The reflexive edge strength of toroidal fullerene
Mohamed Basher · AKCE International Journal of Graphs and Combinatorics · 2022
A toroidal fullerene (toroidal polyhex) is a cubic bipartite graph embedded on the torus such that each face is a hexagon. The total k-labeling is defined as a combination of an edge function χe from the edge set to the set {1,2,…,ke} and a vertex function χv from the vertex set to the set {0,2,…,2kv}, where k=max{ke,2kv}. The total k-labeling of graph Ω such that every two distinctive edges have distinctive weight is called an edge irregular reflexive k-labeling, where for any edge x1x2, the edge weight wt(x1x2) is defined as the summation of the edge label χe(x1x2) itself and its two vertex labels χv(x1) and χv(x2). The reflexive edge strength of the graph Ω symbolized by, res(Ω) is the smallest k for which the graph Ω has an edge irregular reflexive k-labeling. In this paper we determine the exact value of reflexive edge strength of toroidal polyhexes.