Lecture 10: Introduction to Algebraic Graph Theory

Let A(G) denote the adjacency matrix of the graph G. The polynomial pA(G)(x) is usually referred to as the characteristic polynomial of G. For convenience, we use p(G, x) to denote pA(G)(x). The spectrum of a graph G is the set of eigenvalues of A(G) together with their multiplicities. Since A (short for A(G)) is a real symmetric matrix, basic linear algebra tells us a few thing about A and its eigenvalues (the roots of p(G, x)). Firstly, A is diagonalizable and has real eigenvalues. Secondly, if λ is an eigenvalue of A, then the λ-eigenspace has dimension equal to the multiplicity of λ as a root of p(G, x). Thirdly, if n = |V (G)|, then Cn is the direct sum of all eigenspaces of A. Last but not least,