A Harnack inequality for homogeneous graphs and subgraphs

Fan RK Chung Shing-Tung Yau

Convex and Discrete Geometry mathscidoc:1912.43539

Comm. Anal. Geom, 2, (4), 627-640, 1994
Let (V, E) denote a graph with vertex set V= V (T) and edge set E= E (). Suppose a group TL acts on V such that:(i) for all g Ti,{gu, gv} GE if and only if {u, v} E,(ii) for any two vertices and v, there is a g T such that guv. Then we say is a homogeneous graph with the associated group Ti. In other words, is vertex-transitive under the action of Ti and We can identify V with the coset space Ti/X where X={g E Ti: gvv}, for a fixed vertex v, denotes the isotropy group. We note that the Cayley graph is a special case of homogeneous graphs with X trivial. The edge set of a homogeneous graph can be described by an (edge) generating set ii H such that each edge of is of the form {v, gv} for some v V, and g . In this paper we require the generating set to be symmetric, ie, g if and only if g^ 1 K. We say that a homogeneous graph is invariant if for every vertex a K, we have aKa-1= K.
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  title={A Harnack inequality for homogeneous graphs and subgraphs},
  author={Fan RK Chung, and Shing-Tung Yau},
  booktitle={Comm. Anal. Geom},
Fan RK Chung, and Shing-Tung Yau. A Harnack inequality for homogeneous graphs and subgraphs. 1994. Vol. 2. In Comm. Anal. Geom. pp.627-640. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191224203917371774103.
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