21 Spectral 2.pdf - Ma/CS 6b Class 21 Spectral Graph Theory 2 1 2 4 3 Eigenvalues-2,0,0,2 1 2 3 Eigenvalues − 2,0 2 By Adam Sheffer Chromatic number

21 Spectral 2.pdf - Ma/CS 6b Class 21 Spectral Graph Theory...

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2/24/2017 1 Ma/ CS 6 b Class 21: Spectral Graph Theory 2 By Adam Sheffer ? 1 ? 2 ? 3 ? 4 ? 1 ? 2 ? 3 Eigenvalues: 2 , 0 , 2 . Eigenvalues: - 2,0,0,2 . Chromatic number Given a graph 𝐺 , the chromatic number 𝜒 ( 𝐺 ) is the minimum number of colors required to color the vertices of 𝐺 . 𝜒 𝐺 = 3
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2/24/2017 2 Coloring Graphs with Bounded Degrees Claim. Let 𝐺 = (𝑉, 𝐸) be a graph with maximum degree ? , then 𝜒 𝐺 ? + 1 . Proof. At each step choose an arbitrary uncolored vertex ? . Since ? has at most ? neighbors, one of the ? + 1 colors must be OK for ? . Example: ? + 1 - coloring 1 2 3 5 4 6
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2/24/2017 3 Sometimes We Cannot Do Better 𝐾 𝑛 - complete graph with ? vertices. Max degree: ? 1 . 𝜒 𝐾 𝑛 = ? . 𝐶 𝑛 - cycle of odd length ? . Max degree: 2 . 𝜒 𝐾 𝑛 = 3 . Today s Goal The naïve coloring bound implies that the simple graph below requires three colors . Using eigenvalues of graphs, we can obtain a better bound. ? 1 ? 2 ? 3
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2/24/2017 4 Recall: The Spectrum of a Graph Consider a graph 𝐺 = 𝑉, 𝐸 and let 𝐴 be the adjacency matrix of 𝐺 . The eigenvalues of 𝐺 are the eigenvalues of 𝐴 . The characteristic polynomial 𝜙 𝐺 ; 𝜆 is the characteristic polynomial of 𝐴 . The spectrum of 𝐺 is ???? 𝐺 = 𝜆 1 , , 𝜆 ? ? 1 , , ? ? , where 𝜆 1 , … , 𝜆 ? are the distinct eigenvalues of 𝐴 and ? ? is the multiplicity of 𝜆 ? . Example: Spectrum 𝐴 = 0 1 0 1 1 0 1 0 0 1 1 0 0 1 1 0 det 𝜆𝐼 𝐴 = det 𝜆 −1 0 −1 −1 𝜆 −1 0 0 −1 −1 0 𝜆 −1 −1 𝜆 = 𝜆 2 𝜆 2 𝜆 + 2 . ???? 𝐶 4 = 0 2 2 2 1 1 ? 1 ? 2 ? 3 ? 4
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