This example is a square inside a square, having 8 vertices, and exotic solutions. This is the only SCT graph on 8 vertices with exotic solutions, not containing an induced $k$-cycle, for $k \ge 5$.
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Computing the (linearly) stable solutions for K5C5 takes a minute or two:
elapsedTime stablesols = showExoticSolutions G;
The source of this document is in /build/reproducible-path/macaulay2-1.25.05+ds/M2/Macaulay2/packages/Oscillators/Documentation.m2:841:0.