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The Spanning Trees Formulas in a Class of Double Fixed-Step Loop Networks

The Spanning Trees Formulas in a Class of Double Fixed-Step Loop Networks. Talip Atajan, Naohisa Otsuka Tokyo Denki University, Japan Xuerong Yong University of Puerto Rico, USA. This Talk is Composed of. A double fixed-step loop network An oriented spanning tree Reliability of a network

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The Spanning Trees Formulas in a Class of Double Fixed-Step Loop Networks

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  1. The Spanning Trees Formulas in a Class of Double Fixed-Step LoopNetworks Talip Atajan,Naohisa OtsukaTokyo Denki University, JapanXuerong YongUniversity of Puerto Rico, USA

  2. This Talk is Composed of • A double fixed-step loop network • An orientedspanning tree • Reliability of a network • Designing electrical circuits • Matrix tree theorem • Recurrence formula • A series of formulae for special p, q and n • New techniques

  3. 0 1 15 14 2 3 13 4 12 5 11 10 6 7 9 8 DEFINATION A double fixed-step loop network 4

  4. DEFINATION An orientedspanning tree An orientedspanning tree in a digraph D is a rooted tree with the same vertexset as D, that is, thereis a node specified as the root and fromit there is a path to any vertex of D. 0 1 0 1 0 1 3 2 3 3 2 2 G

  5. APPLICATION • Reliability of a network 0 1 7 2 6 5 3

  6. APPLICATION • Designing electrical circuits • Ohm’s Law • Kirchhoff’s Law

  7. Techniques Matrix Tree Theorem 0 1 3 2 G 0 1 3 2 0 1 3 2

  8. Techniques • Recurrence formula

  9. Techniques • The formulae for

  10. 3,5 C 16 0 3 6 9 0 3 0 6 9 1 15 0 3 6 9 12 13 14 2 12 13 12 13 3 13 15 10 15 10 15 4 10 12 2 7 5 2 2 11 7 7 10 6 5 5 4 4 5 4 7 9 8 1 14 11 8 1 14 11 8 1 14 11 8 Our Results 0 0 15 15 1 1 14 14 2 2 3 3 13 13 4 4 12 12 5 11 11 5 10 10 6 6 9 7 9 8 8 7

  11. Our Results • Theorem 2 (Opposed side of Waring's formula)

  12. Our Results

  13. Our Results • Using Newton's identities

  14. Our Results • Theorem 1

  15. 3,5 C 16 0 1 15 14 2 3 13 4 12 5 11 10 6 7 9 8 Our Results 0 13 14 15 11 12 10 2 3 1 9 7 8 6 5 4 0 15 1 14 2 3 13 4 12 11 5 10 6 9 8 7

  16. Thank you for your attandnce END

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