I wonder if the following simple generalization of Johnson and Kneser
graphs has a name? Let the vertex set of the graph $G(n,k,t)$ be the
set of $k$-element subsets of an $n$-set, with two $k$-sets adjacent if their
intersection has cardinality $t$.


So for $t=0$ we have the Kneser graphs, and for $t=k-1$ we have the Johnson graphs.
Does this family of intersection graphs have a name, or a standard notation?



Name for Kneser/Johnson-like graphs?
Chen and Lih call such a $G(n,k,t)$ (with the same notation) a *uniform subset graph*; see ["Hamiltonian uniform subset graphs" (1987)](http://www.sciencedirect.com/science/article/pii/009589568790044X). As you point out, the Johnson graph $J(n,k)$ is then $G(n,k,k-1)$. And the Kneser graph $KG(n,k)$ corresponds to $G(2n+k,n,0)$.


