Question

explain why the two figures show equivalent graphs then draw a third equivalent graph.
why are the two graphs equivalent? select all that apply
a. the vertices are connected in the same way
b. the graphs have the same number of vertices
c. the graphs have a number of edges that matches the number of vertices.
d. the graphs have the same number of edges.

Explanation:

For two graphs to be equivalent (isomorphic), they must have the same number of vertices (so they can be matched one - to - one) and the same number of edges (so the connections can be preserved). Also, the vertices must be connected in the same way (i.e., the adjacency of vertices is preserved).

• Option A: If the vertices are connected in the same way (preserving adjacency), it is a key property of equivalent graphs. For example, if in one graph vertex \(B\) is adjacent to \(C\) and \(K\), and in the other graph the corresponding vertex (mapped from \(B\)) is adjacent to the corresponding vertices (mapped from \(C\) and \(K\)), this is important for equivalence.
• Option B: Having the same number of vertices is a necessary condition. If graph \(G_1\) has \(n_1\) vertices and graph \(G_2\) has \(n_2\) vertices, and \(n_1

eq n_2\), they cannot be equivalent. Here, both graphs have 4 vertices (\(B\), \(C\), \(K\), \(R\)).

• Option C: The statement “The graphs have a number of edges that matches the number of vertices” is not a valid condition for graph equivalence. For example, a cycle graph \(C_4\) (4 vertices and 4 edges) and a complete bipartite graph \(K_{2,2}\) (4 vertices and 4 edges) are not equivalent.
• Option D: Having the same number of edges is a necessary condition. Counting the edges:
• In the first graph, we can count the edges. Let's assume the connections: \(B - C\), \(B - K\), \(B - R\), \(C - R\) (4 edges).
• In the second graph, \(B - C\), \(B - K\), \(B - R\), \(C - R\) (4 edges).

Answer:

A. The vertices are connected in the same way, B. The graphs have the same number of vertices, D. The graphs have the same number of edges.