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I mean multiple loops on the same node for example can I have a graph whose degree level is 6 and it contains only one node?

The One
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  • Sure, why not?${}$ –  Jun 25 '14 at 08:01
  • @Mike Miller Never seen it in any example. It means not simple graph can be everything! is it right? – The One Jun 25 '14 at 08:11
  • I don't know what you mean by everything, and you should be careful and check your definition of graph. Mine is just "set" (of vertices) and "set of pairs of vertices" (edges). –  Jun 25 '14 at 08:12
  • A graph with 7 nodes whose deg levels are 0,1,1,2,3,5,6 is possible? i can just draw a loop from each of the nodes to itself times the deg level excluding the odd ones. @Mike Miller – The One Jun 25 '14 at 08:15
  • No, that doesn't work, for different reasons (loops are counted twice in degree). Yes, there is a graph with those degree levels. –  Jun 25 '14 at 08:17
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    @Mike: No, you be careful! If a graph is a set of vertices and a set of pairs $(v_i,v_j)$ of vertices, then there can only be at most one edge $(v_i,v_j)$ for a given pair of vertices, because a set can't contain two identical elements. – TonyK Jun 25 '14 at 08:35
  • @TonyK I never said ordered (nor meant it), which of course doesn't prevent that mistake :P But I think the moral here (check your definitions) is highlighted by my mistake. (And thank you for the correction.) –  Jun 25 '14 at 08:38
  • Just to clear up some potential confusion: my original comment said "ordered pairs" instead of "pairs". Then I realised my mistake and edited it out. Meanwhile Mike Miller simul-posted his comment. – TonyK Jun 25 '14 at 08:50

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As everyone said, depends on definitions. Wikipedia (and other sources) distinguish multigraphs and pseudographs; both can have multiple edges but only the latter is allowed to have loops. Pseudographs are also called loop-multigraphs. I have not seen a version of pseudograph definition that would restrict the number of loops at the same vertex. The MathWorld article Pseudograph shows the picture of a graph with multiple loops:

pseudograph