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Isoperimetric numbers and spectral radius of some infinite planar graphs. (English) Zbl 0756.05098
Summary: Let $$N$$ be a triangulation of a non-compact, open subset of the 2-sphere, projective plane, torus, or Klein bottle, and let $$G$$ be its (geometric) dual graph. If every 0-simplex of $$N$$ is contained in at least $$k$$ 2- simplices, where $$k\geq 7$$, then the isoperimetric number $$i(G)$$ of $$G$$ is at least $$3(k-6)/(5k-18)$$. If $$G$$ has at most $$m$$ ends then, if $$(k- 3)m\geq k\chi(S)$$, $$i(G)\geq 3(k-6)/[(5-2\chi(S)/k-18]$$, and $$i(G)\geq (k- 6)/(k-4)$$ otherwise. These bounds, except the last one, are shown to be the best possible. Even better bounds are obtained, assuming $$G$$ is cyclically $$t$$-edge connected $$(3<t\leq k)$$. Also nontrivial bounds on the spectral radius of $$G$$ are derived from the above results.

##### MSC:
 05C99 Graph theory 05C50 Graphs and linear algebra (matrices, eigenvalues, etc.) 52B60 Isoperimetric problems for polytopes
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