The point-line incidence graph of a finite projective plane is distance regular with
diameter three and girth six. In fact a bipartite graph with diameter three and girth six
is necessarily the incidence graph of a projective plane. Next, the point-line incidence
graph of a generalized quadrangle with parameters $(s,s)$ is distance regular with diameter four and girth eight. In both cases the parameters of the geometry determine the parameters
of the graph, and the geometries are isomorphic if and only if the graphs are.

If $n$ is a prime power but not a prime and $n \ge9$, there are at least two projective planes of order $n$. For the generalized quadrangles we have the $W(q)$ ones, constructed from a symplectic space of dimension four over $GF(q)$, with parameters $(q,q)$.
There is
a second family of GQ's with these parameters constructed from ovals in projective planes,
but to get get non-isomorphic graphs we need non-isomorphic ovals which means we must
work in characteristic two.

If you'll forgive a self-reference, the graphs corresponding to the classical projective planes and the graphs belong to whiat I've called the $W(q)$ GQ's are described in detail
in my book with Royle.

The standard source of info for distance-regular graphs is the book by Brouwer, Cohen and Neumaier (titled "Distance-Regular Graphs"). The standard for GQ's is by Payne and Thas
(titled "Generalized Quadrangles", seems to be online at http://cage.ugent.be/~bamberg/FGQ.pdf).

Both these families are examples of "generalized polygons", but knowing this might not help much.