Explain and draw why graphs with the given specifications do not exist.

a) Tree, nine vertices, nine edges.

b) Tree, five vertices, total degree 10.

c) Graph, connected, six vertices, five edges, has a non-trivial circuit.

ii. Show that a finite simple graph with more than one vertex has at least two

vertices with the same degree.

iii. Prove that in a tree there exists exactly one path between every pair of vertices.