- 22.214.171.124.132: Are the following graphs bipartite? If you answer is yes, find S an...
- 126.96.36.199.133: Are the following graphs bipartite? If you answer is yes, find S an...
- 188.8.131.52.134: Are the following graphs bipartite? If you answer is yes, find S an...
- 184.108.40.206.135: Are the following graphs bipartite? If you answer is yes, find S an...
- 220.127.116.11.136: Are the following graphs bipartite? If you answer is yes, find S an...
- 18.104.22.168.137: Are the following graphs bipartite? If you answer is yes, find S an...
- 22.214.171.124.138: Can you obtain the answer to Prob. 3 from that to Prob. I?
- 126.96.36.199.139: Find an augmenting path:
- 188.8.131.52.140: Find an augmenting path:
- 184.108.40.206.141: Find an augmenting path:
- 220.127.116.11.142: Augmenting the given matching, find a maximum cardinality matching:...
- 18.104.22.168.143: Augmenting the given matching, find a maximum cardinality matching:...
- 22.214.171.124.144: Augmenting the given matching, find a maximum cardinality matching:...
- 126.96.36.199.145: (Scheduling and matching) Three teachers Xl, X2' -'3 teach four cla...
- 188.8.131.52.146: (Vertex coloring and exam scheduling) What is the smallest number o...
- 184.108.40.206.147: How many colors do you need in vertex coloring the graph in Prob. 5?
- 220.127.116.11.148: Show that all trees can be vertex colored with two colors.
- 18.104.22.168.149: (Harbor management) How many piers does a harbor master need for ac...
- 22.214.171.124.150: What would be the answer to Prob. 18 if only the five 987 ship~ 51>...
- 126.96.36.199.151: (Complete bipartite graphs) A bipartite graph G = (5, T: E) is call...
- 188.8.131.52.152: (Planar graph) A planar graph is a graph that can be drawn on a she...
- 184.108.40.206.153: (Bipartite graph K3,3 not planar) Three factories 1, 2, 3 are each ...
- 220.127.116.11.154: (Four- (vertex) color theorem) The famous Jour-color theorem states...
- 18.104.22.168.155: (Edge coloring) The edge chromatic number xeCG) of a graph G is the...
- 22.214.171.124.156: Vizing's theorem states that for any graph G (without multiple edge...
Solutions for Chapter 23.8: Bipartite Graphs. Assignment Problem~
Full solutions for Advanced Engineering Mathematics | 9th Edition
Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or - sign.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x - x) (x - x) T is positive (semi)definite; :E is diagonal if the Xi are independent.
Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].
0,1,1,2,3,5, ... satisfy Fn = Fn-l + Fn- 2 = (A7 -A~)I()q -A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].
Gram-Schmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.
A sequence of steps intended to approach the desired solution.
Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , Aj-Ib. Numerical methods approximate A -I b by x j with residual b - Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.
Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b - Ax is orthogonal to all columns of A.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).
Nullspace N (A)
= All solutions to Ax = O. Dimension n - r = (# columns) - rank.
Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Q-l. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.
Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.
Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b - Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) -1 AT.
Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.
R = [~ CS ] rotates the plane by () and R- 1 = RT rotates back by -(). Eigenvalues are eiO and e-iO , eigenvectors are (1, ±i). c, s = cos (), sin ().
Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.
Special solutions to As = O.
One free variable is Si = 1, other free variables = o.
Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).
Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.
Tridiagonal matrix T: tij = 0 if Ii - j I > 1.
T- 1 has rank 1 above and below diagonal.