Tomaž Pisanski: More questions than answers about nut graphs

Singular graphs of nullity one admitting a full kernel eigenvector are called nut graphs. Nut graphs constitute a fascinating family of graphs that has found many applications, in particular in mathematical chemistry. It has been promoted mostly by Irene Sciriha and her co-authors. Sciriha almost single-handedly laid down the fundamental theory of nut graphs. The emphasis of this talk is in the entries of the standard kernel eigenvector associated with a nut graph. Namely, the standard kernel eigenvector having the property that its integer entries have no non-trivial common factor is unique up to multiplication by -1.

In this talk we present some questions about nut graphs that have attracted researchers of this topic. We recall some tools that enable one to construct larger nut graphs from smaller ones. They explain some answers to these questions. We also ask some questions that may be of interest to some students.

This is a preparatory talk for another talk that will be delivered in Sombor, Serbia, in June 2022.

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