Spectra of infinite graphs via freeness with amalgamation

Jorge Garza Vargas, Archit Kulkarni

Published 2019-12-20Version 1

We use tools from free probability and $C^*$-algebras to study the spectra of weighted adjacency operators associated to infinite graphs. Special attention is devoted to universal covers of finite undirected weighted graphs, where we obtain new results: (1) we derive a tight bound on the number of components of the spectrum of the universal cover in terms of the number of vertices in the base graph, and (2) by using Voiculescu's notion of freeness with amalgamation we obtain algebraic descriptions of the spectral measure and the spectral radius of the universal cover. We then develop a framework that extends the applicability of the techniques mentioned above to a broader class of graphs. More specifically, we introduce a combinatorial graph product and show that in the non-commutative probability context, it corresponds to the notion of freeness with amalgamation. Finally, we discuss the case of Cayley graphs of amalgamated free products of groups. We show that that these graphs, as well as universal covers, can be expressed using our graph product and can therefore be analyzed using our framework.