Final answer:
Two isomorphic graphs with a square arrangement of vertices and different diagonals as edges can have non-isomorphic spanning trees due to the choice of edges included in the spanning trees.
Step-by-step explanation:
A student asked about an example of two isomorphic graphs whose spanning trees are not isomorphic. First, let's understand that two isomorphic graphs have a one-to-one correspondence between their vertices and edges, preserving adjacency. However, the structure of their spanning trees can differ due to the multiple spanning trees that a graph can have.
Here is a simple example to illustrate this. Consider two separate graphs, both with four vertices arranged in a square. In one graph, the edges form a square with one diagonal (graph A), while in the other graph, the edges form a square with the opposite diagonal (graph B). The graphs are isomorphic because there is a one-to-one correspondence between their vertices and edges. However, when looking at the spanning trees, graph A can have a spanning tree that includes the diagonal, while graph B cannot. If we consider the shape formed by the edges of the spanning trees, one will have a triangle with an extension (in graph A) and the other will simply be a path or a 'T' shape (in graph B).
Therefore, while the original graphs are isomorphic, their spanning trees can be non-isomorphic due to different choices of edges.