Matrika sosednosti neskončnega lokalno končnega grafa določa linearen gosto definiran operator na ▫$\ell^2$▫. Zaprtje tega operatorja imenujemo operator sosednosti, njegov spekter pa spekter grafa. ...Okarakterizirani so grafi, ki imajo omejen ali celo kompakten operator sosednosti. Če dodamo ali zbrišemo končno mnogo povezav, se bistveni spekter ne spremeni. Delno je posplošen izrek o simetričnosti spektra bipartitnih (končnih) grafov. Spektralni radij grafa je enak supremumu spektralnih radijev končnih (resp. vseh) podgrafov grafa. Dokazane so še nekatere druge lastnosti.
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