The Tropical Geometry of Graph Spanning Trees

This paper explores the deep connections between the combinatorial structure of spanning trees in a graph and the geometric framework of tropical geometry. We demonstrate that the set of all spanning trees of a given graph can be represented as a tropical variety, specifically a subvariety of a tropical Grassmannian. This representation is achieved by associating a vector to each spanning tree based on the weights of its edges, and the tropicalization of the classical Tutte polynomial provides the defining equations for this variety. The paper delves into the structure of this tropical variety, known as the Bergman fan of the underlying matroid of the graph. We show that the combinatorial properties of the spanning trees, such as their relationships through edge swaps, are encoded in the polyhedral complex structure of this tropical variety. This geometric viewpoint allows for the application of tools from algebraic and polyhedral geometry to solve combinatorial problems related to spanning trees, such as enumeration and optimization. Furthermore, we discuss the role of valuated matroids in extending this framework to weighted graphs, providing a geometric interpretation for optimization problems like finding the minimum spanning tree. The paper synthesizes foundational concepts from graph theory, matroid theory, and tropical geometry to provide a coherent and modern perspective on the study of spanning trees.