图的Laplacian矩阵

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设G=<V,E>是一个n阶无向简单(无环，无多重边)图，其顶点集和边集分别记为V=V(G)={v₁,v₂,…v_n}和E=E(G)={e_l,e₂,…,e_n)，我们用如下方式刻画图的Laplacian矩阵：

(1) L(G)=D(G) - A(G)，其中A(G)为图G的邻接矩阵，D(C)=diag(d1,d2,…dn)为图G的度矩阵。

(2) 对图G的每一条边e_k=v_iv_j，选择其中一个顶点为其正端点，另一个顶点为其负端点。这样就给G一个定向，对图G的一个定向，我们用Q(G)=(q_ij)_nxm表示其关联矩阵，其中

则图G的Laplacian矩阵的可以表示为L(G)=Q(G)Q^T(G)，虽然关联矩阵Q(G)与图G的定向有关，但L(G)与图的定向无关。

(3) 设f是Rⁿ中的一个n维列向量，则

其中L(G)是一个半正定对称矩阵，其每一特征值是非负的，将L(G)的特征值排序为：

0 = λ₁≤ λ₂≤ … ≤ λ_n

参考资料

Spectral Graph Theory

Lecture 1: Introduction to Graphs, Spectra and Random Walks, Walking on Grids and Lines.

Lecture 2:Random Walks and 2SAT, Markov Chains

Lecture 3:Random Walks on Graphs,Hitting time, Commute time, Cover time, Random Walks and Electrical Networks

Lecture 4: More on Cover time, Graph Connectivity, Start Graphs and Eigenvalues.

Lecture 5:Erdos-Renyi Random Graphs.

Lecture 6:Galton-Watson Process, Giant Component.

Lecture 7:The Laplacian.

Lecture 8:Extremal Eigenvalues and Eigenvectors of the Laplacian and the Adjacency Matrix.

Lecture 9:The Other Eigenvectors and Eigenvalues of the Laplacian.

Lecture 10:Graphic Inequalities and Lowerbounds on the Second Laplacian Eigenvalue.

Lecture 11:Graph Cutting and Cheeger's Inequality.

Lecture 12:Relaxations, Duality and the Connection with lambda_2

Lecture 13:The Unique Games Conjecture and SDP Duality.

Lecture 14:Random Walks and Eigenvalues.

Lecture 15:Pseudorandom Generators and Random Walks on Expanders.

Lecture 16:Expander Graphs and Their Properties.

Lecture 17:Error-Correcting Codes.

Lecture 18:Expander Codes.

Lecture 19:Cayley Graphs.

Lecture 20:Construction of Expanders.

Lecture 21:Diameter and Second Eigenvalue.

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