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The spectral radius of infinite graphs. (English) Zbl 0671.05052
For an infinite graph $$\Gamma$$ with vertex set V and finitely bounded valency, the adjacency operator A is well-defined on $$\ell^ 2(V)$$ and is bounded and self-adjoint. The spectral radius $$\rho$$ ($$\Gamma)$$ is the supremum of $$| <x,Ax>|$$ over $$\| x\| =1$$. Expansion properties of $$\Gamma$$ are measured by the isoperimetric constant i($$\Gamma)$$ defined as the infinuum of the ratio of the number of edges having exactly one endpoint in X and $$| X|$$ over all finite $$X\subset V$$. Some bounds in terms of $$\rho$$ ($$\Gamma)$$ are derived for i($$\Gamma)$$, thus supporting the idea that in infinite graphs the spectral radius is related to expansion properties of the graph.
Reviewer: D.Cvetković

##### MSC:
 05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
##### Keywords:
infinite graph; spectral radius; expansion properties
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