In an exercise I am asked to say how many trees of 6 vertex are there, taking into account that there has to be four vertex of degree 2. After some attempts, I have reached the conclusion that there is none, but I'm not sure it is right. Here is what I have done:
Since the tree has $6$ vertex, there has to be $5$ edges, and taking into account that $2|E| = \sum_{v \in V}{deg}(v)$, I obtain that there are $2$ edges of degree $1$. However, no matter how much I try, I end up with the fact that or there is a vertex of degree $3$ or there are more vertex of degree $1$. Since there are supposedly two vertex of degree $1$, there are two options: they are joint with the same vertex or not. In the first case, that vertex of degree $2$ cannot be joint with any other vertex since the maximum degree according to the exercise is $2$. And since the graph is connected, this means that one of those two vertex of degree $1$ are not actually of that degree. In the second case, a similar reasoning turns up to be something with no concordance with the conditions of the exercise. Does this mean that there aren't any graphs with these restrictions?
2026-03-30 00:24:33.1774830273