A generalized polygon is an incidence plane satisfying two path conditions. More precisely, let 
be an incidence geometry, i.e., a set
with a symmetric, reflexive binary
relation 
,
and let 
and 
be elements of 
.
An incidence plane is an incidence geometry
whose object set is the disjoint
union of two sets 
and 
such that for 
or 
, 
only if 
. Then a generalized polygon satisfies, for all 
,
1. There exists a path of length at most 
from 
to 
, and
2. There exists at most one irreducible path of length less than 
from 
to 
.
(Feit and Higman 1964).
The incidence graph of a generalized polygon has vertex set 
,
so it is a bipartite graph with parts 
and 
. Generalized polygons need not be regular. For example, the
incidence graph of the generalized
quadrangle 
has 27 point vertices of
degree 5 and 45 line vertices of degree
3; in particular, it is neither a regular graph
nor a distance-regular graph, and the unequal
bipartition sizes also prevent it from having a Hamiltonian
path or Hamiltonian cycle (Godsil and Royle
2001, pp. 84-87).
The only cubic generalized polygons are the generalized 2-gon 
(utility graph), generalized triangle 
, generalized quadrangle 
, and generalized
hexagon 
(Feit and Higman 1964, Royle).
