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.
draw a third equivalent graph. choose the correct answer below.
a.

k
b
r
c

b.

k
b
r
c

c.

k
b
r
c

d.

k
r
b
c

Explanation:

Equivalent graphs (also known as isomorphic graphs) must have the same number of vertices, the same number of edges, and the vertices must be connected in the same way. Let's analyze each option:

• Option A: For two graphs to be equivalent, the adjacency relationships (how vertices are connected) must be the same. If vertices are connected in the same way (i.e., the adjacency lists or matrices are the same up to vertex - label permutation), this is a key property of equivalent graphs.
• Option B: The number of vertices is a necessary condition for graph equivalence. If two graphs have a different number of vertices, they cannot be equivalent.
• Option C: The number of edges not matching the number of vertices is not a relevant condition for graph equivalence. For example, a cycle graph \(C_n\) (where \(n\) is the number of vertices) has \(n\) edges, but a complete graph \(K_n\) has \(\frac{n(n - 1)}{2}\) edges. The relationship between the number of edges and vertices is not a criterion for equivalence.
• Option D: The number of edges is a necessary condition. If two graphs have a different number of edges, they cannot be equivalent.

For drawing a third - equivalent graph, we can re - arrange the vertices while maintaining the adjacency relationships. The key is to keep the same vertex - connection pattern. For example, if in the original graphs, vertex \(B\) is connected to \(C\) and \(K\), \(C\) is connected to \(B\) and \(R\), \(K\) is connected to \(B\), and \(R\) is connected to \(C\), we can redraw the graph with vertices in a different geometric arrangement but with the same connections.

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.