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Is a figure with only one vertices and one edge connected to that vertices known as a graph ?

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If that edge is a loop, yes. It is a Multigraph. Otherwise, it is not a graph.

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  • $\begingroup$ Sorry I forgot to say m it’s not a loop $\endgroup$ – Ling Min Hao Jun 12 at 15:29
  • $\begingroup$ Then that's not a graph. Simple graphs with non-empty edge sets need to have at least $2$ vertices. $\endgroup$ – ArsenBerk Jun 12 at 15:30
  • $\begingroup$ Okay thanks , this also holds to subgraph right ? $\endgroup$ – Ling Min Hao Jun 12 at 15:30
  • $\begingroup$ Subgraphs are also graphs so yes. For instance, in a graph with $2$ vertices $x,y$ and an edge $xy$, we can't take a vertex $x$ and an edge $xy$ as a subgraph since it is not even a graph. $\endgroup$ – ArsenBerk Jun 12 at 15:31
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With the strict definition of a simple graph, then there is no graph with one vertex and one edge. However, you could define a generalization of simple graphs which allows "half" edges with only one vertex and is not a loop. As far as the combinatorics, a half edge is no different than a loop, except in a path, it would be a dead end. However, in a drawing of the generalized graph on a surface, the loop would be closed curve, while a half edge would be be an open curve with only one of its endpoints to be considered a vertex. How useful this would be is not obvious to me now, but there is no reason why it could not have some useful purpose, just as graphs themselves have developed a great number of uses.

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  • $\begingroup$ We could see it as half-edges too. The same way the configuration random graph model is built. Then introduce some matching between them... $\endgroup$ – Thomas Lesgourgues Jun 12 at 19:35
  • $\begingroup$ @ThomasLesgourgues Yes, thanks for your comment about half-edges! $\endgroup$ – Somos Jun 12 at 19:37
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That is not a graph. A simple graph with one edge need to have at least 2 vertices.

Perhaps you meant this:

The graph $K_n$-clique is a graph in which all vertices are connected. Perhaps you want $K_2$.

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