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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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References:
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