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Do we have any formula for counting the number of graphs with $n$ vertices, that has exactly $k$ vertices with degree $d$ and the other vertices have different and disjoint degreedegrees? (Different and disjoint are the same, $d_1$ is different or disjoint rather than $d_2$, iff $d_1\neq‎ d_2$.)

For example, for $n=3, k=2, d=1$, we only have one graph($P_3$) with this property. Also, for $n=4, k=2, d=2$, we have the only graph with degree sequence $1, 2, 2, 3$.

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# Counting Special Graphs

Do we have any formula for counting the number of graphs with $n$ vertices, that has exactly $k$ vertices with degree $d$ and the other vertices have different and disjoint degree?

For example, for $n=3, k=2, d=1$, we only have one graph($P_3$) with this property. Also, for $n=4, k=2, d=2$, we have the only graph with degree sequence $1, 2, 2, 3$.