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I want to ask about the relationship between distinct vertices and distinct edges.

I was able to come up with an explanation for "distinct vertices" $\Rightarrow$ "distinct edges".

An edge is considered the same if it has the same endpoints (am I allowed to take this as self-evident?). Therefore, if all vertices are distinct in a walk, no vertex appears twice, so every edge is distinct.

Is this correct? Also, is the converse true/can I make a counterexample?

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  • $\begingroup$ The converse is certainly false. You can have a walk with distinct edges visiting any given vertex many times, if you have enough distinct edges meeting at that vertex. $\endgroup$ – Gerry Myerson Jun 8 '13 at 4:16
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You are correct, and multigraphs would need a stronger definition.

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