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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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