Strongly regular n-e.c. graphs

TL;DR

This paper investigates the properties and bounds of strongly regular n-e.c. graphs, especially those derived from partial geometries, and explores their subgraph containment and construction limitations.

Contribution

It derives new parameter bounds for strongly regular n-e.c. graphs from partial geometries and extends previous bounds on n-e.c. block intersection graphs.

Findings

Derived bounds on parameters of strongly regular n-e.c. graphs from partial geometries.

Provided examples of strongly regular graphs containing all small subgraphs but not n-e.c. for n > 2.

Connected the properties of these graphs to known combinatorial designs and previous work.

Abstract

A result of Erd\"os and R\'enyi shows that for a fixed integer n almost all graphs satisfy the n-e.c. adjacency property. However, there are few explicit constructions of n e.c. graphs for n > 2, and almost all known families of n-e.c. graphs are strongly regular graphs. In this paper we derive parameter bounds on strongly regular n-e.c. graphs constructed from the point sets of partial geometries. This work generalizes bounds on n-e.c. block intersection graphs of balanced incomplete block designs given by McKay and Pike. It also relates to work by Griggs, Grannel, and Forbes' determining 3-e.c. graphs that are block intersection graphs of Steiner triple systems. In addition to these bounds, we give examples of strongly regular graphs that contain every possible subgraph of small order but are not n-e.c. for n > 2.