The girth, odd girth, distance function, and diameter of generalized Johnson graphs

For any non-negative integers $v>k>i$, the {\em generalized Johnson graph}, $J(v,k,i)$, is the undirected simple graph whose vertices are the $k$-subsets of a $v$-set, and where any two vertices $A$ and $B$ are adjacent whenever $\mathrm{\mid }A\cap B\mathrm{\mid }=i$. In this article, we derive formulas for the girth, odd girth, distance function, and diameter of $J(v,k,i)$.