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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.

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