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If $(V, E)$ is a directed graph with $V$ set of vertices and $E$ set of edges, then $E\subset V\times V$, Where $E$ being the set of all possible edges. Will the notation still hold good, if the graph is connected and undirected , i.e., every adjacent vertices are reachable from both ends? I think the notation will still hold because the edge $(V1, V2)$ is same as the edge $(V2, V1)$. Thanks in advance for help.

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  • $\begingroup$ Why would it possibly matter whether I choose to have $E$ be a subset of $V \times V$ or $\binom V2$ or $\mathbb R^{17}$ or anything else, as long as I define the incidence relation between $V$ and $E$ appropriately? I feel like any argument that relies on looking at the structure of an element of $E$ is no longer dealing with purely graph-theoretic properties of a graph. $\endgroup$ Mar 3, 2018 at 23:48

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Technically yes but you would need additional conditions on $E$ and working in this setup for undirected graphs would introduce all sorts of issues which can be avoided by instead requiring $E \subset $$V \choose 2$, where $V \choose 2$ means the subsets of $V$ of size 2.

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