Graph with a vertex of infinite degree elementary equiv. with a graph with vertices of arbitrarily large finite degree

Question

I want to prove that :
Given a graph $G$ with vertices of arbitrarily large finite degree, there is a graph elementary equivalent to $G$ that contains a vertex of infinite degree.
(the degree of a vertex is $n$ if it is connected with $n$ other vertices)
I've tried this way :
Let $\Sigma = EDiag(G) \cup \{deg(v) > n | n \in \omega\}$ where $v$ is a new symbol of constant. Every finite $\Sigma_0 \subset \Sigma$ is satisfied by an expantion of G, hence, for Compactness theorem, $\Sigma$ is satisfiable; so there is a model for $\Sigma$ i.e. a graph $G_1$ with a vertex of infinite degree and such that $G$ elementary embeds in $G_1$.
I don't know how to find the other elementary embedding or how to prove that $Th(G) = Th(G_1)$. Is even the first part correct?