Graph theory problems about vertices and interior angles.

Question

I have three problems of graph theory:

$1.$We have a $10$ gon then maximum number of acute angles that we can make is?

$2.$We have $5$ vertices then how many connected trees we can make?

$3.$How many $4$ vertices connected graphs not including a triangle we can make?

My try:I think $3$ is the answer of first problem,is this correct?

For third problem,I make $3$ such graphs,is this correct?or we can make more such graphs?

For second problem I just know the deffinition but I do not know how to solve this.

Are there formulas to solve these problems?

Thanks.

Answer 1

1. For any simple polygon, the sum of the exterior angles (i.e. the supplements of of the interior angles) equals $360^\circ$. And for a convex polygon, the exterior angles are all nonnegative. Hence at most 3 of the exterior angles can exceed $90^\circ$, i.e. at most three interior angle scan be acute.