Methods from Linear Algebra for the Enumeration of Spanning Trees

Sammanfattning: In this report, we study the enumeration of spanning trees in graphs, using two methods withinlinear algebra, Kirchhoff’s Matrix Tree Theorem and an alternative method, also referred to asLemma 1, derived by S. Klee and M.T Stamps in [KS20]. Along with introducing preliminary tools from linear algebra, we also study the Laplace matrix ofa graph and use its properties in the two methods to derive combinatorical expressions on spanningtree enumeration of different graph families. We discuss properties of the Laplace matrix obtainedfrom different graph structures, and determine which method is more suitable in each case, withrespect to linear algebra. Specifically, complete graphs, Ferrers graphs and Windmill graphs areconsidered.