A spanning forest of a graph 
is a spanning subgraph
that is a forest. For enumeration, every acyclic choice
of edges is counted, retaining all vertices,
including isolated vertices. Some uses additionally
require maximality, giving one spanning tree in
each connected component of 
. The enumeration convention does not impose this requirement.
Let 
count spanning forests under the enumeration convention. Wu et al. (2026)
proved that every nonempty 
-regular graph that is a simple graph on 
vertices, with 
, satisfies
 |
The exponential constant is best possible, approached by sequences of 
-regular graphs whose girth
tends to infinity. This resolves the upper-bound conjecture described by Wu et
al. (2026).
