The Reflexive Edge Strength of Cycles Plus One Edge

Abstract

Let G be a simple graph with a vertex set V (G) and edge set E(G). Given a vertex labeling fV : V (G) → {0; 2; 4; : : : ; 2kv} and an edge labelings fE : E(G) → {1; 2; 3; : : : ; 2ke}. Define a function f by f(x) = fV (x) if x ∈ V (G) and f(x) = fE(x) if x ∈ E(G). We call f be the total k-labeling where k = max{ke; kv}. A total k-labeling f is called an edge irregular reflexive k-labeling of G if every two distinct edge xy and x'y', we have wtf (xy)/ = wtf (x'y') where wtf (uv) = f(u) + f(uv) + f(v) if uv is an edge of G. The reflexive edge strength of G, denoted by res(G) is the minimum k for G which has an edge irregular reflexive k-labeling. In this paper, we give the exact value of res(Cn +e) where Cn +e is a cycle of order n plus one edge which contains a triangle.