# New Strongly Regular Graphs from Finite Geometries via Switching

## TL;DR

This paper demonstrates that certain strongly regular graphs derived from finite geometries are not uniquely determined by their parameters, using a variation of Godsil-McKay switching, and provides new insights into graph and design constructions.

## Contribution

It introduces a novel application of switching to produce non-isomorphic strongly regular graphs from finite geometries, extending previous results and simplifying proofs.

## Key findings

- Certain strongly regular graphs are not determined by their parameters for n ≥ 6.
- The switching method produces many non-isomorphic graphs and designs.
- Provides a linear algebra perspective on graph and design constructions.

## Abstract

We show that the strongly regular graph on non-isotropic points of one type of the polar spaces of type $U(n, 2)$, $O(n, 3)$, $O(n, 5)$, $O^+(n, 3)$, and $O^-(n, 3)$ are not determined by its parameters for $n \geq 6$. We prove this by using a variation of Godsil-McKay switching recently described by Wang, Qiu, and Hu. This also results in a new, shorter proof of a previous result of the first author which showed that the collinearity graph of a polar space is not determined by its spectrum. The same switching gives a linear algebra explanation for the construction of a large number of non-isomorphic designs.