Vertex degrees close to the average degree

01/19/2023

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by   Johannes Pardey, et al.

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Let G be a finite, simple, and undirected graph of order n and average degree d. Up to terms of smaller order, we characterize the minimal intervals I containing d that are guaranteed to contain some vertex degree. In particular, for d_+∈(√(dn),n-1], we show the existence of a vertex in G of degree between d_+-((d_+-d)n/n-d_++√(d_+^2-dn)) and d_+.