Counting Isomorphism Classes of Spanning Trees of Complete Bipartite Graphs

TL;DR

This paper investigates the number of distinct isomorphism classes of spanning trees in complete bipartite graphs, providing bounds based on degree sequences and integer partitions.

Contribution

It introduces new bounds on the count of isomorphism classes of spanning trees in complete bipartite graphs using partition functions.

Findings

Lower bounds based on partition counts $P_a(a+b-1)$ and $P_b(a+b-1)$

An upper bound in terms of the parameters $a$ and $b$

Enhanced understanding of the relationship between degree sequences and spanning tree structures

Abstract

Spanning trees of complete bipartite graphs exhibit a rich interaction between degree sequences and graph structure. In this paper, we obtain lower bounds on the number of isomorphism classes of spanning trees in $K_{a,b}, 2 \leq a \leq b$ in terms of $P_a(a+b-1)$ and $P_b(a+b-1)$ where $P_k(m)$ is the number of integer partitions of $m$ of length $k$. Furthermore, we obtain an upper bound in terms of $a$ and $b$.