[Solved] MA1971 Exercise Set V

1. Show that the harmonic mean is less than or equal to the geometric mean. Hint: Let b_k := 1/a_k and apply the relationship between the geometric mean and the arithmetic mean to the b_k.
2. Draw all of the possible graphs on four vertices. Which ones are connected andwhich are disconnected?
3. A walk is a alternating sequence of vertices and incident edges of the form

{v₀,e₁,v₁,e₂,v₂,,e_n,v_n}

where repetition is allowed. A closed walk is a walk which begins and ends at the same vertex.

• Please give an example of a closed walk with an even number of edges whichcontains no cycle.
• Please show that any closed walk with an odd number of edges must containa cycle.

4. Please show that any planar graph must have a vertex with degree that is less thanor equal to 5.