1. 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)

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

3. DEFINITION A double fixed-step loop network 15 1 14 2 3 13 12 4 5 111 2 3 4 5 6 7 8 9 10 11 12 13 14 15 4

4. DEFINITION An oriented spanning tree
An oriented spanning tree in a digraph D
is a rooted tree with the same vertex set as
D, that is, there is a node specified as the
root and from it there is a path to any vertex
of D. 1 1 1 3 2 3 2 3 2 G

5. APPLICATIONS
Reliability of a network 1 7 6 2 5 3

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

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

8. Techniques
Recurrence formulas

9. Techniques
The formula for

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

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

16. Thank you. Questions?
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