# Vertices of given degree in a random graph

## Abstract

This paper concerns the degree sequence d1 ≥ d2 ≥ … ≥ dn of a randomly labeled graph of order n in which the probability of an edge is p(n) ≦ 1/2. Among other results the following questions are answered. What are the values of p(n) for which d1, the maximum degree, is the same for almost every graph? For what values of p(n) is it true that d2 > d2 for almost every graph, that is, there is a unique vertex of maximum degree? The answers are (essentially) p(n) = o(logn/n/n) and p(n)n/logn → ∞. Also included is a detailed study of the distribution of degrees when 0 < lim n p(n)/log n ≦ lim n p(n)/log n < ∞. Copyright © 1982 Wiley Periodicals, Inc., A Wiley Company

## Publication Title

Journal of Graph Theory

## Recommended Citation

Bollobás, B.
(1982). Vertices of given degree in a random graph.* Journal of Graph Theory**, 6* (2), 147-155.
https://doi.org/10.1002/jgt.3190060209