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Example 4.1: unique graph on 8 vertices with exotic solutions and no induced cycle of length at least 5 -- example 4.1 in arXiv 2312.16069

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 $k \ge 5$.

i1 : needsPackage "Oscillators";
i2 : printingPrecision = 3;
i3 : G = graph{
         {0,4},{1,4},{1,5},
         {3,5},{2,6},{3,6},
         {0,7},{2,7},{4,5},
         {5,6},{6,7},{4,7}}

o3 = Graph{0 => {4, 7}      }
           1 => {4, 5}
           2 => {6, 7}
           3 => {5, 6}
           4 => {0, 1, 5, 7}
           5 => {4, 1, 3, 6}
           6 => {5, 3, 2, 7}
           7 => {0, 4, 2, 6}

o3 : Graph
i4 : stablesols = {{0, 0, -1, .707, -.707, -.707, .707, -1, 1, 0, -.707, -.707, .707, .707},
         {1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0},
         {0, 0, -1, .707, -.707, -.707, .707, 1, -1, 0, .707, .707, -.707, -.707}}

o4 = {{0, 0, -1, .707, -.707, -.707, .707, -1, 1, 0, -.707, -.707, .707,
     ------------------------------------------------------------------------
     .707}, {1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0}, {0, 0, -1, .707,
     ------------------------------------------------------------------------
     -.707, -.707, .707, 1, -1, 0, .707, .707, -.707, -.707}}

o4 : List

Computing the (linearly) stable solutions for K5C5 takes a minute or two:

elapsedTime stablesols = showExoticSolutions G;

See also


The source of this document is in /build/reproducible-path/macaulay2-1.25.05+ds/M2/Macaulay2/packages/Oscillators/Documentation.m2:841:0.