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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 at Mayaguez, USA (Presented at the SIAM Workshop on Analytic Algorithmics and Combinatorics (Jan 2009, New York).

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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 Loop Networks Talip Atajan,Naohisa OtsukaTokyo Denki University, JapanXuerong YongUniversity of Puerto Rico at Mayaguez, USA(Presented at the SIAM Workshop on Analytic Algorithmics and Combinatorics (Jan 2009, New York)

  2. We Will Talk About • A double fixed-step loop network • An orientedspanning tree • Reliability of a network • Designing electrical circuits • Modified matrix tree theorem • Recurrence formulas • New techniques and open problems

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

  4. DEFINITION 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. APPLICATIONS • Reliability of a network 0 1 7 2 6 5 3

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

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

  8. Techniques • Recurrence formulas

  9. Techniques • The formula 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. Questions? END

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