Spectral graph theory
Cospectral mates

A pair of graphs are said to be cospectral mates if they have the same spectrum, but are non-isomorphic.

The smallest pair of cospectral mates is { K, C ∪ K }, comprising the 5-vertex star and the graph union of the 4-vertex cycle and the single-vertex graph, as reported by Collatz and Sinogowitz in 1957.

The smallest pair of polyhedral cospectral mates are enneahedra with eight vertices each.
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