Let $\mathcal{G}$ denote the set of all countable infinite (simple unlabeelled) graphs.
$|\mathcal{G}|\supseteq|\mathbb{R}|$
To see this, we construct an injective function $f:2^\mathbb{N}\rightarrow\mathcal{G}$ (I take $\mathbb{N}=\{1,2,\ldots\}$), because $|2^\mathbb{N}|=|\mathbb{R}|$. Let $P_n$ denote the path graph on $n$ vertices. Note that $P_1$ is the trivial edgeless graph. For two graphs $G_1,G_2$, denote their sum by $G_1\oplus G_2$ (graphically, we just draw both graphs next to another, without connecting them). This operation is commutative and associative and we can without problem generalize it to any set $S$ of graphs to get $\bigoplus_{G\in S}G$. Furthermore for two Graphs $G_1,G_2$ let $G_1\vee G_2$ denote the join of $G_1$ and $G_2$. Then for any graph $G$, $G\vee P_1$ is the graph obtained by adding a new vertex and edges connecting the new vertex with all vertices of $G$.
Now define $f$ by $f(A)=\left(\left(\bigoplus_{n\in A}P_n\right)\vee P_1\right)\oplus \bigoplus_{n\in\mathbb{N}\setminus A} P_n$
Visually, we draw each number in the plane (by their path graph) and then glue all numbers/graphs together which are in the subset $A$.
$|\mathcal{G}|\subseteq|\mathbb{R}|$
We can describe a directed multigraph with a tuple $(V,E,S,T)$ where $V$ is the set of vertices, $E$ is the set of edges and $S,T$ are functions from $E$ to $V$, describing the source and target vertex of every edge. The set of all countable directed multigraphs can then be described by $\mathcal{G}'=\{(\mathbb{N},\mathbb{N},S,T):S,T\in \mathbb{N}^\mathbb{N}\}=\{\mathbb{N}\}\times\{\mathbb{N}\}\times\mathbb{N}^\mathbb{N}\times\mathbb{N}^\mathbb{N}$
Note that this set contains a lot of graphs that are isomorphic to one another. There is also a natural inclusion map from $\mathcal{G}$ into $\mathcal{G}'$, which can be obtained by assigning the natural numbers to the vertices and edge each and a direction to every edge, so $|\mathcal{G}|\subseteq|\mathcal{G}'|$. Furthermore it is known that $|\mathbb{N}^\mathbb{N}|=|\mathbb{R}|$, so $|\mathcal{G}'|=|\{1\}\times\{1\}\times\mathbb{R}\times\mathbb{R}|=|\mathbb{R}^2|=|\mathbb{R}|$.