Counting Closed Walks and Spanning Trees in Graphs via Linear Algebra and Matrices 1 Adjacency Matrices and Counting Closed Walks

The material of this section is based on Chapter 1 of Richard Stanley’s notes “Topics in Algebraic Combinatorics”, which can be found at http://math.mit.edu/∼rstan/algcomb.pdf. Recall that an m-by-n matrix is an array of numbers (m rows and n columns), and we can multiply an mby-n matrix and an n-by-p matrix together to get an m-by-p matrix. The resulting matrix has entries obtained by taking each row of the first matrix, and each column of the second and taking their dot product. For example,