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 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.

"A sequence $S$ is potentially $K_4-e$ graphical if it has a realization containing a $K_4-e$ as a subgraph. Let $\sigma(K_4-e,n)$ denote the smallest degree sum such that every $n$ -term graphical sequence $S$ with $\sigma(S)\geq\sigma(K_4-e,n)$ is potentially $K_4-e$ graphical. Gould, Jacobson, Lehel raised the problem of determining the value of $\sigma(K_4-e,n)$ . In this paper, we prove that $\sigma(K_4-e,n)=2[(3n-1)/2]$ for $n\geq7$ and $n=4,5$ , and $\sigma(K_4-e,6)=20$ .''

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

In a graph G, for a subset S of the vertex set, the induced subgraph determined by S has edge set consisting of all edges of G with both endpoints in S. The (edge) boundary, denoted by S consists of all edges containing one endpoint in S and one endpoint not in S.