Prime and co-prime edge anti-magic vertex labeling of families of unicyclic graphs in sensor network

Abstract

A finite simple undirected graph containing p vertices and q edges is called as prime anti-magic if it has an injection from the vertex set of integers {1, 2, ... , p} satisfying that for each edge uv, the labels given to u and v are relatively coprime and the induced edge labels generated by f(uv) = f(u)+f(v) are all distinct. If the vertices have been assigned a prime labeling, then the graph is called co-prime. In this article, we study some new classes of unicyclic graphs that admits prime and co-prime anti-magic labelings. Also, they are satisfied with the condition of odd prime anti-magic labeling. Thus, this prime and co-prime anti-magic labeling could serve as a surveillance or security model for various kinds of buildings.