Reflexive edge strength on tadpole graph
Nadia Indarwati Setia Budi, Diari Indriati, Tri Atmojo Kusmayadı, Titin Sri Martini · AIP conference proceedings · 2022
Labeling of graph G can be defined as a mapping from the set of vertices with symbol V(G), the set of edges with symbol E(G), or a union of both into non-negative integers. Given a graph G = (V, E) a labeling ηv is the label vertex V(G) which v = {,0, 2, …, 2kv} and ηe is the label edge E(G) which e = {1, 2, …, ke}, where k is the maximum between 2kv and ke is called an edge irregular reflexive k-labeling if for each edge x1x2 and x′1x′2 have a different weight. The weight of an edge on graph G is the sum of the labels of two vertices that are adjacent and the label of edge incident to that vertices. The reflexive edge strength or res(G) is the smallest k for which such labeling exists. In this case, we will discuss res of the tadpole graph Tr,s with r ≥ 3, s ≡ 0, 1, 2 (mod 6), r represents the length of the cycles and s represents the length of the path. There are 2 formulas res(Tr,s) with r ≥ 3, s ≡ 0, 1, 2 (mod 6) is res(Tr,s)=[r+s3]+1 if (r + s) ≡ 2, 3 (mod 6) and res(Tr,s)=[r+s3] if (r + s) ≢ 2,3 (mod 6).