Geometrical approach to Seidel's switching for strongly regular graphs

TL;DR

This paper presents a simplified geometric framework for understanding Seidel's switching in strongly regular graphs, characterizing when switching preserves or alters graph parameters based on induced subgraph regularity.

Contribution

It introduces a geometric approach to Seidel's switching, providing clear conditions for when the resulting graph remains strongly regular or changes parameters.

Findings

Switching preserves strong regularity if the induced subgraph is appropriately regular.

Switching alters parameters if the induced subgraph has specific regularity and size.

Geometric perspective simplifies understanding of Seidel's switching in strongly regular graphs.

Abstract

In this paper, we simplify the known switching theorem due to Bose and Shrikhande as follows. Let $G=(V,E)$ be a primitive strongly regular graph with parameters $(v,k,\lambda,\mu)$. Let $S(G,H)$ be the graph from $G$ by switching with respect to a nonempty $H\subset V$. Suppose $v=2(k-\theta_1)$ where $\theta_1$ is the nontrivial positive eigenvalue of the $(0,1)$ adjacency matrix of $G$. This strongly regular graph is associated with a regular two-graph. Then, $S(G,H)$ is a strongly regular graph with the same parameters if and only if the subgraph induced by $H$ is $k-\frac{v-h}{2}$ regular. Moreover, $S(G,H)$ is a strognly regualr graph with the other parameters if and only if the subgraph induced by $H$ is $k-\mu$ regular and the size of $H$ is $v/2$. We prove these theorems with the view point of the geometrical theory of the finite set on the Euclidean unit sphere.