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Molecular graphs of unsaturated carbon frameworks or hydrocarbons pruned ofhydrogen atoms, are chemical graphs. A chemical graph is a connected simplegraph of maximum degree $3$ or less. A nut graph is a connected simple graphwith a singular adjacency matrix that has one zero eigenvalue and a non-trivialkernel eigenvector without zero entries. Nut graphs have no vertices of degree$1$: they are leafless. The intersection of these two sets, the chemical nutgraphs, is of interest in applications in chemistry and molecular physics,corresponding to structures with fully distributed radical reactivity andomniconducting 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 verticesof degree $3$. With the aid of systematic local constructions that producelarger nut graphs from smaller, the combinations $(v_3, v_2)$ corresponding torealisable chemical nut graphs are characterised. Apart from a finite set ofsmall 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. Ofthese combinations, only $(20,0)$ cannot be realised by a planar chemical nutgraph. The main result characterises the ranges of edge counts for chemical nutgraphs of all orders $n$.

Charting the space of chemical nut graphs