Spanning Trees in 2-trees

TL;DR

This paper studies the enumeration of spanning trees in 2-trees, providing bounds, explicit counts, and characterizations of extremal cases, revealing a Fibonacci number pattern for the maximum.

Contribution

It introduces an inductive method to list spanning trees in 2-trees and characterizes the extremal 2-trees with the most and fewest spanning trees.

Findings

The 2-tree with the fewest spanning trees has $n-2$ vertices of degree 2 and exactly $n2^{n-3}$ spanning trees.

The 2-trees with the most spanning trees have exactly two vertices of degree 2, with the count equal to Fibonacci number $F_{2n-2}$.

An inductive approach is used to enumerate spanning trees without repetition and establish bounds.

Abstract

A spanning tree of a graph $G$ is a connected acyclic spanning subgraph of $G$. We consider enumeration of spanning trees when $G$ is a $2$-tree, meaning that $G$ is obtained from one edge by iteratively adding a vertex whose neighborhood consists of two adjacent vertices. We use this construction order both to inductively list the spanning trees without repetition and to give bounds on the number of them. We determine the $n$-vertex $2$-trees having the most and the fewest spanning trees. The $2$-tree with the fewest is unique; it has $n-2$ vertices of degree $2$ and has $n2^{n-3}$ spanning trees. Those with the most are all those having exactly two vertices of degree $2$, and their number of spanning trees is the Fibonacci number $F_{2n-2}$.