Provide three different spanning trees for the following graph: Prove that G is a tree if and only if there is a unique simple path between any two distinct vertices. Prove that if all vertices have...

Provide three different spanning trees for the following graph:

Prove that G is a tree if and only if there is a unique simple path between any two distinct vertices.

Prove that if all vertices have degree > = 2, then G contains a polygon.

Prove that a tree with >¼ 2 vertices must have > = 2 pendant vertices, which are vertices of degree 1. 27. Prove that if T is a tree with a vertex of degree k > 1, then T must have > = k pendant vertices.