Vertex degrees close to the average degree

TL;DR

This paper characterizes minimal degree intervals in graphs that are guaranteed to contain at least one vertex degree, based on the average degree and graph order, providing bounds for certain degree ranges.

Contribution

It introduces a precise characterization of degree intervals in graphs relative to the average degree, extending understanding of degree distributions in finite graphs.

Findings

Existence of a vertex with degree in a specific interval for certain average degrees.

Explicit bounds for vertex degrees based on average degree and graph size.

Theoretical framework for degree distribution analysis in simple graphs.

Abstract

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 $d_+\in \left(\sqrt{dn},n-1\right]$, we show the existence of a vertex in $G$ of degree between $d_+-\left(\frac{(d_+-d)n}{n-d_++\sqrt{d_+^2-dn}}\right)$ and $d_+$.