Experimental Realisation of Nut Graphs Using Coaxial Cable Networks
Maxine M. McCarthy, David M. Whittaker, Patrick W. Fowler, Barry T. Pickup
Match Communications in Mathematical and in Computer Chemistry1/1/2026·DOI·Read paper
Abstract
A nut graph is a connected simple graph with nullity one that
has a full kernel eigenvector. In the context of π systems, chemical
nut graphs are identical with the subcubic nut graphs. Chemical
nut graphs are of interest for at least two reasons: the possibility of
fully distributed radical reactivity arising from partial occupation of
their sole non-bonding orbital, and their identification with the class
of strong omniconductors of nullity one in source-and-sink models of
ballistic molecular conduction. In this note, a mapping between the
H¨uckel tight-binding theory of π-conjugated carbon frameworks and
the Telegraph equations for networks of coaxial cables is described.
This mapping motivates experimental realisation of chemical nut
graphs, where the radio-frequency resonances of the cable network
correspond to the spectrum of a weighted version of the molecular
graph. In particular, measurements with a vector network analyser give direct access to the nullstate of the cable network, and
hence to a simulation of the non-bonding molecular orbital of the
π system. We demonstrate the distributed nature of the kernel
eigenvector, which underpins the description of chemical reactivity,
omni-conduction and electrical properties, for the cable models of
the smallest chemical nut graph, and the three smallest nut graphs.
We also show the feasibility of larger cable networks obtainable by
known constructions for infinite families of nut graphs
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