Is it necessary that in two undirected graphs if we need to prove that vertices have one to one correspondence then graph should have same number of edges? What about same number of degree?
Can anyone give me a example when one to one correspondence exist without same number of edges?
How about these graphs - do they have one-one correspondence just because they have same number of vertices?
EDIT: Please note I am only looking for ONE-ONE correspondence and NOT Isomorphism
No, these graphs are not isomorphic. It is not sufficient that there be an equal number of nodes, or even an equal number of nodes and edges. There must exist (at least) one bijective mapping from $G(V, E)$ to $G'(V', E')$ such that two nodes $v_1, v_2 \in V$ are connected by an edge $e \in E$ if and only if the corresponding nodes $v'_1, v'_2 \in V'$ are connected by an edge $e' \in E'$.