Consider an undirected graph G = (V, E) with an injective cost
function c: E →...
Consider an undirected graph G = (V, E) with an injective cost
function c: E → N. Suppose T is a minimum spanning tree of G for
cost function c. If we replace each edge cost c(e), e ∈ E, with
cost c'(e) = c(e)2 for G, is T still a minimum spanning
tree of G? Briefly justify your answer.
4. Let a = 24, b = 105 and c = 594.
(a) Find the prime...
4. Let a = 24, b = 105 and c = 594.
(a) Find the prime factorization of a, b and c.
(b) Use (a) to calculate d(a), d(b) and d(c), where, for any
integer n, d(n) is the number of positive divisors of n;
(c) Use (a) to calculate σ(a), σ(b) and σ(c), where, for any
integer n, σ(n) is the sum of positive divisors of n;
(d) Give the list of positive divisors of a, b and c.
Let G = (V, E) be an undirected and connected graph with
Laplacian matrix L.
(a)...
Let G = (V, E) be an undirected and connected graph with
Laplacian matrix L.
(a) How are the eigenvalues of L2 related to the
eigenvalues of L?
(b) If instead of running the consensus protocol, x ̇ = −Lx, one
runs the protocol from Homework 1, given by
x ̇ = −L2x, will consensus still be achieved? Justify
your answer.
(c) Assuming both x ̇ = −Lx and x ̇ = −L2x converge,
which protocol converges faster? Justify your...
Let S = {A, B, C, D, E, F, G, H, I, J} be the set...
Let S = {A, B, C, D, E, F, G, H, I, J} be the set consisting of
the following elements:
A = N, B = 2N , C = 2P(N) , D = [0, 1), E = ∅, F = Z × Z, G = {x
∈ N|x 2 + x < 2}, H = { 2 n 3 k |n, k ∈ N}, I = R \ Q, J =
R.
Consider the relation ∼ on S given...
let
A = { a, b, c, d , e, f, g} B = { d,...
let
A = { a, b, c, d , e, f, g} B = { d, e , f , g}
and C ={ a, b, c, d}
find :
(B n C)’
B’
B n C
(B U C) ‘