Spectrum of the infinite biregular tree (with an example)

Question

This question concerns the spectrum of the universal covers of graphs. In particular, some results in:

1. Angel, Omer; Friedman, Joel; Hoory, Shlomo The non-backtracking spectrum of the universal cover of a graph. Trans. Amer. Math. Soc. 367 (2015), no. 6, and
2. Nagnibeda, Tatiana Random walks, spectral radii, and Ramanujan graphs. Random walks and geometry, 487–500, Walter de Gruyter, Berlin, 2004 .

It's well known that in the case of infinite $k$-regular trees the spectrum is: $$ [-2 \sqrt{k-1}, 2 \sqrt{k-1}] . $$ Moreover, in the case of an infinite $(k,l)$-biregular tree the spectrum is: $$ \{ \lambda \in \mathbb{R}\ \big| \ |\sqrt{k - 1} - \sqrt{ l - 1} |\leq |\lambda| \leq |\sqrt{k - 1} + \sqrt{ l - 1} | \} \cup \{ 0\}\ , $$ as one can see, for example, in

• Mohar, Bojan; Woess, Wolfgang A survey on spectra of infinite graphs. Bull. London Math. Soc. 21 (1989), and
• Li, Wen-Ch'ing Winnie; Solé, Patrick Spectra of regular graphs and hypergraphs and orthogonal polynomials. European J. Combin. 17 (1996) .

However, very little is known about the spectrum of the universal cover of irregular graphs, in general. (1) and (2) are the only sources I found that give an expression or algorithm to compute values in that spectrum. Are there other sources that you may be aware of?

EDIT: this second question doesn't really make sense, because $0$ is in the spectrum of the infinite $(3,2)$-regular tree. A second question is that, while trying to apply the algorithm described in (1) to the $(3,2)$-complete bipartite graph, I wasn't able to see why $0$ is not in the spectrum (of the adjacency operator) of the infinite $(3,2)$-biregular tree. So,

• What should be the ratio system $\{ r_e\}$, so that the corresponding Perron eigenvalue $\alpha(r) < 1$?
• Given that $r_e = \frac{G(u,v)}{G(u,u)}$, with $e = (u,v)$, and the Green function is invariant under automorphisms of the base graph, can we assume that $r_e$ is also invariant under the same automorphisms?
• Is the Green function associated to the adjacency operator unique? And does that say that the, if it exists, the ratio systems are unique?

I appreaciate any insight from people, because computing this simple example has got me very confused. Thank you in advance!