Gauss-Bonnet Formula, Finiteness Condition, and Characterizations of Graphs Embedded in Surfaces

@article{Chen2008GaussBonnetFF,
title={Gauss-Bonnet Formula, Finiteness Condition, and Characterizations of Graphs Embedded in Surfaces},
author={Beifang Chen and Guantao Chen},
journal={Graphs and Combinatorics},
year={2008},
volume={24},
pages={159-183}
}

Let G be an infinite graph embedded in a closed 2-manifold, such that each open face of the embedding is homeomorphic to an open disk and is bounded by finite number of edges. For each vertex x of G, define the combinatorial curvature KG(x) = 1− d(x) 2 + ∑ σ∈F (x) 1 |σ| as that of [9], where d(x) is the degree of x, F (x) is the multiset of all open faces σ in the embedding such that the closure σ̄ contains x (the multiplicity of σ is the number of times that x is visited along ∂σ), and |σ| is… CONTINUE READING