On inclusive distance vertex irregularity strength of book graph

Abstract

<p>The concept of distance vertex irregular labeling of graphs was introduced by Slamin in 2017. The <em>distance vertex irregular labeling</em> on a graph <em>G</em> with <em>v</em> vertices is defined as an assignment <em>λ</em> : V → {1, 2, ..., <em>k</em>} so that the weights calculated at vertices are distinct. The weight of a vertex <em>x</em> in <em>G</em> is defined as the sum of the labels of all the vertices adjacent to <em>x</em> (distance 1 from <em>x</em>). The distance vertex irregularity strength of graph <em>G</em>, denoted by <em>dis(G)</em>, is defined as the minimum value of the largest label <em>k</em> over all such irregular assignments. Bong, Lin and Slamin generalized the concept to inclusive and non-inclusive distance irregular labeling. The difference between them depends on the way to calculate the weight on the vertex whether the vertex label we calculate its weight is included or not. The <em>inclusive distance vertex irregularity strength</em> of <em>G</em>, is defined as the minimum of the largest label <em>k</em> over all such inclusive irregular assignments. In this paper, we determine the inclusive distance vertex irregularity strength of some particular classes of graphs such as book graph.</p>