graph with a vertex setVand an edge setE. An edge irregular totalk-labelling is a functionf : V ᴗE → {1,2,…,k}such that for any two different edgesxyandx’y’inE, their weights are distinct. The weight of edgexyis the sum of label of edgexy, labels of vertexxand of vertexy. The minimumkfor which the graphGadmits an edge irregular totalk-labelling is called the total edge irregularity strength ofG, denoted bytes(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.
Highlights
Graph labelling is a function of the set of integers to the set of elements on the graph with certain conditions [1]
If graph G can be labelled with an irregular edge k-labelling, the minimum k is called irregularity strength of G(denoted by s(G))
Bača et al defined an edge irregular total k-labelling of graph G as a function f from the union of the set of vertices and the set of edges to the set {1,2, ... , k} such that any two different edges of G have different weights [3]
Summary
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]. If graph G can be labelled with an irregular edge k-labelling, the minimum k is called irregularity strength of G(denoted by s(G)). Bača et al defined an edge irregular total k-labelling of graph G as a function f from the union of the set of vertices and the set of edges to the set {1,2, ... If the graph G can be labelled with a total irregular k-labelling, the minimum k is called the total edge irregularity strength G (denoted by tes(G)). In previous research [20], we have shown the tes 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. We investigate the formula of an edge irregular total k-labelling and determine the tes of quadruplet and quintuplet book graphs
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