On Distance Irregular Labeling of Disconnected Graphs

Faisal Susanto, Kristiana Wijaya, Prasanti Mia Purnama, Slamin

A distance irregular $ k $-labeling of a graph $ G $ is a function $ f:V(G)\rightarrow\{1,2,…,k\} $ such that the weights of all vertices are distinct. The weight of a vertex $ v $, denoted by $ wt(v) $, is the sum of labels of all vertices adjacent to $ v $ (distance $ 1 $ from $ v $), that is, $ wt(v)=\sum_{u\in N(v)}f(u) $. If the graph $ G $ admits a distance irregular labeling then $ G $ is called a distance irregular graph. The distance irregularity strength of $ G $ is the minimum $ k $ for which $ G $ has a distance irregular $ k $-labeling and is denoted by $ \dis(G) $. In this paper, we derive a new lower bound of distance irregularity strength for graphs with $ t $ pendant vertices. We also determine the distance irregularity strength of some families of disconnected graphs namely disjoint union of paths, suns, helms and friendships.