1. **Problem Statement:** We need to find a nontrivial connected graph $G$ which satisfies the following properties: (a) Every bridge of $G$ is adjacent to an edge that is not a bridge. (b) Every edge of $G$ that is not a bridge is adjacent to a bridge. (c) $G$ contains two nonadjacent bridges. (d) Every two edges of $G$ that are not bridges are adjacent. 2. **Understanding the Problem:** - A **bridge** in a graph is an edge whose removal increases the number of connected components. - Two edges are **adjacent** if they share a vertex. 3. **Constructing the Graph:** - To satisfy (c), we need at least two bridges which do not share a vertex. - Condition (d) requires that any pair of non-bridge edges must share a vertex. - Condition (a) requires every bridge to be adjacent to at least one non-bridge edge. - Condition (b) requires every non-bridge edge to be adjacent to a bridge. 4. **Graph Example:** Consider a graph $G$ composed of a triangle ($3$-cycle) $C = \{v_1, v_2, v_3\}$ and two pendant edges attached to two different vertices in the triangle: - Add vertices $v_4$ connected to $v_1$ and $v_5$ connected to $v_2$. Graph edges: - Triangle edges (non-bridges): $e_1 = v_1v_2$, $e_2 = v_2v_3$, $e_3 = v_3v_1$ - Pendant edges (bridges): $e_4 = v_1v_4$, $e_5 = v_2v_5$ 5. **Verification:** - Bridges: $e_4$ and $e_5$ (removal disconnects $v_4$ or $v_5$). - Non-bridges: $e_1, e_2, e_3$ (triangle cycle). - (a) Bridges $e_4$ adjacent to $e_1, e_3$ (non-bridges), $e_5$ adjacent to $e_1, e_2$ - (b) Non-bridges $e_1$ adjacent to $e_4, e_5$ (bridges), $e_2$ adjacent to $e_5$, $e_3$ adjacent to $e_4$. - (c) Bridges $e_4$ and $e_5$ are not adjacent (do not share a vertex). - (d) All non-bridge edges ($e_1, e_2, e_3$) share vertices pairwise in the triangle. 6. **Final answer:** The graph composed of a triangle with two pendant edges at two distinct vertices satisfies all conditions.