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CS 598: Spectral Graph Theory. Lecture 3 Extremal Eigenvalues and Eigenvectors of the Laplacian and the Adjacency Matrix. Today More on Courant-Fischer and Rayleigh quotients Applications of Courant-Fischer Adjacency matrix vs. Laplacian Perron-Frobenius Courant-Fischer Refresher (1) xT Ax k max min T n S R ,dim( S ) k xS x x xT Ax k n min max T S R ,dim( S ) n k 1 xS x x Sylvester’s Law of Inertia xT Ax k max min T n S R ,dim( S ) k xS x x Courant-Fischer Refresher (2) xT Lx k min max T S of dimk xS x x xT Lx k max min T S of dimn k 1 xS x x Courant-Fischer for Laplacian Applying Courant-Fischer for the Laplacian we get : 0, v 1 1 2 min x 1, x 0 1 T (x i xT Lx k min max T S of dim k xS x x x j )2 x Lx ( i , j )E min x 1, x 0 xT x xi 2 iV max max x0 T x Lx max x0 xT x (x ( i , j )E x j )2 x iV i 2 i Useful for getting bounds, if calculating spectra is cumbersome. To get upper bound on λ2, just need to produce vector with small Rayleigh Quotient. Similarly, t o get lower bound on λmax, just need to produce vector with large Rayleigh Quotient Example 1 Lemma1: Let G=(V,E) be a graph with some vertex w having degree d. Then max d Lemma 2: We can also improve on that. Under same assumptions, we can show: max d 1 Proof: see blackboard Example 1 Lemma1: Let G=(V,E) be a graph with some vertex w having degree d. Then max d Lemma 2: We can also improve on that. Under same assumptions, we can show: max d 1 Lemma 2 is tight, take star graph (ex) Example 2 The Path graph Pn on n vertices has 12 2 2 n Already knew that, but this is easier and more general. Proof: see blackboard Example 3 2 2 n Example 3 node 1 node 2 node 4 node 5 node 3 node 1 node 6 0 -1 1 1 1 -1 1 1 1 1 -1 -1 -1 -1 -1 Lower bounds are harder, we will see some in two lectures (different technique) Adjacency Matrix vs. Laplacian Adjacency Matrix Refresher i j 1, if (i, j ) edge Aij 0 if no edge between i, j j G = {V,E} • Unweighted graphs for simplicity i A has n eigenvalues (counting multiplicities) {a1 a2 … an } 1 Adjacency Matrix vs. Laplacian for d-regular graphs Bounds on the Eigenvalues of Adjacency Matrix Bounds on the Eigenvalues of Adjacency Matrix Courant-Fischer for Adjacency Matrix Refresher xT Ax k max min T S of dim k xS x x xT Ax 1 max T x 0 x x Will see next how to apply Courant-Fischer for the adjacency matrix to get another bound on the first eigenvalue as well as a relation to graph coloring Bounding Adjacency Matrix Eigenvalues xT Ax 1 max T x 0 x x Bounding Adjacency Matrix Eigenvalues Chromatic Number Chromatic Number Adjacency Matrix: The PerronFrobenius Theorem Adjacency Matrix: The PerronFrobenius Theorem Proof in 3 parts. We next show Lemma 1: If all entries of an nxn matrix A are positive, then it has a positive eigenvector v with corresponding positive eigenvalue α, that is Av=αv. Geometric proof on blackboard Adjacency Matrix: The PerronFrobenius Theorem Adjacency Matrix: The PerronFrobenius Theorem Laplacian: The Perron-Frobenius Theorem Theory can also be applied to Laplacians and any matrix with non-positive off-diagonal entries. It involves the eigenvector with smallest eigenvalue. Perron-Frobenius for Laplacians:Let M be a matrix with non-positive off-diagonal entries s.t. the graph of the no-zero off-diagonal entries is connected. Then the smallest eigenvalue has multiplicity 1 and the corresponding eigenvector is strictly positive. Proof on blackboard.