 23.8.23.1.132: Are the following graphs bipartite? If you answer is yes, find S an...
 23.8.23.1.133: Are the following graphs bipartite? If you answer is yes, find S an...
 23.8.23.1.134: Are the following graphs bipartite? If you answer is yes, find S an...
 23.8.23.1.135: Are the following graphs bipartite? If you answer is yes, find S an...
 23.8.23.1.136: Are the following graphs bipartite? If you answer is yes, find S an...
 23.8.23.1.137: Are the following graphs bipartite? If you answer is yes, find S an...
 23.8.23.1.138: Can you obtain the answer to Prob. 3 from that to Prob. I?
 23.8.23.1.139: Find an augmenting path:
 23.8.23.1.140: Find an augmenting path:
 23.8.23.1.141: Find an augmenting path:
 23.8.23.1.142: Augmenting the given matching, find a maximum cardinality matching:...
 23.8.23.1.143: Augmenting the given matching, find a maximum cardinality matching:...
 23.8.23.1.144: Augmenting the given matching, find a maximum cardinality matching:...
 23.8.23.1.145: (Scheduling and matching) Three teachers Xl, X2' '3 teach four cla...
 23.8.23.1.146: (Vertex coloring and exam scheduling) What is the smallest number o...
 23.8.23.1.147: How many colors do you need in vertex coloring the graph in Prob. 5?
 23.8.23.1.148: Show that all trees can be vertex colored with two colors.
 23.8.23.1.149: (Harbor management) How many piers does a harbor master need for ac...
 23.8.23.1.150: What would be the answer to Prob. 18 if only the five 987 ship~ 51>...
 23.8.23.1.151: (Complete bipartite graphs) A bipartite graph G = (5, T: E) is call...
 23.8.23.1.152: (Planar graph) A planar graph is a graph that can be drawn on a she...
 23.8.23.1.153: (Bipartite graph K3,3 not planar) Three factories 1, 2, 3 are each ...
 23.8.23.1.154: (Four (vertex) color theorem) The famous Jourcolor theorem states...
 23.8.23.1.155: (Edge coloring) The edge chromatic number xeCG) of a graph G is the...
 23.8.23.1.156: 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
ISBN: 9780471488859
Solutions for Chapter 23.8: Bipartite Graphs. Assignment Problem~
Get Full SolutionsSince 25 problems in chapter 23.8: Bipartite Graphs. Assignment Problem~ have been answered, more than 49005 students have viewed full stepbystep solutions from this chapter. Advanced Engineering Mathematics was written by and is associated to the ISBN: 9780471488859. Chapter 23.8: Bipartite Graphs. Assignment Problem~ includes 25 full stepbystep solutions. This textbook survival guide was created for the textbook: Advanced Engineering Mathematics, edition: 9. This expansive textbook survival guide covers the following chapters and their solutions.

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.

Covariance matrix:E.
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].

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

GramSchmidt 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.

Iterative method.
A sequence of steps intended to approach the desired solution.

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. 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.

Multiplier eij.
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 = Ql. 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.

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , 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.