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I have the graph $V=\{a,b,c\}$ and $E=\{(b,b),(c,c),(a,b)\}$.

From the definition of isolated vertex:

"an isolated vertex has no edge",

I can tell that vertex $c$ is not an isolated vertex. What confused me is that if $c$ is not connected to the other 2 point, is this graph a connected graph? Because this graph has no isolated vertex.

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  • $\begingroup$ Welcome to MSE. Your question is phrased as an isolated problem, without any further information or context. This does not match many users' quality standards, so it may attract downvotes, or closed. To prevent that, please edit the question. This will help you recognise and resolve the issues. Concretely: please provide context, and include your work and thoughts on the problem. These changes can help in formulating more appropriate answers. $\endgroup$ Sep 27 '21 at 8:24
  • $\begingroup$ What is the definition of connectedness? $\endgroup$ Sep 27 '21 at 9:18
  • $\begingroup$ Cf. math.stackexchange.com/questions/2051323/… $\endgroup$ Sep 27 '21 at 9:20
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Your graph indeed has no isolated vertices. It is not connected: its connected components are $\{a,b\}$ and $\{c\}$.

A graph that has an isolated vertex is not connected, but the converse is not true: a graph can be disconnected while having no isolated vertices. For example, you can take the disjoint union of two connected graphs: the result has two connected components and no isolated vertices. If this is not clear to you, then maybe you are confused about the definition of connectivity.

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