Question

Let1≤ m ≤ n. Show that

(a). Km ∩ Kn = Km,

(b). Km ∪ Kn = Kn,

(c). Km ⊆ Kn.

Answer #1

11.43. Let 1 ≤ m ≤ n. Show that
(a). Km ∩ Kn =
Km,
(b). Km ∪ Kn =
Kn,
(c). Km ⊆ Kn.

Find the diameters of Kn (Connected graph with n vertices), Km,n
(Bipartite graph with m and n vertices), and Cn (Cycle graph with n
vertices). For each, clearly explain your reasoning.

GRAPH THEORY:
Show that Kn is not magic when n is a multiple of
4.
HINT: First find a formula in terms of n for what the common sum
would be, then substitute 4k fo n and say what goes wrong.

A and B are two m*n matrices. a. Show that B is invertible. b.
Show that Nullsp(A)=Nullsp(BA)

. Let n ∈ N. Prove (by induction) that n =
2knmn for some nonnegative kn ∈ Z
and some odd mn ∈ N. (Again, kn and mn may
depend on n.)

Consider the complete bipartite graph Kn,n with 2n vertices. Let
kn be the number of edges in Kn,n. Draw K1,1, K2,2 and K3,3 and
determine k1, k2, k3. Give a recurrence relation for kn and solve
it using an initial value.

MATH 353
6. Find a necessary and sufficient condition on m and
n for Km,n to be Hamiltonian. Prove your
answer.

The beam is subjected to a uniformly distributed load
w= 12 kN/m. The beam is a two-span continuous beam. The
length of each span is L = 2 m. The beam is made of steel.
Young's modulus (E) of steel is 200 GPa. The value of the moment of
inertia is 21 * 106 mm4. Use the force method
given in Week 6. Determine the reaction at the middle support
B.
Select one:
a. 6.00 kN
b. 18.00 kN
c....

MATH 353
2. (a) For which values of n > 1, if any, is Kn an
Euler graph? Explain.
(b) For which values of n > 1, if any, does Kn
have an Euler path? Explain.

For what values of m, n ∈ Z>0 does the complete bipartite
graph Km,n have a perfect matching? Prove it

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