On 2-locally Seidel graphs

The $i$-neighbourhood of a vertex $a$ of a graph $\Gamma$ is the subgraph $\Gamma_i(a)$ induced by $\Gamma$ on the set of all vertices of $\Gamma$ that lie at distance $i$ from $a$. Let $\mathcal F$ denote a class of graphs. A graph $\Gamma$ is called an $i$-locally $\mathcal F$-graph if $\Gamma_i(a)$ lies in $\mathcal F$ for any vertex $a$ of $\Gamma$. In this paper we classify the connected regular graph in which the 2-neighbourhoods are Seidel graphs. (Recall that a Seidel graph is a strongly regular graph that has eigenvalue $-2$). The class of Seidel graphs consists of the complete multipartite graphs with parts of order 2, lattice and triangular graphs, as well as the Shrikhande, Chang, Petersen, Clebsch, and Schlafli graphs.