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Search: id:A123590
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| A123590 |
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Tripartite straight linked graphs as matrices producing polynomials and their triangular sequence: Matrix model (A120658 ): M(n,m,9)={{0, 1, 1, 1, 0, 0, 1, 0, 0}, {1, 0, 1, 0, 1, 0, 0, 1, 0}, {1, 1, 0, 0, 0, 1, 0, 0, 1}, {1, 0, 0, 0, 1, 1, 1, 0, 0}, {0, 1, 0, 1, 0, 1, 0, 1, 0}, {0, 0, 1, 1, 1, 0, 0, 0, 1}, {1, 0, 0, 1, 0, 0, 0, 1, 1}, {0, 1, 0, 0, 1, 0, 1, 0, 1}, {0, 0, 1, 0, 0, 1, 1, 1, 0}} This Model is striaght hyper-connections between 3 generalized K(n) complete graphs. |
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| 1, 1, -1, 0, -2, 1, 2, 3, 0, -1, 0, -4, -5, 0, 1, 0, 0, 0, 6, 0, -1, 0, 0, 12, -4, -9, 0, 1, 0, 12, -10, -24, 8, 12, 0, -1, 1, 4, -15, -8, 35, -4, -14, 0, 1, 64, -144, 0, 168, -36, -81, 12, 18, 0, -1, -128, 96, 320, -200, -284, 116, 121, -20, -22, 0, 1, 0, 40, -52, -236, 170, 354, -112, -158, 18, 25, 0, -1, 1280, -1536, -2304, 2432
(list; graph; listen)
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OFFSET
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1,5
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COMMENT
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The Large roots count: Table[x /. NSolve[CharacteristicPolynomial[An[d], x] == 0, x][[d]], {d, 2, 20}] {2.`, 2.`, 2.5615528128088303`, 2.449489742783178`, 3.`, 3.4880262221757476`, 3.552081133571793`, 3.9999999995851967`, 4.4586794310874645`, 4.597458186284443`, 5.`, 5.444061970030621`, 5.6239192478734195`, 5.999999274025329`, 6.43569176446824`, 6.641461869097823`, 6.999999682622629`, 7.415010662974701`, 7.654010866523878`}
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REFERENCES
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F. Chung and R. L. Graham, Erdos on Graphs, AK Peters Ltd., MA, 1998
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LINKS
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Eric Weisstein's World of Mathematics, Complete Graph
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FORMULA
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m(n,m,d)=If[m == n + Floor[d/3], 1, If[m == n - Floor[d/3], 1,If[m == n + Floor[2*d/3], 1, If[m == n - Floor[2*d/3],1, If[ n <= Floor[d/3] && m <= Floor[d/3] && (n < m || n > m), 1, If[ n > Floor[d/3] && n < Floor[2*d/3] + 1 && m > Floor[d/3] && m < Floor[2*d/3] + 1 && (n < m ||n > m), 1, If[ n > Floor[2*d/3] && m > Floor[2*d/3] && (n < m || n > m), 1, If[n == m, 0, 0]]]]]]]]
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EXAMPLE
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Triangular sequence:
{1},
{1, -1},
{0, -2, 1},
{2, 3, 0, -1},
{0, -4, -5, 0, 1},
{0, 0, 0, 6,0, -1},
{0, 0, 12, -4, -9, 0, 1},
{0, 12, -10, -24, 8, 12, 0, -1},
{1, 4, -15, -8, 35, -4, -14, 0, 1},
{64, -144, 0, 168, -36, -81, 12, 18, 0, -1},
{-128, 96, 320, -200, -284, 116, 121, -20, -22, 0, 1},
{0, 40, -52, -236,170, 354, -112, -158, 18,25, 0, -1},
{1280, -1536, -2304, 2432, 2016, -1440, -1008, 360, 261, -32, -30, 0, 1}, {-1920, -256, 5920, 1152, -6536, -1968, 3222, 1320, -666, -348, 46, 35, 0, -1}
Polynomials:
1
1 - x,
-2 x + x^2,
2 + 3 x - x^3,
-4 x - 5 x^2 + x^4,
6 x^3 - x^5,
12 x^2 - 4 x^3 - 9 x^4 + x^6,
12 x - 10 x^2 - 24 x^3 + 8 x^4 + 12 x^5 - x^7,
1 + 4 x - 15 x^2 - 8 x^3 + 35 x^4 - 4 x^5 - 14 x^6 + x^8,
64 - 144 x + 168 x^3 - 36 x^4 - 81 x^5 + 12 x^6 + 18 x^7 - x^9
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MATHEMATICA
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M[n_, m_, d_] = If[ m == n + Floor[d/3], 1, If[m == n - Floor[d/3], 1, If[m == n +Floor[2*d/3], 1, If[m == n - Floor[2*d/3], 1, If[ n <= Floor[d/3] && m <= Floor[d/3] && (n < m || n > m), 1, If[ n > Floor[d/3] && n < Floor[2*d/3] + 1 && m > Floor[d/3] && m <Floor[2*d/3] + 1 && (n < m || n > m), 1, If[ n > Floor[2*d/3] && m > Floor[2*d/3] && (n < m || n > m), 1, If[n == m, 0, 0]]]]]]]]; An[d_] := Table[M[n, m, d], {n, 1, d}, {m, 1, d}]; Join[An[1], Table[CoefficientList[CharacteristicPolynomial[An[d], x], x], {d, 1, 20}]]; Flatten[%]
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CROSSREFS
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Cf. A120658.
Sequence in context: A127510 A129390 A129391 this_sequence A092872 A113125 A088239
Adjacent sequences: A123587 A123588 A123589 this_sequence A123591 A123592 A123593
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KEYWORD
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uned,sign
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AUTHOR
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Roger Bagula and Gary Adamson (rlbagulatftn(AT)yahoo.com), Nov 12 2006
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