Classification of Ryser Graphs

The object of study is $(2,r_1,r_2)$-regular graphs in which the union of the neighborhoods of two different vertices $u$ and $w$ contains $r_1$ or $r_2$ vertices, depending on whether $u$ and $w$ are adjacent. It is proved that such graphs either are strongly regular or decompose into the direct sum of a complete multipartite graph and a clique. Earlier, the case $r_1=r_2$ was studied by other authors.