2/23/2017 Ma/CS 6b Class 20: Spectral Graph Theory By Adam Sheffer Recall: Parity of a Permutation 푆푛 – the set of permutations of 1,2, … , 푛 . A permutation 휎 ∈ 푆푛 is even if it can be written as a composition of an even number of transpositions. Otherwise, 휎 is odd. 1, 휎 is even 푠푔푛 휎 = ቊ −1, 휎 is odd 1 2/23/2017 Determinants 푆푛 – the set of permutations of 1,2, … , 푛 . 푀 – an 푛 × 푛 matrix. 푚푖푗 – 푗’th cell in 푖’th row of 푀. The determinant of 푀 is det 푀 = 푀 = ෍ 푚1,휎(1) ⋯ 푚푛,휎 푛 ⋅ 푠푔푛 휎 휎∈푆푛 Simple Example det 푀 = 푀 = ෍ 푚1,휎(1) ⋯ 푚푛,휎 푛 ⋅ 푠푔푛 휎 휎∈푆푛 푎 푏 = 푎푑 − 푏푐. 푐 푑 2 2/23/2017 Eigenvalues and Eigenvectors 퐴 – an 푛 × 푛 matrix of real numbers. The eigenvalues of 퐴 are the numbers 휆 such that 퐴푥 = 휆푥 for some nonzero vector 푥 ∈ ℝ푛. ◦ A vector 푥 that satisfies 퐴푥 = 휆푥 for some eigenvalue 휆 is an eigenvector of 퐴. 2 1 Example. 퐴 = . 1 2 ◦ Eigenvalue 휆1 = 3 with eigenvectors of the form 푎, 푎 . ◦ Eigenvalue 휆2 = 1 with eigenvalues of the form 푎, −푎 . Characteristic Polynomial 퐴 – an 푛 × 푛 matrix of real numbers. The characteristic polynomial of 퐴 is 휙 퐴; 휆 = det 휆퐼 − 퐴 . ◦ The roots of 휙 퐴; 휆 are exactly the eigenvalues of 퐴. ◦ The algebraic multiplicity of an eigenvalue 휆푖 is the multiplicity of 휆푖 in 휙 퐴; 휆 . ◦ (Do not confuse with the geometric multiplicity, which is the dimension of the space of eigenvectors associated with 휆). 3 2/23/2017 Example: Multiplicities 2 0 0 0 퐴 = 1 2 0 0 0 1 3 0 0 0 1 3 휆 − 2 0 0 0 det 휆퐼 − 퐴 = det −1 휆 − 2 0 0 0 −1 휆 − 3 0 0 0 −1 휆 − 3 = 휆 − 2 2 휆 − 3 2. So we have the eigenvalues 휆1 = 2 and 휆2 = 3, each with (algebraic) multiplicity 2. Recall: Adjacency Matrix Consider a graph 퐺 = 푉, 퐸 . ◦ We order the vertices as 푉 = 푣1, 푣2, … , 푣푛 . ◦ The adjacency matrix of 퐺 is a symmetric 푛 × 푛 matrix 퐴. The cell 퐴푖푗 contains the number of edges between 푣푖 and 푣푗. 푣1 0 1 0 0 1 1 0 1 0 1 푣 푣 퐴 = 0 1 0 1 0 5 2 0 0 1 0 1 1 1 0 1 0 푣4 푣3 4 2/23/2017 The Spectrum of a Graph Consider a graph 퐺 = 푉, 퐸 and let 퐴 be the adjacency matrix of 퐺. ◦ The eigenvalues of 퐺 are the eigenvalues of 퐴. ◦ The characteristic polynomial 휙 퐺; 휆 is the characteristic polynomial of 퐴. ◦ The spectrum of 퐺 is 휆 , … , 휆 푠푝푒푐 퐺 = 1 푡 , 푚1, … , 푚푡 where 휆1, … , 휆푡 are the distinct eigenvalues of 퐴 and 푚푖 is the multiplicity of 휆푖. Example: Spectrum 푣1 푣2 0 1 0 1 퐴 = 1 0 1 0 0 1 0 1 1 0 1 0 푣4 푣3 휆 −1 0 −1 det 휆퐼 − 퐴 = det −1 휆 −1 0 0 −1 휆 −1 −1 0 −1 휆 = 휆2 휆 − 2 휆 + 2 . 0 2 −2 푠푝푒푐 퐶 = 4 2 1 1 5 2/23/2017 Spectral Graph Theory Spectral graph theory is the study of properties of a graph in relationship to the characteristic polynomial, eigenvalues, and eigenvectors of matrices associated with the graph, such as its adjacency matrix. ◦ Beyond being useful in graph theory, it is also used in research in quantum chemistry. Slight Change of Notation Instead of multiplicities, let 휆1, … , 휆푛 be the not necessarily distinct eigenvalues of 퐺. 2 −1 For example, if the spectrum is , 2 2 we write 휆1 = 휆2 = 2 and 휆3 = 휆4 = −1 (instead of 휆1 = 2, 푚1 = 2, 휆2 = −1, 푚2 = 2). 6 2/23/2017 The Spectral Theorem Every adjacency matrix 퐴 is symmetric and real. Theorem. Any real symmetric 푛 × 푛 matrix has real eigenvalues and 푛 orthonormal eigenvectors. ◦ The algebraic and geometric multiplicities are the same in this case. 푛 ◦ We have 휙 퐴; 휆 = ς푖=1 휆 − 휆푖 . ◦ The multiplicity of an eigenvalue 휆 is 푛 − 푟푎푛푘 휆퐼 − 퐴 . Changing the Diagonal 퐴 – an 푛 × 푛 real symmetric matrix. 푐 – a real constant. Adding 푐 to each cell in the diagonal of 퐴 increases each of the eigenvalues of 퐴 by 푐. ◦ Since 훼 + 푐 is a root of det 휆퐼 − (푐퐼 + 퐴) if and only if 훼 is a root of det(휆퐼 − 퐴). 7 2/23/2017 The Spectrum of a Complete Graph Problem. What is the spectrum of 퐾푛? ◦ 1푛×푛 – the all 1’s 푛 × 푛 matrix. ◦ The adjacency matrix is 퐴 = 1푛×푛 − 퐼. ◦ What is the spectrum of 1푛×푛? 푛 0 . 1 푛 − 1 (Eigenvectors 1,1, … , 1 , 1, −1,0, … , 0 , etc.) ◦ Using the argument from the previous slide, we have 푛 − 1 −1 푠푝푒푐 퐾 = . 푛 1 푛 − 1 Trace 퐴 – 푛 × 푛 adjacency matrix of a graph 퐺. 푛 The trace of 퐴 is 푡푟푎푐푒 퐴 = σ푖=1 퐴푖푖. In 휙 퐴; 휆 = det 휆퐼 − 퐴 , the coefficient of 휆푛−1 is −푡푟푎푐푒 퐴 . 푛 Since 휙 퐴; 휆 = ς푖=1 휆 − 휆푖 , we have 푛 푡푟푎푐푒 퐴 = σ푖=1 휆푖 Since 퐺 is simple, we get 푛 푡푟푎푐푒 퐴 = ෍ 퐴푖푖 = 0. 푖=1 푛 ◦ Thus, we have σ푖=1 휆푖 = 0. 8 2/23/2017 Complete Bipartite Graphs Problem. What is the spectrum of 퐾푚,푛? 0 1 ◦ We have 퐴 = 푚×푚 푚×푛 . 1푛×푚 0푛×푛 ◦ We have 푟푎푛푘 퐴 = 2 so the multiplicity of 휆 = 0 is 푚 + 푛 − 푟푎푛푘 0 ⋅ 퐼 − 퐴 = 푚 + 푛 − 2. ◦ Denote the remaining eigenvalues as 휆2 and 휆3. We have 0 = 푡푟푎푐푒 퐴 = 휆2 + 휆3. Searching for 휆2 We know that the spectrum of 퐾푚,푛 is 0 휆 −휆 2 2 . 푚 + 푛 − 2 1 1 ◦ We thus have 푚+푛 푚+푛−2 2 2 휙 퐾푚,푛; 휆 = ෑ 휆 − 휆푖 = 휆 휆 − 휆2 푖=1 0 1 ◦ Recall that 퐴 = 푚×푚 푚×푛 . 1푛×푚 0푛×푛 ◦ Thus the coefficient of 휆푚+푛−2 in det 휆퐼 − 퐴 is − σ푖≠푗 퐴푖푗퐴푗푖 = −푚푛. 9 2/23/2017 Finding 휆2 푚+푛 푚+푛 2 푚+푛−2 휙 퐺; 휆 = ෑ 휆 − 휆푖 = 휆 − 휆2휆 . 푖=1 The coefficient of 휆푚+푛−2 in det 휆퐼 − 퐴 is − σ푖≠푗 퐴푖푗퐴푗푖 = −푚푛. So the spectrum of 퐾푚,푛 is 0 푚푛 − 푚푛 . 푚 + 푛 − 2 1 1 Which Famous American Serial Killer is Related to Mathematics Charles Manson The Unabomber Ted Bundy 10 2/23/2017 Recall: Minors A minor of a matrix 퐴 is the determinant of a square sub-matrix, cut down from 퐴 by removing one or more of its rows or columns. A 푘 × 푘 minor is the determinant of a 푘 × 푘 submatrix of 퐴. The Coefficient of 휆푛−2 퐴 – 푛 × 푛 adjacency matrix of a (simple) graph 퐺 = (푉, 퐸). ◦ The leading coefficient of 휙 퐺; 휆 is 1. ◦ The coefficient of 휆푛−1 is −푡푟푎푐푒 퐴 = 0. What is the coefficient of 휆푛−2? ◦ As in the example of 퐾푚,푛, it is − σ푖≠푗 퐴푖푗퐴푗푖 (sum of the 2 × 2 minors with the same row and column indices). ◦ Such a minor is -1 if 푣푖, 푣푗 ∈ 퐸, and 0 otherwise. ◦ Therefore, the coefficient of 휆푛−2 is −|퐸|. 11 2/23/2017 퐴 – 푛 × 푛 adjacency matrix of a graph 퐺. What is the coefficient of 휆푛−3 in 휙 퐺; 휆 ? ◦ It is the sum of the 3 × 3 minors with the same row and column indices). ◦ Such a 3 × 3 minor is nonzero iff for the corresponding indices 푖, 푗, 푘 we have that 푣푖, 푣푗, 푣푘 form a cycle of length three. Then 0 푒푖푗 푒푖푘 0 −1 −1 det 푒푖푗 0 푒푗푘 = det −1 0 −1 푒푖푘 푒푗푘 0 −1 −1 0 = −2. ◦ Thus, the coefficient of 휆푛−3 is −2 times the number of copies of 퐾3 in 퐺. A Generalization 퐴 – 푛 × 푛 adjacency matrix of a graph 퐺. The coefficient of 휆푛−푖 in 휙 퐺; 휆 is the sum of the 푖 × 푖 minors of 퐴 with the same row and column indices (times −1 푖). 12 2/23/2017 Powers of 퐴 퐴 – 푛 × 푛 adjacency matrix of a graph 퐺. Recall that cell 푖, 푗 of 퐴푘 is the number of paths of length 푘 between 푣푖 and 푣푗. If 휆 is an eigenvalue of 퐴, what do we know about 퐴푘? ◦ 푣 – an eigenvector of 휆. ◦ 퐴푘푣 = 휆퐴푘−1푣 = 휆2퐴푘−2푣 = ⋯ = 휆푘푣. ◦ So 휆푘 is an eigenvalue of 퐴푘, with corresponding eigenvector 푣. Adding Isolated Vertices 퐴 – 푛 × 푛 adjacency matrix of a graph 퐺. What happens to the spectrum of 퐺 when adding an isolated vertex? ◦ We add a row and a column of zeros to 퐴. ◦ Recall that the multiplicity of an eigenvalue 휆 is 푛 − 푟푎푛푘 휆퐼 − 퐴 . ◦ Specifically, the multiplicity of the eigenvalue 0 is 푛 − 푟푎푛푘(퐴). ◦ Since adding the new row and column does not change 푟푎푛푘(퐴), this increases the multiplicity of 0 by 1. 13 2/23/2017 Eigenvalues of Bipartite Graphs Lemma. Let 퐺 be a bipartite graph and let 휆 be a nonzero eigenvalue of 퐺 with multiplicity 푚. Then −휆 is also an eigenvalue of 퐺 with multiplicity 푚. 푣1 푣2 푣4 푣3 Example: Spectrum 푣1 푣2 0 1 0 1 퐴 = 1 0 1 0 0 1 0 1 1 0 1 0 푣4 푣3 휆 −1 0 −1 det 휆퐼 − 퐴 = det −1 휆 −1 0 0 −1 휆 −1 −1 0 −1 휆 = 휆2 휆 − 2 휆 + 2 .