## The number of spanning trees in a class of double fixed-step loop networks.(English)Zbl 1155.05032

Summary: We develop a method to count the number of spanning trees in certain classes of double fixed-step loop networks with nonconstant steps. More specifically our technique finds the number of spanning trees in $$\vec C_n^{p,q}$$, the double fixed-step loop network with $$n$$ vertices and jumps of size $$p$$ and $$q$$, when $$n = d_{1}m$$, and $$q = d_{2}m + p$$ where $$d_{1}$$, $$d_{2}$$, and $$p$$ are arbitrary parameters and $$m$$ is a variable.

### MSC:

 05C20 Directed graphs (digraphs), tournaments 05C05 Trees 05C30 Enumeration in graph theory 68R10 Graph theory (including graph drawing) in computer science
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### References:

 [1] Bermond, Distributed loop computer networks: A survey, J Parallel Distrib Comput 24 pp 2– (1995) [2] Biggs, Algebraic graph theory (1993) · Zbl 0284.05101 [3] Boesch, Spanning tree formulas and Chebyshev polynomials, Graph Combinatorics 2 pp 191– (1986) · Zbl 0651.05028 [4] Cvetkovič, Spectra of graphs: Theory and applications (1995) [5] Erdös, Distributed loop networks with minimum transmission delay, Theoret Comput Sci 100 pp 223– (1992) [6] Fàbrega, Fault-tolerant routings in double fixed-step networks, Discrete Appl Math 78 pp 61– (1997) · Zbl 0890.68099 [7] Golin, Counting spanning trees and other structures in non constant-jump circulant graphs, 15th Annu Int Symp Algorithms and Computation pp 508– (2004) · Zbl 1116.05303 [8] Harary, Graph theory (1969) [9] Hwang, A survey on double loop networks, Proc DIMACS Workshop on Reliability of Computer and Communication Networks pp 143– (1989) [10] Jennings, First course in numerical methods (1964) · Zbl 0127.08102 [11] Kirchhoff, Über die Auflösung der gleichungen, auf welche man bei der untersuchung der linearen verteilung galvanischer Ströme geführt wird, Ann Phys Chem 72 pp 497– (1847) [12] Liu, Distributed loop computer networks, Adv Comput 17 pp 163– (1978) [13] Lonc, On the number of spanning trees in directed circulant graphs, Networks 37 pp 129– (2001) · Zbl 0974.05043 [14] Vohra, Counting spanning trees in the graphs of Kleitman and Golden and a generalization, J Franklin Inst 318 pp 349– (1984) · Zbl 0569.05016 [15] Yong, An asymptotic behavior of the complexity of double fixed step loop networks, Appl Math J Chinese Univ Ser B 12 pp 233– (1997) [16] Zhang, Asymptotic enumeration theorems for the numbers of spanning trees and Eulerian trails in circulant digraphs and graphs, Sci China Ser A 43 pp 264– (1999) [17] Zhang, The number of spanning trees in circulant graphs, Discrete Math 223 pp 337– (2000) · Zbl 0969.05036
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