Does the single source shortest path always have the shortest edge in the graph? Why or why not?

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*by single source shortest path, I mean the path that is the solution to the traveling salesperson problem

Does it also have the second shortest edge?

Also, what other general properties can be said about the single source path?

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No. Hint: you can find a counterexample with only three nodes and three arcs, with weights $1$, $2$, and $2$.