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McKay trees

Given a finite group $G$ and its representation $ρ$, the corresponding McKay graph is a graph $Γ(G,ρ)$ whose vertices are the irreducible representations of $G$; the number of edges between two vertices $π,τ$ of $Γ(G,ρ)$ is $dim Hom_G(π\otimes ρ, τ) $. The collection of all McKay graphs for a given group $G$ encodes, in a sense, its character table. Such graphs were also used by McKay to provide a bijection between the finite subgroups of $SU(2)$ and the affine Dynkin diagrams of types $A, D, E$, the bijection given by considering the appropriate McKay graphs. In this paper, we classify all (undirected) trees which are McKay graphs of finite groups and describe the corresponding pairs $(G,ρ)$; this classification turns out to be very concise. Moreover, we give a partial classification of McKay graphs which are forests, and construct some non-trivial examples of such forests.