  A nut graph is a graph on at least 2 vertices whose adjacency matrix has
nullity 1 and for which non-trivial kernel vectors do not contain a zero.
Chemical graphs are connected, with maximum degree at most three. We present a
new algorithm for the exhaustive generation of non-isomorphic nut graphs. Using
this algorithm, we determined all nut graphs up to 13 vertices and all chemical
nut graphs up to 22 vertices. Furthermore, we determined all nut graphs among
the cubic polyhedra up to 34 vertices and all nut fullerenes up to 250
vertices.
  Nut graphs are of interest in chemistry of conjugated systems, in models of
electronic structure, radical reactivity and molecular conduction. The relevant
mathematical properties of chemical nut graphs are the position of the zero
eigenvalue in the graph spectrum, and the dispersion in magnitudes of kernel
eigenvector entries ($r$: the ratio of maximum to minimum magnitude of
entries). Statistics are gathered on these properties for all the nut graphs
generated here. We also show that all chemical nut graphs have $r \ge 2$ and
that there is at least one chemical nut graph with $r = 2$ for every order $n
\ge 9$ (with the exception of $n = 10$).
