HW 2: due Thursday, September 20 in class
Note: for the rest of this class, assume all graphs are SIMPLE unless other-
wise stated.
1. Show that in any group of two or more people, there are always two
with exactly the same number of friends in the group.
2. Show that if Gis a tree with ∆ ≥k, then Ghas at least kvertices of
degree 1. (Recall ∆ is the maximum degree of G).
3. A center of Gis a vertex usuch that maxv∈Vd(u, v) is as small as
possible. Show that a tree has either exactly one center or two adjacent
centers.
4. Let Gbe connected and let e∈E. Show that eis in every spanning
tree of Gif and only if eis a cut edge of G.
5. Prove that the graph obtained from Knby deleting an edge has (n−
2)nn−3spanning trees. (Recall that Knis the complete graph on n
vertices).