1) a) Let k ≥ 2 and let G be a k-regular bipartite
graph. Prove that G has no cut-edge. (Hint: Use the bipartite
version of handshaking.)
b) Construct a simple, connected, nonbipartite 3-regular graph
with a cut-edge. (This shows that the condition “bipartite” really
is necessary in (a).)
2) Let F_n be a fan graph and Let a_n = τ(F_n) where τ(F_n) is
the number of spanning trees in F_n. Use deletion/contraction to
prove that a_n = 3a_n-1 - a_n-2...
Let (?,?,?) be a probability space and suppose that ?∈? is an
event with ?(?)>0.
Prove that the function ?:?→[0,1] defined by ?(?)=?(?|?) is a
probability on (?,?).
Let S={1,2,3,6} and define the relation ~ on S2 by
(m,n) ~ (k,l) for m+l=n+k
Show that it is an equivalent relation
Find the number of different equivalent classes and write all
of them
Prove the following statements!
1. Let S = {0, 1, . . . , 23} and define f : Z→S by f(k) = r
when 24|(k−r). If g : S→S is defined by
(a) g(m) = f(7m) then g is injective and
(b) g(m) = f(15m) then g is not injective.
2. Let f : A→B and g : B→C be injective. Then g ◦f : A→C is
injective.
3. Let f : A→B and g : B→C be surjective....