On edge irregularity strength of cycle-star graphs

For a simple graph G, a vertex labeling φ:V(G) → \1, 2,…,k\ is called k-labeling. The weight of an edge $uv$ in G, written wφ(uv), is the sum of the labels of end vertices u and v, i.e., wφ(uv)=φ(u)+φ(v). A vertex k-labeling is defined to be an edge irregular k-labeling of the graph G if for every two distinct edges u and v, wφ(u) ≠ wφ(v). The minimum k for which the graph G has an edge irregular k-labeling is called the edge irregularity strength of G, written $es(G)$. In this paper, we study the edge irregular k-labeling for cycle-star graph CSk,n-k and determine the exact value for cycle-star graph for 3 ≤ k ≤ 7 and n-k ≥ 1. Finally, we make a conjecture for the edge irregularity strength of CSk,n-k for k ≥ 8 and n-k ≥ 1.