Final Answer:
The three statements a, c, and d imply one another in an undirected graph G on n nodes. If G is connected (a), then it contains n - 1 edges (c) and is acyclic (d). Conversely, if G has n - 1 edges (c), it is necessarily connected (a) and acyclic (d).
Step-by-step explanation:
In an undirected graph G with n nodes, the number of edges, denoted by |E(G)|, corresponds to the connectivity and presence of cycles. Statement c, |E(G)| = n - 1, is linked to both graph connectivity and acyclicity. For a connected graph, the minimum number of edges required is n - 1 to ensure there are no disconnected nodes. This condition forms a spanning tree, guaranteeing connectivity without cycles.
A connected graph with n nodes and n - 1 edges forms a tree. A tree is by definition acyclic, fulfilling statement d. Conversely, in an acyclic graph, the maximum number of edges possible is n - 1, indicating a connected graph, satisfying statement a. Additionally, the absence of cycles in a graph with n - 1 edges ensures it is acyclic, supporting statement d.
Furthermore, considering a graph without cycles (statement b), if there are no cycles, it implies a tree structure, meeting statements c and d. Conversely, in a graph with n - 1 edges forming a tree, there can be no cycles, aligning with statement b. Therefore, each statement logically implies the other two, forming an interdependent relationship among connectivity, edge count, and acyclicity in an undirected graph.