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This is a quick question and I know all about these graphs what I am supposed to know. But I am unable to explain myself in words. Like I know for regular graph the vertex must have same degree and bipartite graph is a complete bipartite iff it contain all the elements m.n(say) I am looking for a mathematical explanation.

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In the complete bipartite graph $K_{m,n}$, the vertices have degree $m$ or degree $n$ (and both of these degrees are reached). Thus, if you want it to be regular, a sufficient and necessary condition is $n=m$.

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