The number of bounded‐degree spanning trees

Abstract

AbstractFor a graph , let be the number of spanning trees of with maximum degree at most . For , it is proved that every connected ‐vertex ‐regular graph with satisfies where approaches 1 extremely fast (e.g., ). The minimum degree requirement is essentially tight as for every there are connected ‐vertex ‐regular graphs with for which . Regularity may be relaxed, replacing with the geometric mean of the degree sequence and replacing with that also approaches 1, as long as the maximum degree is at most . The same holds with no restriction on the maximum degree as long as the minimum degree is at least .