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In this paper we study spectra of Laplacians of infinite weighted graphs.Instead of the assumption of local finiteness we impose the condition ofsummability of the weight function. Such graphs correspond to reversible Markovchains with countable state spaces. We adopt the concept of the Cheegerconstant to this setting and prove an analogue of the Cheeger inequalitycharacterising the spectral gap. We also analyse the concept of the dualCheeger constant originally introduced in \cite{B14}, which allows estimatingthe top of the spectrum. In this paper we also introduce a new combinatorialinvariant, k$(G,m)$, which allows a complete characterisation of bipartitegraphs and measures the asymmetry of the spectrum (the Hausdorff distancebetween the spectrum and its reflection at point $1\in \Bbb R$). We comparek$(G, m)$ to the Cheeger and the dual Cheeger constants. Finally, we analyse infull detail a class of infinite complete graphs and their spectra.

Spectra of infinite graphs with summable weight functions