3.1.1. Locally finite graphs

Let $X = (V_{X},\mathbf {E}_{X})$ be a graph in the sense of Serre (see [Reference Serre48] and also [Reference Sunada52]), where $V_X$ and $\mathbf {E}_X$ are two sets, to be interpreted as the sets of vertices and (directed) edges of X. In particular, each edge $e \in \mathbf {E}_X$ has an origin $o(e) \in V_X$ and a terminus $t(e) \in V_X$ , giving rise to the incidence map

$$ \begin{align*} \begin{aligned} \mathrm{inc} \colon &\mathbf{E}_{X} \to V_{X} \times V_{X} \\ &e \mapsto (o(e),t(e)) \end{aligned} \end{align*} $$

and to the inversion map $\mathbf {E}_{X} \rightarrow \mathbf {E}_{X}$ , denoted by $e \mapsto \bar {e}$ , such that

$$ \begin{align*}(o(\overline{e}),t(\overline{e})) = (t(e),o(e))\end{align*} $$

and $\overline {\overline {e}} = e \neq \overline {e}$ for every $e \in \mathbf {E}_X$ .

A graph X is called finite if both $V_{X}$ and $\mathbf {E}_{X}$ are finite sets. Moreover, a graph X is called locally finite if for each vertex $v \in V_{X}$ , the set of edges with origin at v, defined as

$$ \begin{align*} \mathbf{E}_{X,v} = \{ e \in \mathbf{E}_{X} \, {:} \, o(e) = v\} \end{align*} $$

is finite. In this case, one defines the valency (or degree) of a vertex $v \in V_{X}$ to be

$$ \begin{align*}\mathrm{val}_{X}(v) = |\mathbf{E}_{X,v}|. \end{align*} $$

Any finite graph is in particular locally finite.

3.1.2. Paths and loops

Let us recall that a path $c = e_{1} \cdot \cdots \cdot e_{m}$ in a graph X consists of a sequence of directed edges $e_{i} \in \mathbf {E}_{X}$ such that $t(e_{i}) = o(e_{i+1})$ for each index $i \in \{1,\ldots ,m-1\}$ . The origin and the terminus of the path $c = e_1 \cdot \cdots \cdot e_m$ are defined as $o(c) = o(e_{1})$ and $t(c) = t(e_{m})$ , respectively.

A graph X is called connected if given any two vertices $v_{1}, v_{2} \in V_{X}$ , there exists a path c in X such that $o(c) = v_{1}$ and $t(c) = v_{2}$ . Finally, a loop based at a vertex $v_0 \in X$ is a path c in X such that $o(c) = t(c) = v_0$ .

This allows one to define the fundamental group of X based at a vertex $v_0 \in V_X$ , which is denoted by $\pi _1(X,v_0)$ , as the set of loops based at $v_0$ , considered modulo homotopy (see [Reference Sunada52, Section 3.5] for the precise definition of this equivalence relation in the context of graphs), endowed with the group operation given by the concatenation of paths (see [Reference Sunada52, Section 5.3] for details).

3.1.3. Galois covers of graphs

Let Y and X be two graphs. A morphism of graphs $f:Y \rightarrow X$ is called a cover (or a covering map) if the following two conditions are satisfied:

1. (1) $f:V_{Y} \rightarrow V_{X}$ is surjective;

2. (2) for all $w \in V_{Y}$ , the restriction $f|_{\mathbf {E}_{Y,w}}$ induces a bijection

   $$ \begin{align*}f|_{\mathbf{E}_{Y,w}}:\mathbf{E}_{Y,w} \stackrel{\approx}{\rightarrow} \mathbf{E}_{X,f(w)}. \end{align*} $$

We will often refer to $Y/X$ as a cover if the covering map is understood from the context. Given a cover $f:Y \rightarrow X$ , one defines as usual $\mathrm {Aut}_{f}(Y/X)$ to be the subgroup of $\mathrm {Aut}(Y)$ consisting of the automorphisms $\iota \in \mathrm {Aut}(Y)$ satisfying $f \circ \iota = f$ . Again, we will often write $\mathrm {Aut}(Y/X)$ instead of $\mathrm {Aut}_{f}(Y/X)$ if f is understood.

Let us also recall that a cover $f:Y \rightarrow X$ is called Galois if the following two conditions are satisfied:

1. (1) the graph Y is connected (and hence also X);

2. (2) the group $\mathrm {Aut}(Y/X)$ acts transitively on the fibre $f^{-1}(v)$ for all $v \in V_{X}$ .

If $Y/X$ is a Galois cover, we write $\mathrm {Gal}(Y/X)$ instead of $\mathrm {Aut}(Y/X)$ . In this case, one has the usual Galois correspondence between subgroups of $\mathrm {Aut}(Y/X)$ and equivalence classes of intermediate covers of $Y/X$ .

3.1.4. Voltage assignments

Let X be a graph and let G be a group. A voltage assignment on X with values in G is defined to be a function $\alpha :\mathbf {E}_{X} \rightarrow G$ satisfying

$$ \begin{align*} \alpha(\bar{e}) = \alpha(e)^{-1} \end{align*} $$

for every $e \in \mathbf {E}_{X}$ . Each such voltage assignment can be defined starting from an orientation of X, which is a subset $S \subseteq \mathbf {E}_X$ such that for each edge $e \in \mathbf {E}_X$ , either e or $\overline {e}$ belong to S, but not both. Then, to get a voltage assignment as above, it suffices to define it on any orientation S and set $\alpha (\overline {s}) := \alpha (s)^{-1}$ for every $s \in S$ .

3.1.5. Covers from voltage assignments

Given a graph X, a group G and a voltage assignment

$$ \begin{align*}\alpha \colon \mathbf{E}_X \to G,\end{align*} $$

one can construct a new graph $X(G,\alpha )$ as follows:

• • the vertices of $X(G,\alpha )$ are given by $V_X \times G$ ;

• • the (directed) edges of $X(G,\alpha )$ are given by $\mathbf {E}_X \times G$ ;

• • the origin, terminus and inverse maps are defined as

  $$ \begin{align*} \begin{aligned} o(e,\sigma) &= (o(e),\sigma), \\ t(e,\sigma) &= (t(e),\sigma \cdot \alpha(e)), \\ \overline{(e,\sigma)} &= (\bar{e},\sigma \cdot \alpha(e)) \end{aligned} \end{align*} $$
  for each edge $(e,\sigma ) \in \mathbf {E}_X \times G$ .

It is easy to see that if X is locally finite, then so is $X(G,\alpha )$ , and that the map of graphs

$$ \begin{align*}p \colon X(G,\alpha) \rightarrow X\end{align*} $$

defined as $p(v,\sigma ) := v$ on each vertex $(v,\sigma ) \in V_X \times G$ , and as $p(e,\sigma ) := e$ on each edge $(e,\sigma ) \in \mathbf {E}_X \times G$ , is actually a covering map. Moreover, this covering map is Galois whenever $X(G,\alpha )$ is connected and, in this case, $\operatorname {Gal}(X(G,\alpha )/X) \cong G$ canonically.

To conclude, let us observe that the construction of $X(G,\alpha )$ is functorial with respect to $\alpha $ . More precisely, for each morphism of groups $f \colon G \to H$ , one gets a new voltage assignment $\beta := f \circ \alpha $ with values in H, and a natural map of graphs

(3-1) $$ \begin{align} f_\ast \colon X(G,\alpha) \to X(H,\beta), \end{align} $$

which is defined on each vertex $(v,\sigma ) \in V_X \times G$ as $f_{\ast }(v,\sigma ) := (v,f(\sigma ))$ , and on each edge $(e,\sigma ) \in \mathbf {E}_X \times G$ as $f_\ast (e,\sigma ) := (e,f(\sigma ))$ . Finally, it is easy to see that this morphism $f_\ast $ is a covering map whenever f is surjective.

3.1.6. Monodromy representations

Let X be a graph and $\alpha \colon \mathbf {E}_X \to G$ be a voltage assignment with values in a group G. Then, the monodromy representation attached to $\alpha $ at a vertex $v_0 \in V_X$ is given by the following map:

$$ \begin{align*} \begin{aligned} \rho_{\alpha,v_0} \colon \pi_{1}(X,v_{0}) &\to G \\ [e_1 \cdot \cdots \cdot e_n] &\mapsto \alpha(e_1) \cdot \cdots \cdot \alpha(e_n), \end{aligned} \end{align*} $$

which is easily seen to be a well-defined morphism of groups. Moreover, this map can be used to detect when the graph $X(G,\alpha )$ is connected, and thus when the natural covering map $p \colon X(G,\alpha ) \to X$ is Galois, as we recall in the following theorem, which is proven in [Reference Ray and Vallières45, Section 2.3.1].

Remark 3.3. Let X be a connected graph, $v_0 \in V_X$ a vertex of X, and $\alpha \colon \mathbf {E}_X \to G$ a voltage assignment such that $\rho _{\alpha ,v_0}$ is surjective. Fix moreover a universal cover $\pi \colon \widetilde {X} \twoheadrightarrow X$ and a vertex $w_0 \in \widetilde {X}$ such that $\pi (w_0) = v_0$ . Thanks to the universal property of the universal cover, proved for example in [Reference Sunada52, Theorem 5.10], it can be shown that the intermediate Galois cover of $\widetilde {X} \to X$ given by $X(G,\alpha )$ corresponds to the subgroup $\varphi _{w_0}(\ker (\rho _{\alpha ,v_0})) \trianglelefteq \mathrm {Gal}(\widetilde {X}/X)$ , where

$$ \begin{align*}\varphi_{w_{0}}: \pi_{1}(X,v_{0}) \stackrel{\sim}{\longrightarrow}\mathrm{ Gal}(\widetilde{X}/X)\end{align*} $$

is the usual group isomorphism.

3.1.7. Systems of Galois covers

Let X be a graph and $\alpha \colon \mathbf {E}_X \to G$ a voltage assignment taking values in a group G. Given a group homomorphism $f \colon G \to H$ and a vertex $v_0 \in V_X$ , the monodromy representation attached at $v_0$ to the Galois cover $f_\ast $ defined in (3-1) is given by $f \circ \rho _{\alpha ,v_0}$ . This shows in particular that if the graph $X(G,\alpha )$ is connected, the graph $X(H,f \circ \alpha )$ will be connected whenever f is surjective. Moreover, any morphism of groups $f \colon G \to H$ induces another morphism of groups

(3-2) $$ \begin{align} \mathrm{ker}(f) \rightarrow \mathrm{Aut}(X(G,\alpha)/X(H,f \circ \alpha)), \end{align} $$

which sends each $\tau \in \ker (f)$ to the automorphism $\phi _\tau \colon X(G,\alpha ) \to X(G,\alpha )$ defined by setting $\phi _\tau (v,\sigma ) := (v,\tau \cdot \sigma )$ for each vertex $(v,\sigma ) \in V_X \times G$ , and $\phi _\tau (e,\sigma ) := ({e,\tau \cdot \sigma })$ for each edge $(e,\sigma ) \in \mathbf {E}_X \times G$ . The morphism of groups (3-2) is actually an isomorphism whenever f is surjective, as follows from the unique lifting theorem [Reference Sunada52, Theorem 5.1].

In particular, the previous discussion shows that any voltage assignment

$$ \begin{align*}\alpha \colon \mathbf{E}_X \to G\end{align*} $$

for which $X(G,\alpha)$ is connected induces a system of Galois covers indexed by the lattice of quotients of the group G. As we will see in the upcoming sections of this paper, it is interesting to study how various graph invariants evolve when moving across this system.

3.2.1. Ihara zeta functions

To do so, we will make use of another invariant of a finite connected graph X, namely its Ihara zeta function, which we denote by $Z_X(u)$ . This is a rational function of u, which can be explicitly computed thanks to the Ihara determinant formula, which we recall below in (3-3), and is proven in [Reference Bass2, Reference Kotani and Sunada29]. More precisely, given an ordering $V_X = \{v_1,\ldots ,v_g\}$ of the vertices of X, we let $A_X := (a_{i,j}) \in \mathbb {Z}^{g \times g}$ denote the adjacency matrix of X, which is defined by setting $a_{i,j} := \# \{ e \in \mathbf {E}_X \colon o(e) = v_i, \ t(e) = v_j \}$ . Moreover, we let $D_X := (d_{i,j}) \in \mathbb {Z}^{g \times g}$ denote the valency (or degree) matrix of X, which is a diagonal matrix defined by setting $d_{i,i} := \mathrm {val}_X(v_i)$ for each $i \in \{1,\ldots ,g\}$ . Then, we can write the Ihara zeta function $Z_X(u)$ using the following explicit formula:

(3-3) $$ \begin{align} Z_{X}(u) = \frac{1}{(1-u^{2})^{-\chi(X)} \cdot \mathrm{det}(\mathrm{Id}_g - A_X u + (D_X - \mathrm{Id}_g)u^{2})}, \end{align} $$

where $\mathrm {Id}_g$ denotes the $g \times g$ identity matrix and $\chi (X) := \lvert V_X \rvert - \lvert \mathbf {E}_X \rvert /2$ is the Euler characteristic of X. In particular, we have that $Z_X(u)^{-1} = (1-u^2)^{-\chi (X)} \cdot h_X(u)$ , where

$$ \begin{align*} h_{X}(u) := \mathrm{det}(\mathrm{Id}_g - A_X u + (D_X - \mathrm{Id}_g)u^{2}) \in \mathbb{Z}[u] \end{align*} $$

is a polynomial.

3.2.2. Hashimoto’s formula

This explicit formula can be used to relate the Ihara zeta function to the number of spanning trees of X. More precisely, given a finite connected graph X, one has

(3-4) $$ \begin{align} h_{X}'(1) = -2 \chi(X) \kappa(X), \end{align} $$

as was proven by Hashimoto in [Reference Hashimoto20, Theorem B] (see also [Reference Bass2, Part II, Sections 5 and 6]). Such a formula, which can be considered as an analogue of the class number formula in the context of graph theory, admits an equivariant generalisation.

3.2.3. Artin–Ihara L-functions

Given a Galois cover of finite connected graphs $Y/X$ , one can associate to any linear complex representation $\rho \colon \mathrm {Gal}(Y/X) \to \mathrm {GL}_n(\mathbb {C})$ an Artin–Ihara L-function $L_{Y/X}(u,\rho )$ . This admits an explicit description analogous to (3-3). More precisely, let $d_\rho \in \mathbb {N}$ denote the degree of the representation $\rho $ , and fix an ordering $V_X = \{v_1,\ldots ,v_g\}$ of the vertices of X and a section $\iota \colon V_{X} \rightarrow V_{Y}$ of the projection $V_{Y} \to V_{X}$ . Then, [Reference Terras53, Theorem 18.15] shows that the Artin–Ihara L-function $L_{Y/X}(u,\rho )$ can be explicitly computed thanks to the following formula:

$$ \begin{align*} L_{Y/X}(u,\rho) = \frac{1}{(1-u^2)^{- \chi(X) \cdot d_\rho} \cdot \det(\mathrm{Id}_{gd_{\rho}} - A_\rho u + Q_\rho u^2)}, \end{align*} $$

where $A_\rho , Q_\rho \in \mathbb {C}^{g d_\rho \times g d_\rho }$ are two explicit matrices, whose definitions we now recall. Given $\sigma \in G$ ,

$$ \begin{align*}A(\sigma) := (\#\{ e \in \mathbf{E}_Y \colon o(e) = \iota(v_i), \ t(e) = \sigma(\iota(v_j)) \})_{i,j = 1,\ldots,g},\end{align*} $$

and we define

$$ \begin{align*} A_\rho := \sum_{\sigma \in G} A(\sigma) \otimes \rho(\sigma) \quad \text{and} \quad Q_\rho := (D_X \otimes \mathrm{Id}_{d_\rho}) - \mathrm{Id}_{g \cdot d_\rho}, \end{align*} $$

where $\otimes $ denotes the Kronecker product of matrices. For more details, we refer the interested reader to [Reference Terras53, Definition 18.13]. As before, this explicit formula allows one to write $L_{Y/X}(u,\rho )^{-1} = (1-u^2)^{- \chi (X) \cdot d_\rho } \cdot h_{Y/X}(u,\rho )$ , where

$$ \begin{align*} h_{Y/X}(u,\rho) := \det(\mathrm{Id}_{gd_{\rho}} - A_\rho u + Q_\rho u^2) \in \mathbb{C}[u] \end{align*} $$

is a polynomial. In particular, if $\mathrm {Gal}(Y/X)$ is abelian and $\psi $ is a character of $\mathrm {Gal}(Y/X)$ , we have that $L_{Y/X}(u,\psi )^{-1} = (1-u^2)^{-\chi (X)} \cdot h_{Y/X}(u,\psi )$ , where

(3-5) $$ \begin{align} h_{Y/X}(u,\psi) := \det(\mathrm{Id}_{g} - A_\psi u + (D_X - \mathrm{Id}_g) u^2), \end{align} $$

because $d_\psi = 1$ and $Q_\psi = D_X - \mathrm {Id}_g$ , as follows easily from [Reference Terras53, Definition 18.13].

3.2.4. Spanning trees and abelian covers

To conclude this subsection, let us recall that the Artin–Ihara L-functions satisfy the Artin formalism (see [Reference Bass2, Reference Stark and Terras51]). This implies that for every Galois cover of finite graphs $Y/X$ , with Galois group $G := \mathrm {Gal}(Y/X)$ , the Ihara zeta function $Z_Y(u)$ admits the following factorisation:

$$ \begin{align*} Z_Y(u) = \prod_{\rho \in \mathrm{Irr}(G)} L_{Y/X}(u,\rho)^{d_\rho}, \end{align*} $$

where $\mathrm {Irr}(G)$ denotes the set of equivalence classes of complex irreducible representations of a finite group G (see [Reference Terras53, Corollary 18.11]). Therefore, we see easily that

(3-6) $$ \begin{align} h_{Y}(u) = \prod_{\rho \in \mathrm{Irr}(G)} h_{Y/X}(u,\rho)^{d_\rho}, \end{align} $$

using the relation $\chi (Y) = \lvert G \rvert \cdot \chi (X)$ between the Euler characteristics of Y and X, which is explained at the end of [Reference Sunada52, Section 5.1], and the classical identity $\lvert G \rvert = \sum _{\rho \in \mathrm {Irr}(G)} d_\rho ^2$ , proven for example in [Reference Serre49, Section 2.4, Corollary 2]. Finally, if $\rho _0$ denotes the trivial representation of G and $\chi (X) \neq 0$ ,

$$ \begin{align*} \lvert G \rvert \cdot \kappa(Y) = \kappa(X) \cdot \prod_{\rho \neq \rho_0} h_{Y/X}(1,\rho)^{d_\rho}, \end{align*} $$

thanks to (3-4) and (3-6), combined with the fact that

$$ \begin{align*} h_{Y/X}(1,\rho_0) = h_{X}(1) = 0, \end{align*} $$

which holds because the Laplacian matrix $D_X - A_X$ is singular (as explained in [Reference Corry and Perkinson6, Proposition 2.8]). In particular, if $Y/X$ is a Galois cover of finite graphs such that $\chi (X) \neq 0$ and $G := \mathrm {Gal}(Y/X)$ is abelian,

(3-7) $$ \begin{align} |G| \cdot \kappa(Y) = \kappa(X) \prod_{\psi \neq \psi_{0}}h_{Y/X}(1,\psi), \end{align} $$

where $\psi _0$ denotes the trivial character of G and $\psi \in G^\vee $ runs over all nontrivial characters of G.

3.3.1. The Ihara polynomial

More precisely, let X be a finite connected graph such that $\chi (X) \neq 0$ , and let us start with a voltage assignment $\alpha :\mathbf {E}_{X} \rightarrow G$ . As before, let us fix an ordering of the vertices of X, given by $V_{X} = \{v_{1},\ldots ,v_{g}\}$ . Then, we can define the matrix

$$ \begin{align*} A_{\alpha}(t) := \bigg(\sum_{\substack{e \in \mathbf{E}_{X} \\ \mathrm{inc}(e) = (v_{i},v_{j})}} t^{\alpha(e)}\bigg) \in \mathbb{Z}[t^G]^{g \times g}, \end{align*} $$

which we use to introduce the Ihara polynomial

(3-8) $$ \begin{align} \mathcal{I}_{\alpha}(t) = \mathrm{det}(D_X - A_{\alpha}(t)) \in \mathbb{Z}[t^{G}], \end{align} $$

where $D_X$ denotes, as before, the valency (or degree) matrix of X. When no confusion seems to occur, we will just write $\mathcal {I}_\alpha $ instead of $\mathcal {I}_\alpha (t)$ .

We note in particular that the Ihara polynomial is self-reciprocal. In other words, we have the following identity:

(3-9) $$ \begin{align} \mathcal{I}_{\alpha}\bigg( \frac{1}{t}\bigg) = \mathcal{I}_{\alpha}(t), \end{align} $$

which comes from the fact that the transpose of the matrix $A_\alpha (t)$ equals the matrix $A_\alpha (t^{-1})$ by definition. Moreover, for every morphism of groups $f \colon G \to H$ , we have by definition that

(3-10) $$ \begin{align} \mathcal{I}_\beta = f_\ast(\mathcal{I}_\alpha), \end{align} $$

where $\beta := f \circ \alpha $ and $f_\ast \colon \mathbb {Z}[t^G] \to \mathbb {Z}[t^H]$ denotes the morphism of rings induced by f.

3.3.2. The Ihara polynomial and the number of spanning trees

Suppose now that X is a finite connected graph such that $\chi (X) \neq 0$ , and fix a voltage assignment

$$ \begin{align*}\alpha \colon \mathbf{E}_X \to G\end{align*} $$

with values in some finite abelian group G, such that the induced graph $X(G,\alpha )$ is connected. The following result expresses how the number of spanning trees changes from X to $X(G,\alpha )$ , using the Ihara polynomial $\mathcal {I}_\alpha $ .

Proposition 3.5. For every finite connected graph X, such that $\chi (X) \neq 0$ , and every voltage assignment $\alpha \colon \mathbf {E}_X \to G$ with values in a finite abelian group G, such that the associated graph $X(G,\alpha )$ is connected,

$$ \begin{align*} \lvert G \rvert \cdot \kappa(X(G,\alpha)) = \kappa(X) \cdot \prod_{\psi \neq \psi_0} \mathcal{I}_\alpha(\psi(1)), \end{align*} $$

where $\mathcal {I}_\alpha (\psi (1)) := \psi (\mathcal {I}_\alpha ) \in \mathbb {C}$ is obtained by applying to $\mathcal {I}_\alpha $ the natural linear extension of the character $\psi $ to the group ring $\mathbb {Z}[t^G]$ .

Proof. First of all, observe that $\mathcal {I}_\alpha (\psi (1)) = \det (D_X - \widetilde {A}_\psi )$ , where we define

$$ \begin{align*} \widetilde{A}_{\psi} := \bigg(\sum_{\substack{e \in \mathbf{E}_{X} \\ \mathrm{inc}(e) = (v_{i},v_{j})}} \psi(\alpha(e)) \bigg) \in \mathbb{C}^{g \times g} \end{align*} $$

for every character $\psi \in G^\vee $ . In particular, one can prove that $\widetilde {A}_\psi = A_\psi $ , as explained in [Reference McGown and Vallières40, Corollary 5.3]. Therefore, we see from the definition of the polynomial $h_{Y/X}(u,\psi )$ , which was given in (3-5), that $\mathcal {I}_\alpha (\psi (1)) = h_{Y/X}(1,\psi )$ for every character $\psi \in G^\vee $ . To conclude the proof, it is just sufficient to substitute this equality in the explicit expression

$$ \begin{align*} |G| \cdot \kappa(Y) = \kappa(X) \prod_{\psi \neq \psi_{0}}h_{Y/X}(1,\psi), \end{align*} $$

which was recalled in (3-7).

3.3.3. $\mathbb {Z}$ -towers of graphs and Pierce–Lehmer sequences

From now on, we will specialise to the case of $\mathbb {Z}$ -towers of graphs. More precisely, we will consider a finite connected graph X such that $\chi (X) \neq 0$ , endowed with a voltage assignment ${\alpha \colon \mathbf {E}_X \to \mathbb {Z}}$ with values in the additive group of the integers, such that the derived graph $X_\infty := X(\mathbb {Z},\alpha )$ is connected, which is equivalent to saying that there exists a vertex $v_0 \in V_X$ such that the monodromy representation $\rho _{\alpha ,v_0} \colon \pi _1(X,v_0) \to \mathbb {Z}$ is surjective, as explained in Theorem 3.2. In this case, we have a natural isomorphism $\mathbb {Z}[t^{\mathbb {Z}}] \cong \mathbb {Z}[t^{\pm 1}]$ . Therefore, we see from (3-9) that the Ihara polynomial can be written as

$$ \begin{align*} \mathcal{I}_{\alpha}(t) = c_0 + c_1 (t + t^{-1}) + \cdots + c_b (t^b + t^{-b}) \end{align*} $$

for some $c_0,\ldots ,c_b \in \mathbb {Z}$ such that $c_b \neq 0$ . Clearing denominators, we can define

$$ \begin{align*} I_\alpha(t) := t^b \mathcal{I}_\alpha(t) \in \mathbb{Z}[t], \end{align*} $$

which is a polynomial of degree $2 b$ such that $I_\alpha (0) \neq 0$ . Finally, we define $e := \mathrm {ord}_{t = 1}(I_\alpha )$ , and we write

(3-11) $$ \begin{align} I_\alpha(t) = (t-1)^e J_\alpha(t) \end{align} $$

for some polynomial $J_\alpha \in \mathbb {Z}[t]$ such that $J_\alpha (0) \cdot J_\alpha (1) \neq 0$ .

Now, one can associate to the voltage assignment $\alpha \colon \mathbf {E}_X \to \mathbb {Z}$ a system of finite graphs $X_n := X(\mathbb {Z}/n \mathbb {Z},\pi _n \circ \alpha )$ , where $\pi _n \colon \mathbb {Z} \twoheadrightarrow \mathbb {Z}/n \mathbb {Z}$ is the natural quotient map. In particular, each of these graphs will be connected, because $X_\infty $ is assumed to be connected, and the maps $\pi _n$ are surjective. Moreover, the number of spanning trees of each graph $X_n$ can be computed in terms of a Pierce–Lehmer sequence associated to the polynomial $J_\alpha $ , as the following result shows.

Theorem 3.6. Let X be a finite connected graph such that $\chi (X) \neq 0$ and fix a voltage assignment $\alpha \colon \mathbf {E}_X \to \mathbb {Z}$ such that the graph $X_\infty := X(\mathbb {Z},\alpha )$ is connected. Moreover, for every $n \in \mathbb {N}$ , we let $X_n := X(\mathbb {Z}/n \mathbb {Z},\pi _n \circ \alpha )$ , where $\pi _n \colon \mathbb {Z} \twoheadrightarrow \mathbb {Z}/n \mathbb {Z}$ is the natural quotient map. Then, the number of spanning trees of $X_n$ can be computed as

(3-12) $$ \begin{align} \kappa(X_{n}) = (-1)^{b(n-1)} \cdot \kappa(X) \cdot n^{e-1} \cdot \frac{\Delta_{n}(J_{\alpha})}{\Delta_{1}(J_{\alpha})}, \end{align} $$

where the integers $b := -{\mathrm {ord}}_{t = 0}(\mathcal {I}_\alpha ) \geq 0$ and $e := {\mathrm {ord}}_{t = 1}(\mathcal {I}_\alpha ) \geq 1$ are defined in terms of the Ihara polynomial $\mathcal {I}_\alpha \in \mathbb {Z}[t^{\pm 1}]$ , whose definition was recalled in (3-8). Moreover, $\{ \Delta _n(J_\alpha ) \}_{n \in \mathbb {N}}$ is the Pierce–Lehmer sequence, defined as in (2-4), which is associated to the polynomial $J_\alpha (t) := t^b \cdot (t-1)^{-e} \cdot \mathcal {I}_\alpha (t) \in \mathbb {Z}[t]$ .

Proof. First of all, observe that $e \geq 1$ because $\mathcal {I}_\alpha (1) = \det (D_X - A_X)$ , where $D_X$ and $A_X$ are respectively the degree and adjacency matrices associated to X, whose difference $D_X - A_X$ is singular, as explained in [Reference Corry and Perkinson6, Proposition 2.8]. Now, applying Proposition 3.5 to the Galois cover $X_n/X$ ,

(3-13) $$ \begin{align} n \cdot \kappa(X_n) = \kappa(X) \cdot \prod_{\psi \neq \psi_0} \mathcal{I}_{\alpha_n}(\psi(1)), \end{align} $$

where $\alpha _n := \pi _n \circ \alpha $ for every $n \in \mathbb {N}$ . Moreover, it is easy to see using (3-10) that

(3-14) $$ \begin{align} \prod_{\psi \neq \psi_0} \mathcal{I}_{\alpha_n}(\psi(1)) = \prod_{\psi \neq \psi_0} \mathcal{I}_\alpha(\psi(1)) = \prod_{\zeta \in W_n^\ast} \mathcal{I}_\alpha(\zeta), \end{align} $$

where $W_n^\ast := W_n \setminus \{1\}$ denotes the set of nontrivial roots of unity whose order divides n. Now, let us observe that

(3-15) $$ \begin{align} \prod_{\zeta \in W_n^\ast} \mathcal{I}_\alpha(\zeta) &= \prod_{\zeta \in W_n^\ast} (\zeta^{-b} \cdot I_\alpha(\zeta)) = (-1)^{b (n-1)} \prod_{\zeta \in W_n^\ast} I_\alpha(\zeta) \nonumber\\ &= (-1)^{b (n-1)} \mathrm{Res}\bigg( I_\alpha(t), \frac{t^n-1}{t-1} \bigg) \end{align} $$

as follows from the definition of resultant recalled in Section 2.1. Thus,

(3-16) $$ \begin{align} n \cdot \kappa(X_{n}) = (-1)^{b(n-1)} \cdot \kappa(X) \cdot \mathrm{Res} \bigg(I_{\alpha}(t),\frac{t^{n} - 1}{t-1} \bigg) \end{align} $$

by combining (3-13) with (3-14) and (3-15). To conclude, it suffices to observe that

$$ \begin{align*} \begin{aligned} \mathrm{Res} \bigg(I_{\alpha}(t),\frac{t^{n} - 1}{t-1} \bigg) &= \mathrm{Res} \bigg(t-1,\frac{t^{n} - 1}{t-1} \bigg)^e \cdot \mathrm{Res} \bigg(J_{\alpha}(t),\frac{t^{n} - 1}{t-1} \bigg) \\ &= n^e \cdot \mathrm{Res} \bigg(J_{\alpha}(t),\frac{t^{n} - 1}{t-1} \bigg) = n^e \cdot \frac{\Delta_n(J_\alpha)}{\Delta_1(J_\alpha)}, \end{aligned} \end{align*} $$

thanks to the multiplicative property of resultants.

Remark 3.9. Since the polynomial $J_{\alpha }$ is a reciprocal polynomial, it is known that the quantity $\lvert \Delta _{n}(J_{\alpha })/\Delta _{1}(J_{\alpha }) \rvert $ is a square when n is odd and $J_{\alpha }$ is a monic polynomial, as explained for instance in [Reference Einsiedler, Everest and Ward10, Section 2]. It follows that if p and $\ell $ are two distinct rational primes with $\ell $ odd and $J_{\alpha }$ is monic, then

$$ \begin{align*}\mathrm{ord}_{p}(\kappa(X_{\ell^{k}})) = \mathrm{ord}_{p}(\kappa(X)) + \mathrm{ord}_{p}(\Delta_{\ell^{k}}(J_{\alpha})/\Delta_{1}(J_{\alpha}) ), \end{align*} $$

and the parity of the number $\mathrm {ord}_{p}(\kappa (X_{\ell ^{k}}))$ , for all $k \ge 1$ , depends only on the parity of $\mathrm {ord}_{p}(\kappa (X))$ . This remark explains the parity of the p-adic valuation of the number of spanning trees in various $\mathbb {Z}_{\ell }$ -towers appearing in [Reference McGown and Vallières39, Reference McGown and Vallières40, Reference Vallières57] and [Reference Lei and Vallières35], since in each case, the tower in question is constructed from a voltage assignment $\alpha \colon \mathbf {E}_X \to \mathbb {Z}_\ell $ such that $\alpha (\mathbf {E}_X) \subseteq \mathbb {Z}$ . We point out as well that (3-12) is compatible with various results in the literature, such as [Reference Kwon, Mednykh and Mednykh31, Theorem 5.5], [Reference Mednykh42, Theorem 5.5] and [Reference Mednykh and Mednykh41, Theorem 3].

3.5.1. Archimedean asymptotics

We will start from the Archimedean asymptotics of the number of spanning trees, which are provided by the following corollary of Theorem 3.6.

Corollary 3.10. Let X be a finite connected graph such that $\chi (X) \neq 0$ and fix a voltage assignment $\alpha \colon \mathbf {E}_X \to \mathbb {Z}$ such that $X(\mathbb {Z},\alpha )$ is connected. Then, if the polynomial $J_{\alpha }$ defined by (3-11) does not have any root on the unit circle of $\mathbb {C}$ ,

$$ \begin{align*} \kappa(X_{n}) \sim n^{e-1} \frac{\kappa(X)}{|\Delta_{1}(J_{\alpha})|} M_{\infty}(I_{\alpha})^{n} \end{align*} $$

as $n \to +\infty $ .

Remark 3.12. It is reasonable to ask what happens when the Ihara polynomial $\mathcal {I}_\alpha $ has some roots on the Archimedean unit circle. In this case, it is easy to see that these roots will prevent one from getting a precise asymptotic for the growth of $\kappa (X_n)$ . To see this, fix some $\alpha = e^{2 \pi i \theta } \in \mathbb {C}$ with $\theta \in \mathbb {R} \setminus \mathbb {Q}$ . Then, the sequence

$$ \begin{align*}\lvert \alpha^n - 1 \rvert_\infty^2 = 2 (1 - \cos(2 \pi n \theta))\end{align*} $$

is distributed on the interval $[0,4]$ according to the probability density function $({2}/\pi) {\sqrt {4 x-x^2}}$ , thanks to Weyl’s equidistribution theorem [Reference Kuipers and Niederreiter30, Ch. 1, Example 2.1], and to the explicit computation of the probability density function of the random variable $2 (1-\cos (2 \pi U))$ , where U is a random variable that is uniformly distributed in the interval $[0,1]$ , which follows from the general transformation formula for probability density functions (see [Reference DeGroot and Schervish8, Theorem 3.8.4]). Therefore, we see that any asymptotic expansion for the growth of $\kappa (X_n)$ would have to feature some oscillating term, which takes into account this equidistribution phenomenon.

The previous Corollary 3.10 allows us to recover the asymptotics for the number of spanning trees of two particular examples of $\mathbb {Z}$ -towers, which were thoroughly studied in [Reference Mednykh and Mednykh41, Reference Mednykh42].

Example 3.13. Consider the following orientation $S = \{s_1,s_2,s_3\}$ on the dumbbell graph:

and fix a function $\alpha :S \rightarrow \mathbb {Z}$ such that $\alpha (s_2) = 0$ . This defines a voltage assignment on the dumbbell graph X, with values in $\mathbb {Z}$ . Moreover, the derived covers $X_n := X(\mathbb {Z}/n \mathbb {Z},\pi _n \circ \alpha )$ , where $\pi _n \colon \mathbb {Z} \twoheadrightarrow \mathbb {Z}/n \mathbb {Z}$ denotes the canonical projection, are given by the I-graphs $I(n,k,l)$ , where $k := \alpha (s_1)$ and $l := \alpha (s_3)$ . In particular, if $k = 1$ , one gets the family of generalised Petersen graphs $\mathrm {GP}(n,l)$ .

Now, one sees from Theorem 3.2 that the graph $X(\mathbb {Z},\alpha )$ is connected if and only if $(k,l) = 1$ , which we will assume for the rest of this example. Then, we can compute the Ihara polynomial associated to the voltage assignment $\alpha $ , which is given by

$$ \begin{align*} \mathcal{I}_\alpha(t) = (3 - t^{k} - t^{-k})(3 - t^{l} - t^{-l}) - 1 = t^{-(k+l)} \cdot I_{\alpha}(t) \end{align*} $$

where $I_{\alpha }(t) = (3 t^k - t^{2 k} - 1) \cdot (3 t^l - t^{2 l} - 1) - t^{k+l}$ . Since $I_\alpha (1) = I^{\prime }_\alpha (1) = 0$ , while $I^{\prime \prime }_\alpha (1) \neq 0$ , we see that $e := {\mathrm {ord}}_{t=1}(\mathcal {I}_\alpha ) = 2$ , which allows us to write

$$ \begin{align*} I_{\alpha}(t) = (t-1)^{2} \cdot J_{\alpha}(t) \end{align*} $$

for some $J_\alpha \in \mathbb {Z}[t]$ such that $J_\alpha (1) \neq 0$ . A simple calculation shows that $J_\alpha $ has no roots on the unit circle, as explained for instance in [Reference Mednykh42, Lemma 5.2]. Moreover, it is easy to see that

$$ \begin{align*} |\Delta_1(J_{\alpha})| = |J_{\alpha}(1)| = |I_{\alpha}^{\prime\prime}(1)|/2 = k^2 + l^2 \end{align*} $$

and that $\kappa (X) = 1$ . Therefore, Corollary 3.10 shows that

$$ \begin{align*} \kappa(I(n,k,l)) \sim \frac{n}{k^2 + l^2} \cdot M_\infty(I_{\alpha})^n \end{align*} $$

as $n \to \infty $ , which is precisely [Reference Mednykh42, Theorem 6.1].

Example 3.14. Consider the graph X consisting of a bouquet with k loops for some $k \in \mathbb {N}$ , and take an orientation $S = \{s_{1},\ldots ,s_{k} \}$ of X. Moreover, fix any function $\alpha :S \rightarrow \mathbb {Z}$ such that

$$ \begin{align*}1 \leq \alpha(s_1) < \alpha(s_2) < \cdots < \alpha(s_k) \end{align*} $$

and write $a_i := \alpha (s_i)$ for every $i \in \{1,\ldots ,k\}$ . Then, for n large enough, the derived graph $X_n := X(\mathbb {Z}/n\mathbb {Z},\pi _n \circ \alpha )$ is the circulant graph $C_{n}(a_{1},\ldots ,a_{k})$ . Note in particular that the example described in Section 3.4 belongs to this more general family.

Once again, it is easy to see by Theorem 3.2 that the graph $X(\mathbb {Z},\alpha )$ is connected if and only if $(a_{1},\ldots ,a_{k}) = 1$ , which we will assume for the rest of this example. Then, we can compute the Ihara polynomial associated to the voltage assignment $\alpha $ , and we obtain

$$ \begin{align*} \mathcal{I}_{\alpha}(t) = 2k - \sum_{i=1}^{k}(t^{a_{i}} + t^{-a_{i}}) = t^{-a_k} I_\alpha(t), \end{align*} $$

where $I_\alpha (t) := 2 k t^{a_k} - \sum _{i=1}^k (t^{a_k+a_i} + t^{a_k-a_i}) \in \mathbb {Z}[t]$ . In particular, we easily see that

$$ \begin{align*}I_{\alpha}(1) = I_{\alpha}'(1) = 0 \neq I_{\alpha}"(1),\end{align*} $$

which implies that $e := {\mathrm {ord}}_{t = 1}(\mathcal {I}_\alpha ) = 2$ , and that $I_{\alpha }(t) = (t-1)^{2} \cdot J_{\alpha }(t)$ for some $J_{\alpha } \in \mathbb {Z}[t]$ satisfying $J_{\alpha }(1) \neq 0$ . Moreover, one can easily show that $J_\alpha $ does not have any root on the unit circle of $\mathbb {C}$ , as explained in [Reference Mednykh and Mednykh41, Lemma 2]. Finally, we see that $\kappa (X) = 1$ and

$$ \begin{align*}\lvert \Delta_1(J_{\alpha}) \rvert = \lvert J_{\alpha}(1) \rvert = \lvert I_{\alpha}^{"}(1) \rvert/2 = \sum_{i=1}^{k}a_{i}^{2} \end{align*} $$

which, thanks to Corollary 3.10, implies that

$$ \begin{align*} \kappa(C_{n}(a_{1},\ldots,a_{k})) \sim \frac{n}{\sum_{i=1}^{k}a_{i}^{2}} \cdot M_\infty(I_{\alpha})^n \end{align*} $$

as $n \to \infty $ , which is [Reference Mednykh and Mednykh41, Theorem 5] in the particular case when $d=1$ .

3.5.2. p-adic asymptotics

Let us look at the asymptotics of the p-adic valuations of the number of spanning trees in a $\mathbb {Z}$ -tower, for a fixed prime p. As in the Archimedean setting, we start from the case when the Ihara polynomial $\mathcal {I}_\alpha $ does not have any nontrivial root lying on the unit circle of $\mathbb {C}_p$ .

Corollary 3.15. Let X be a finite connected graph such that $\chi (X) \neq 0$ , and fix a voltage assignment $\alpha \colon \mathbf {E}_X \to \mathbb {Z}$ such that $X(\mathbb {Z},\alpha )$ is connected. Fix moreover a rational prime $p \in \mathbb {N}$ . Then, if the polynomial $J_{\alpha }$ defined by (3-11) has no root on the unit circle of $\mathbb {C}_{p}$ ,

$$ \begin{align*}|\kappa(X_{n})|_{p} = |n|_{p}^{e-1} \frac{|\kappa(X)|_{p}}{|\Delta_{1}(J_{\alpha})|_{p}} M_{p}(I_{\alpha})^{n} \end{align*} $$

for every $n \in \mathbb {N}$ , where $e := \mathrm {ord}_{t = 1}(\mathcal {I}_\alpha )$ is the order of vanishing at $t = 1$ of the Ihara polynomial associated to $\alpha $ .