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

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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

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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).

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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

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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

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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).

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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).

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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.

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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.

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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

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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|.

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 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).

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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.

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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 .

0 2 −2 푠푝푒푐 퐺 = 2 1 1

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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.  Proof. ◦ If the two sides of 퐺 have different sizes, we even them out by adding isolated vertices. These do not change the spectrum of 퐺 except for increasing the multiplicity of 0.

 푛 – the number of vertices in each part.  We can write the adjacency matrix of 퐺 as 0푛×푛 퐵 푇 where 퐵 is an 푛 × 푛 matrix. 퐵 0푛×푛 푥  푣 = 푦 – an eigenvector of 휆 (where both 푥 and 푦 have 푛 coordinates). Then 휆푥 0 퐵 푥 퐵푦 = 휆푣 = 퐴푣 = = . 휆푦 퐵푇 0 푦 퐵푇푥 ′ 푥  Then for 푣 = −푦 we have 퐵 −푦 −휆푥 퐴푣′ = = . 퐵푇푥 휆푦  That is, 푣′ is an eigenvector of −휆. Also, if 휆 has 푘 independent eigenvectors, so does −휆.

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More Bipartite Graphs

 Theorem. Let 퐺 be a graph with 푛 vertices. Then the following statements are equivalent: ◦ 퐺 is bipartite. ◦ The nonzero eigenvalues of 퐺 occur in pairs: 휆푖 = −휆푗. 푛 푡 ◦ σ푖=1 휆푖 = 0 for any odd positive integer 푡.

 Proof. ◦ The previous lemma shows that the first statement implies the second.

Proof

 Assume that the eigenvalues of 퐺 occur in pairs: 휆푖 = −휆푗. ◦ For any positive odd integer 푡, we have 푡 푡 휆푖 = −휆푗 . 푛 푡 ◦ Thus σ푖=1 휆푖 = 0.

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Proof (cont.)

 Assume that for any positive odd integer 푛 푡 푡, we have σ푖=1 휆푖 = 0. 푡 푡 ◦ Recall that the 휆푖 ’s are the eigenvalues of 퐴 . 푡 푛 푡 푡 ◦ Since 푡푟푎푐푒 퐴 = σ푖=1 휆푖 = 0, and 퐴 contains only non-negative elements, the main diagonal of 퐴푡 is all zeros. ◦ That is, there are no odd-length cycles in 퐺. ◦ Recall from 6a that a 퐺 is bipartite iff it contains no odd-length cycles.

The End

 Before Ted Kaczynski was the Unabomber, he was a mathematics professor at Berkeley. ◦ At the time, he was the youngest professor ever to get hired by Berkeley. ◦ His specialty was complex analysis.

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