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Charting the space of chemical nut graphs

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 $n$.