Spectra of Infinite Graphs via Freeness with Amalgamation

Abstract We use tools from free probability to study the spectra of Hermitian operators on infinite graphs. Special attention is devoted universal covering trees finite For these graphs, we derive a new variational formula for spectral radius and provide proofs results due Sunada Aomoto using probability. With goal extending applicability techniques beyond trees, introduce combinatorial product operation graphs show that, in noncommutative context, it corresponds notion freeness with amalgamation. that Cayley amalgamated products groups, as well can be constructed our graph product.