Bridge to Higher Mathematics, MA 1971
Exercise Set V
1. Show that the harmonic mean is less than or equal to the geometric mean. Hint:
Let bk := 1/ak and apply the relationship between the geometric mean and the
arithmetic mean to the bk.
2. Draw all of the possible graphs on four vertices. Which ones are connected and
which are disconnected?
3. A walk is a alternating sequence of vertices and incident edges of the form
{v0, e1, v1, e2, v2, ..., en, vn}
where repetition is allowed. A closed walk is a walk which begins and ends at the
same vertex.
(a) Please give an example of a closed walk with an even number of edges which
contains no cycle.
(b) Please show that any closed walk with an odd number of edges must contain
a cycle.
4. Please show that any planar graph must have a vertex with degree that is less than
or equal to 5.
