Cores of Geometric Graphs

TL;DR

This paper investigates the core structures of various classes of strongly regular graphs, extending known results to new types like generalized quadrangles, Steiner systems, and orthogonal arrays, with conditions on size.

Contribution

It proves that point and line graphs of generalized quadrangles are cores or have complete cores, and extends this to large block graphs of Steiner systems and orthogonal arrays.

Findings

Point and line graphs of generalized quadrangles are cores or have complete cores.

For large point sets, block graphs of Steiner systems and orthogonal arrays are cores or have complete cores.

Results extend previous work on rank-3 graphs to broader classes of strongly regular graphs.

Abstract

Cameron and Kazanidis have recently shown that rank-3 graphs are either cores or have complete cores, and they asked whether this holds for all strongly regular graphs. We prove that this is true for the point graphs and line graphs of generalized quadrangles and that when the number of points is sufficiently large, it is also true for the block graphs of Steiner systems and orthogonal arrays.