The spectra of graph substitutions

Let $(X,E_X)$ and $(V,E_V)$ be finite connected graphs without loops. We assume that $V$ has two distinguished vertices $a,b$ and an automorphism $\gamma$ which exchanges $a$ and~$b$. The $V$-edge substitution of $X$ is the graph $X[V]$ where each edge $[x,y] \in E_X$ is replaced by a copy of $V$, identifying $x$ with $a$ and $y$ with $b$ or vice versa. (The latter choice does not matter; it yields isomorphic graphs.) The aim is to describe the spectrum of $X[V]$ in terms of the spectra of $X$ and $V$. Instead of the spectra of the adjacency matrices, we consider the versions which are normalised by dividing each row by the row sum (the vertex degree). These are stochastic, reversible matrices, and our approach applies more generally to reversible transition matrices corresponding to arbitrary positive edge weights invariant under $\gamma$. We write $P$ for the transition matrix over $X$ and $Q$ for the one over $V$. Together, they induce the matrix $P_*$ over $X[V]$. The main part of the spectrum of $P_*$ is the response of the natural frequencies of $X$ to substituting $V$, given by a functional equation coming from a rational function induced by~$Q$. A second part comes from specific eigenvalues of $Q$, if present. Finally, there is the part of $\mathsf{spec}(P_*)$ whose eigenfunctions have $X$ as a nodal set. The results depend on issues like whether $X$ has circles of even length and on the eigenvalues of the restriction of $Q$ to $V \setminus \{ a,b\}$, which are classified into 4 possible types. Quite subtle is the issue of determining the multiplicities of the latter as eigenvalues of $P_*$ in terms of the input.