Some unicyclic graphs and its vertex coloring edge-weighting.

An unicyclic graph is a graph containing exactly one cycle. Let G = (V, E) be a simple, nontrivial, finite, and connected graph with vertex set V(G) and edge sets E(G). Let k ∈ N, a mapping fw: V(G) → N defined by fw(v) = Σv∈e w(e) is called as a k - edge - weighting of graph G. A vertex coloring is an edge-weighting w where fw(u) ≠ fw(v) for any edges uv. We define the vertex coloring as the minimum k such that for every graph G has a vertex-coloring k - edge - weighting and we denoted it by μ(G). In this paper, we analyze the exact value of vertex coloring edge weighting of some unicyclic graphs and it's lower bound. [ABSTRACT FROM AUTHOR]