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.
Highlights
It is well-known that a simple graph of an order of at least two must contain a pair of vertices with the same degree
We describe a desired labeling scheme that proves the exact value of the irregularity strength of fan graphs
We describe a labeling scheme with symmetrical distribution of even weights and odd weights of vertices ui. We use this symmetrical distribution of the weights to prove that the weight of the centre w is always greater than the weights of ui. It proves that the labeling scheme is a desired vertex irregular edge labeling that proves the exact value of the irregularity strength of fan graphs
Summary
It is well-known that a simple graph of an order of at least two must contain a pair of vertices with the same degree. K} of positive integers to the edges of a graph G of order n is called a modular irregular assignment of G if the weight function θ : V ( G ) → Zn defined by θ (v) = wt φ (v) =. We use this symmetrical distribution of the weights to prove that the weight of the centre w is always greater than the weights of ui It proves that the labeling scheme is a desired vertex irregular edge labeling that proves the exact value of the irregularity strength of fan graphs. By modifications of this irregular assignment we obtain labelings that imply the results for the modular irregularity strength of fan graphs
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