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We modify the definition of Ricci curvature of Ollivier of Markov chains on graphs to study the properties of the Ricci curvature of general graphs, Cartesian product of graphs, random graphs, and some special class of graphs.

We prove a Harnack inequality for Dirichlet eigenfunctions of abelian homogeneous graphs and their convex subgraphs. We derive lower bounds for Dirichlet eigenvalues using the Harnack inequality. We also consider a randomization problem in connection with combinatorial games using Dirichlet eigenvalues. 2000 John Wiley & Sons, Inc. J Graph Theory 34: 247257, 2000

We investigate quantum graphs with infinitely many vertices and edges without the common restriction on the geometry of the underlying metric graph that there is a positive lower bound on the lengths of its edges. Our central result is a close connection between spectral properties of a quantum graph and the corresponding properties of a certain weighted discrete Laplacian on the underlying discrete graph. Using this connection together with spectral theory of (unbounded) discrete Laplacians on infinite graphs, we prove a number of new results on spectral properties of quantum graphs. Namely, we prove several self-adjointness results including a Gaffney-type theorem. We investigate the problem of lower semiboundedness, prove several spectral estimates (bounds for the bottom of spectra and essential spectra of quantum graphs, CLR-type estimates) and study spectral types.

This erratum will correct the classification of Theorem 1 in Lin-Lu-Yau, Comm. Anal. Geom., 2014, that misses the Triplex graph.