A note on the relation between two properties of random graphs

TL;DR

This paper investigates the relationship between t-e.c. and pseudo-random properties in random graphs, providing explicit constructions of t-e.c. graphs that are not pseudo-random, highlighting gaps in their correlation.

Contribution

It offers the first explicit construction of infinite t-e.c. graph families that are not pseudo-random, addressing a gap in understanding their relationship.

Findings

Constructed infinite families of t-e.c. graphs not pseudo-random.

Highlighted the gap between t-e.c. and pseudo-random properties.

Provided the first explicit examples of such graphs.

Abstract

The $t$-e.c. and pseudo-random property are typical properties of random graphs. In this note, we study the gap between them which has not been studied well. As a main result, we give the first explicit construction of infinite families of t-e.c. graphs which are not families of best possible pseudo-random graphs.