## Abstract

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

Original language | English |
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Journal | Random Structures and Algorithms |

DOIs | |

State | Accepted/In press - 2022 |

### Bibliographical note

Publisher Copyright:© 2022 Wiley Periodicals LLC.

## Keywords

- bounded degree
- counting
- spanning tree

## ASJC Scopus subject areas

- Software
- Mathematics (all)
- Computer Graphics and Computer-Aided Design
- Applied Mathematics