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SCT graphs with exotic solutions -- finding graphs of small size with exotic solutions

We first construct the ideal $I_G$ for a specific graph $G$ on 5 vertices. We use the 5-cycle as the specific example.

i1 : needsPackage "Oscillators"

o1 = Oscillators

o1 : Package
i2 : needsPackage "NautyGraphs"

o2 = NautyGraphs

o2 : Package
i3 : Gstrs = generateGraphs(5, OnlyConnected => true, MinDegree => 2);
i4 : Gs = Gstrs/stringToGraph

o4 = {Graph{0 => {3, 4}   }, Graph{0 => {3, 4}      }, Graph{0 => {2, 4}   
            1 => {3, 4}            1 => {3, 4}               1 => {3, 4}   
            2 => {3, 4}            2 => {3, 4}               2 => {0, 4}   
            3 => {0, 1, 2}         3 => {0, 1, 2, 4}         3 => {1, 4}   
            4 => {0, 1, 2}         4 => {3, 0, 1, 2}         4 => {2, 0, 3,
     ------------------------------------------------------------------------
       }, Graph{0 => {2, 3}}, Graph{0 => {2, 3, 4}}, Graph{0 => {2, 3, 4}
                1 => {3, 4}         1 => {3, 4}            1 => {3, 4}   
                2 => {0, 4}         2 => {0, 4}            2 => {0, 4}   
                3 => {0, 1}         3 => {0, 1}            3 => {0, 1, 4}
     1}         4 => {2, 1}         4 => {2, 0, 1}         4 => {2, 0, 3,
     ------------------------------------------------------------------------
       }, Graph{0 => {2, 3, 4}   }, Graph{0 => {2, 3, 4}}, Graph{0 => {2, 3,
                1 => {3, 4}               1 => {2, 3, 4}         1 => {2, 3,
                2 => {0, 3, 4}            2 => {0, 1, 4}         2 => {0, 1,
                3 => {2, 0, 1, 4}         3 => {0, 1}            3 => {0, 1,
     1}         4 => {2, 0, 3, 1}         4 => {2, 0, 1}         4 => {2, 0,
     ------------------------------------------------------------------------
     4}   }, Graph{0 => {2, 3, 4}   }, Graph{0 => {1, 2, 3, 4}}}
     4}            1 => {2, 3, 4}            1 => {0, 2, 3, 4}
     4}            2 => {0, 1, 3, 4}         2 => {1, 0, 3, 4}
     4}            3 => {2, 0, 1, 4}         3 => {1, 0, 2, 4}
     1, 3}         4 => {2, 0, 1, 3}         4 => {1, 0, 2, 3}

o4 : List
i5 : printingPrecision = 3

o5 = 3
i6 : for G in Gs list showExoticSolutions G;
-- warning: experimental computation over inexact field begun
--          results not reliable (one warning given per session)
 -- 3.56s elapsed
 -- 3.17s elapsed
 -- 4.55s elapsed
 -- 5.85s elapsed
 -- 6.04s elapsed
 -- 6.56s elapsed
 -- 7.37s elapsed
 -- 5.68s elapsed
 -- 6.75s elapsed
 -- 9.26s elapsed
 -- 6.44s elapsed
warning: some solutions are not regular: {57, 58, 59, 60, 61, 62, 63, 64, 66, 67, 69, 70, 73, 78, 81, 82, 83, 84, 85, 86, 89, 91, 92}
warning: some solutions are not regular: {52, 55, 57, 58, 59, 60, 61, 63, 65, 66, 68, 71, 72, 73, 74, 77, 78, 79, 83, 84, 85, 87, 89, 91, 92, 95, 96, 98, 99, 100}
-- found extra exotic solutions for graph Graph{0 => {2, 3}} --
                                                1 => {3, 4}
                                                2 => {0, 4}
                                                3 => {0, 1}
                                                4 => {2, 1}
+-----+----+----+-----+-----+-----+-----+-----+
|-.809|.309|.309|-.809|.588 |-.951|.951 |-.588|
+-----+----+----+-----+-----+-----+-----+-----+
|1    |1   |1   |1    |0    |0    |0    |0    |
+-----+----+----+-----+-----+-----+-----+-----+
|-.809|.309|.309|-.809|-.588|.951 |-.951|.588 |
+-----+----+----+-----+-----+-----+-----+-----+
-- angles (in degrees), first angle is zero and omitted --
+---+---+---+---+
|144|288|72 |216|
+---+---+---+---+
|0  |0  |0  |0  |
+---+---+---+---+
|216|72 |288|144|
+---+---+---+---+
warning: some solutions are not regular: {24, 25, 29, 30, 31, 34, 35, 42, 44, 46, 51, 52, 55}
warning: some solutions are not regular: {15, 16, 18, 19, 20, 21, 22, 23, 24, 26, 27, 28, 30, 31, 32, 33, 34, 35, 36, 37}
warning: some solutions are not regular: {20, 27, 29, 30, 31}
warning: some solutions are not regular: {46, 51, 52, 55, 56, 65, 67, 70, 73, 75, 76}

The source of this document is in /build/reproducible-path/macaulay2-1.25.06+ds/M2/Macaulay2/packages/Oscillators/Documentation.m2:812:0.