  Molecular graphs of unsaturated carbon frameworks or hydrocarbons pruned of
hydrogen atoms, are chemical graphs. A chemical graph is a connected simple
graph of maximum degree $3$ or less. A nut graph is a connected simple graph
with a singular adjacency matrix that has one zero eigenvalue and a non-trivial
kernel eigenvector without zero entries. Nut graphs have no vertices of degree
$1$: they are leafless. The intersection of these two sets, the chemical nut
graphs, is of interest in applications in chemistry and molecular physics,
corresponding to structures with fully distributed radical reactivity and
omniconducting behaviour at the Fermi level. A chemical nut graph consists of
$v_2 \ge 0$ vertices of degree $2$ and an even number, $v_3 > 0$, of vertices
of degree $3$. With the aid of systematic local constructions that produce
larger nut graphs from smaller, the combinations $(v_3, v_2)$ corresponding to
realisable chemical nut graphs are characterised. Apart from a finite set of
small cases, and two simply defined infinite series, all combinations $(v_3,
v_2 )$ with even values of $v_3 > 0$ are realisable as chemical nut graphs. Of
these combinations, only $(20,0)$ cannot be realised by a planar chemical nut
graph. The main result characterises the ranges of edge counts for chemical nut
graphs of all orders $n$.
