1. **Problem 1.1a:** Find all edges incident on vertex $v_1$. Edges incident on $v_1$ are those connected to $v_1$. Given: - $e_1$ is a loop on $v_1$. - $e_2$ connects $v_1$ to $v_2$. - $e_4$ connects $v_1$ to $v_5$. So, edges incident on $v_1$ are $e_1$, $e_2$, and $e_4$. 2. **Problem 1.1b:** Find all vertices adjacent to $v_3$. Vertices adjacent to $v_3$ are those connected directly by an edge: - $e_3$ connects $v_2$ to $v_3$. - $e_5$ connects $v_5$ to $v_3$. - $e_6$ connects $v_2$ to $v_3$ (parallel to $e_3$). - $e_7$ connects $v_3$ to $v_5$ (parallel to $e_5$). So, vertices adjacent to $v_3$ are $v_2$ and $v_5$. 3. **Problem 1.1c:** Find all loops. A loop is an edge connecting a vertex to itself. - $e_1$ is a loop on $v_1$. So, the only loop is $e_1$. 4. **Problem 1.1d:** Find all parallel edges. Parallel edges connect the same pair of vertices. - $e_3$ and $e_6$ both connect $v_2$ to $v_3$. - $e_5$ and $e_7$ both connect $v_3$ to $v_5$. So, parallel edges are $(e_3, e_6)$ and $(e_5, e_7)$. 5. **Problem 1.1e:** Find all isolated vertices. An isolated vertex has no edges connected. - $v_4$ is isolated. **Final answers for Problem 1.1:** - a) $e_1$, $e_2$, $e_4$ - b) $v_2$, $v_5$ - c) $e_1$ - d) $(e_3, e_6)$ and $(e_5, e_7)$ - e) $v_4$