Study of Exponential Growth Constants of Directed Heteropolygonal Archimedean Lattices
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We infer upper and lower bounds on the exponential growth constants \(\alpha (\Lambda )\), \(\alpha _0(\Lambda )\), and \(\beta (\Lambda )\) describing the large-n behavior of, respectively, the number of acyclic orientations, acyclic orientations with a unique source vertex, and totally cyclic orientations of arrows on bonds of several n-vertex heteropolygonal Archimedean lattices \(\Lambda \). These are, to our knowledge, the best bounds on these growth constants. The inferred upper and lower bounds on the growth constants are quite close to each other, which enables us to infer rather accurate estimates for the actual exponential growth constants. Our new results for heteropolygonal Archimedean lattices, combined with our recent results for homopolygonal Archimedean lattices, are consistent with the inference that the exponential growth constants \(\alpha (\Lambda )\), \(\alpha _0(\Lambda )\), and \(\beta (\Lambda )\) on these lattices are monotonically increasing functions of the lattice coordination number. Comparisons are made with the corresponding growth constants for spanning trees on these lattices. Our findings provide further support for the Merino–Welsh and Conde–Merino conjectures.
KeywordsAcyclic orientations Cyclic orientations Directed graphs
This research was supported in part by the Taiwan Ministry of Science and Technology grant MOST 103-2918-I-006-016 (S.-C.C.) and by the U.S. National Science Foundation grant No. NSF-PHY-16-1620628 (R.S.).
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