we need due to Allard and Froese. In Section \[sec4\], we establish a formula giving a kernel representation of the resolvent associated to the operator we consider and which is crucial for our analysis. In Section \[sec5\], we define and characterize the resonances near the spectral thresholds, while in Section \[sec6\] we give the proof of our main results. Section \[sa\] gathers useful tools on the characteristic values concept of finite meromorphic operator-valued functions.

Presentation of the model {#secd}
=========================

We consider an infinite graph $\mathbb{G} = (\mathcal{V},\mathcal{E})$ with vertices $\mathcal{V}$ and edges $\mathcal{E}$, and we let $\ell^2(\mathcal{V})$ be the Hilbert space $$\label{eq1,0}
\ell^2(\mathcal{V}) := \bigg\lbrace \phi : \mathcal{V} \rightarrow {{\mathbb{C}}}: 
\Vert \phi \Vert^2 := \sum_{v \in \mathcal{V}} \big\vert \phi(v) 
\big\vert^2 < \infty \bigg\rbrace,$$ with the inner product $$\label{eq,ps}
\langle \phi,\psi \rangle := \sum_{v \in \mathcal{V}} \phi(v) \overline{\psi}(v).$$ On $\ell^2(\mathcal{V})$, we consider the symmetric Schrödinger operator ${{-\Delta}}$ defined by $$\label{eq1,1}
({{-\Delta}}\phi)(v) := - \sum_{w \, : \, w \sim v} \Big( \phi(w) - \phi(v) \Big),$$ where $w \sim v$ means that the vertices $w$ and $v$ are connected by an edge. If we define on $\ell^2(\mathcal{V})$ the symmetric operator $L$ by $$\label{eq1,2}
(L \phi)(v) := \sum_{w \, : \, w 