I am given the definition of graph isomorphism as follows:
Let $G$ be a graph with vertex set $V_G$ and edge set $E_G$, and let $H$ be a graph with vertex set $V_H$ and edge set $E_H$. Then $G$ is isomorphic to $H$ if there are one-to-one correspondences
$$\alpha : V_G \rightarrow V_H \text{ and } \beta : E_G \rightarrow E_H$$
such that, for any edge $e \in E_G$,
$$ e \text{ joins vertex } v \text{ to vertex } w \iff \beta(e) \text{ joins vertex } \alpha(v) \text{ to vertex } \alpha(w).$$
In this case, we write $G \cong H$.
From said definition, does it naturally follow that any two isomorphic graphs must have the same number of edges and vertices?
Best Answer
As already noted in comments:
$\alpha$ and $\beta$ are explicitly specified to be "one-to-one correspondences", which is a way of saying "bijections". Since there is a bijection between $V_G$ and $V_H$, they have the same number of elements. Similarly, since there is a bijection between $E_G$ and $E_H$, they have the same number of elements.