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The irregularity and modular irregularity strength of fan graphs

Abstract:

For a simple graph G with no isolated edges and at most, one isolated vertex, a labeling ϕ: E(G) → {1, 2, …, k} of positive integers to the edges of G is called irregular if the weights of the vertices, defined as wtϕ (v) = ∑u∈N(v) ϕ(uv), are all different. The irregularity strength of a graph G is known as the maximal integer k, minimized over all irregular labelings, and is set to ∞ if no such labeling exists. In this paper, we determine the exact value of the irregularity strength and the modular irregularity strength of fan graphs.