Eigenvalue inequalities for graphs and convex subgraphs

  title={Eigenvalue inequalities for graphs and convex subgraphs},
  author={Fan R. K. Chung and Shing-Tung Yau},
For an induced subgraph S of a graph, we show that its Neumann eigenvalue λS can be lower-bounded by using the heat kernel Ht(x, y) of the subgraph. Namely, λS ≥ 1 2t ∑ x∈S inf y∈S Ht(x, y) √ dx √ dy where dx denotes the degree of the vertex x. In particular, we derive lower bounds of eigenvalues for convex subgraphs which consist of lattice points in an d-dimensional Riemannian manifolds M with convex boundary. The techniques involve both the (discrete) heat kernels of graphs and improved… CONTINUE READING