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The number of walks of $k$ steps from the node $\mathsf{0}$ to the node$\lambda$ on the representation graph (McKay quiver) determined by a finitegroup $\mathsf{G}$ and a $\mathsf{G}$-module $\mathsf{V}$ is the multiplicityof the irreducible $\mathsf{G}$-module $\mathsf{G}_\lambda$ in the tensor power$\mathsf{V}^{\otimes k}$, and it is also the dimension of the irreduciblemodule labeled by $\lambda$ for the centralizer algebra$\mathsf{Z}_k(\mathsf{G}) = {\mathsf{End}}_\mathsf{G}(\mathsf{V}^{\otimes k})$.This paper explores ways to effectively calculate that number using thecharacter theory of $\mathsf{G}$. We determine the corresponding Poincar\'eseries. The special case $\lambda = \mathsf{0}$ gives the Poincar\'e series forthe tensor invariants $\mathsf{T}(\mathsf{V})^\mathsf{G} = \bigoplus_{k=0}^\infty (\mathsf{V}^{\otimes k})^\mathsf{G}$. When $\mathsf{G}$ is abelian,we show that the exponential generating function for the number of walks is aproduct of generalized hyperbolic functions. Many graphs (such as circulantgraphs) can be viewed as representation graphs, and the methods presented hereprovide efficient ways to compute the number of walks on them.

Walks on Graphs and Their Connections with Tensor Invariants and Centralizer Algebras