Isomorphism and Subgraphs · Challenge problem

Let $G$ have vertices $1,\ldots,7$ and edges $13,16,17,23,25,27,34,35,37$. Let $H$ have vertices $1,\ldots,7$ and edges $12,16,23,24,26,27,34,36,45$. These are isomorphic. Vertex $1$ of $G$ is one of three vertices of degree $3$ in $G$, but it is the only degree-$3$ vertex of $G$ adjacent to BOTH a degree-$5$ vertex and a degree-$1$ vertex. Under the (unique) isomorphism $\varphi:G\to H$, find $\varphi(1)$.