Question

determine whether the graph is a tree. if the graph is not a tree, give the reason why.
the graph is a tree.
the graph is not a tree because it is disconnected.
the graph is not a tree because it has one or more circuits.

Explanation:

Step1: Recall the definition of a tree

A tree is a connected acyclic (no circuits) graph with \( n \) vertices and \( n - 1 \) edges.

Step2: Analyze the given graph

• Connectivity: All vertices (A, B, C, D) are connected through vertex B. So the graph is connected.
• Circuits: A circuit is a path where the start and end vertices are the same, and no other vertex is repeated. In this graph, there are no such paths. For example, from A to B to C and back to A: there is no edge from C to A. Similarly for other combinations.
• Number of vertices and edges: There are 4 vertices (A, B, C, D) and 3 edges (A - B, B - C, B - D). For a tree with \( n = 4 \) vertices, the number of edges should be \( n - 1=4 - 1 = 3 \), which matches.

Answer:

The graph is a tree.