The following example has two pentagons glued on an edge, and this example has 4 exotic solutions, and it is the unique graph on 8 vertices with more than 2 exotic solutions.
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The computation of these (linearly) stable solutions takes perhaps a minute:
elapsedTime stablesolsPentPent = showExoticSolutions PentPent
The following example has a hexagon and pentago glued identified on a common edge, and this example has 9 vertices and 6 exotic solutions.
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The computation of exotic solutions for this example takes several minutes:
elapsedTime stablesolsHexPent = showExoticSolutions HexPent
The source of this document is in /build/reproducible-path/macaulay2-1.25.05+ds/M2/Macaulay2/packages/Oscillators/Documentation.m2:906:0.