Counting the number of spanning trees in a class of double fixed-step loop networks.(English)Zbl 1202.05062

Summary: A double fixed-step loop network, $${\vec C}_n^{p,q}$$, is a digraph on $$n$$ vertices $$0,1,2,\dots,n-1$$ and for each vertex $$i$$ ($$0<i\leq n-1$$), there are exactly two arcs going from vertex $$i$$ to vertices $$i+p$$, $$i+q$$ $$\pmod n$$. Let $$p<q<n$$ be positive integers such that $$(q-p)† n$$ and $$(q-p)|(k_0n-p)$$ or $$(q-p)|n$$ (where $$k_0=\min \{k|(q-p)|(kn-p)$$, $$k=1,2,3,\dots\}$$ and $$\gcd(q,p)=1$$. In this work we derive a formula for the number of spanning trees, $$T({\vec C}_n^{p,q})$$, with constant or nonconstant jumps and prove that $$T({\vec C}_n^{p,q})$$ can be represented asymptotically by the $$m$$th-order ‘Fibonacci’ numbers. Some special cases give rise to the formulas obtained recently in [Z. Lonc, K. Parol and J.M. Wojciechowski, Networks 37, No. 3, 129–133 (2001; Zbl 0974.05043); X. Yong and F. Zhang, Appl. Math., Ser. B (Engl. Ed.) 12, No. 2, 233–236 (1997; Zbl 0880.05027); Y. Wang, S.-C. Fang and J.E. Lavery, J. Comput. Appl. Math. 201, No. 1, 69–87 (2007; Zbl 1110.65015)].

MSC:

 05C30 Enumeration in graph theory 68M10 Network design and communication in computer systems 68R10 Graph theory (including graph drawing) in computer science 90C35 Programming involving graphs or networks
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References:

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