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Lecture 10: Introduction to Algebraic Graph Theory

Let A(G) denote the adjacency matrix of the graph G. The polynomialpA(G)(x) is usually referred to as thecharacteristic polynomial of G. For convenience, we use p(G, x) to denotepA(G)(x). Thespectrum of a graphG is the set of eigenvalues of A(G) together with their multiplicities. Since A (short forA(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 theλ-eigenspace has dimension equal to the multiplicity of λ as a root ofp(G, x). Thirdly, if n = |V (G)|, thenCn is the direct sum of all eigenspaces of A. Last but not least,