Final answer:
It is true that isomorphic graphs can have different vertex labels, since isomorphism focuses on graph structure rather than vertex names. Similarities include the same number of vertices and edges, and an identical structure. The primary differences are related to the labels of the vertices, and isomorphic graphs are more similar in terms of their mathematical properties.
Step-by-step explanation:
If two graphs are isomorphic, then it is true that they can have different vertex sets. Isomorphism in graphs refers to a correspondence between the graphs that is based on the structure, not the particular names or labels of the vertices. So, one graph may have vertices labeled A, B, C, while another isomorphic graph could have vertices labeled 1, 2, 3, etc. What matters is that there is a one-to-one correspondence between the vertex sets and that the edges are preserved between the vertices under this correspondence.
Two main similarities between such isomorphic graphs would involve them having the same number of vertices and edges, and the same structure, meaning that the connectivity and relations between nodes are identical. Differences are primarily superficial and related to the labeling of vertices, as mentioned, or potentially the graphical representation, but they do not affect the inherent structure of the graph.
Overall, since isomorphism emphasizes the preservation of structural properties over the labeling or graphical representation, isomorphic graphs are more similar than different when it comes to their core mathematical properties.