Hello everyone interested. This is a question that seems simple but has me troubled. I have a tree $G$ with an unknown number of vertices $\{v_j\}$. I know that there exists a vertex with degree $\deg(v_1)= 5$, two vertices with degrees $\deg(v_2),\deg(v_3)=4$, three vertices with degrees $\deg(v_4),\deg(v_5),\deg(v_6)=2$ and the rest of the vertices are leaves. How do I find the number of vertices?
By trying to plot the graph, I get the number $n$ of vertices must be over $6$, but I am confused on what the exact number is, or how the leaves come into the equation.
Any suggestions?
Thanks in advance.
Hint: If you sum the degrees in a graph you get what is called the Handshake lemma: $$\sum _{v_i}deg(v_i)=2E,$$ In a tree, as it contains no cycles(Hence just has $1$ face), Euler characteristic formula says that $V-E+F=2.$
Can you conclude?