The possible degree sequence of the simple graph with 4 vertices is $(0,0,0,0)$ or $(0,0,1,1)$ or $(1,1,1,1)$ or $(0,1,1,2)$ or $(0,2,2,2)$ or $(1,1,2,2)$. Then I do not understand how to prove the given question. Anyone, please help me .
2026-03-27 00:04:01.1774569841
Prove that two simple graphs on 4 vertices are isomorphic if and only if they have the same degree sequence.
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All the $0$'s can be ignored since they will correspond to isolated vertices and isolated vertices are always isomorphic.
Same for the $3$'s, those will be vertices connected to all other vertices.
This means all vertices with degree $0$ or $3$ can be removed from the graph without changing the question.
What you are left with is either two disjunct linegraphs of length $2$ or one linegraph that is completely determined by the degree sequence.