 13.3.1: Use Kruskals algorithm to find a minimum spanning tree in the conne...
 13.3.2: Use Kruskals algorithm to find a minimum spanning tree in the conne...
 13.3.3: What does Kruskals algorithm produce when no two weights in a conne...
 13.3.4: What does Kruskals algorithm produce if all weights in a connected ...
 13.3.5: Use Prims algorithm to find a minimum spanning tree in the connecte...
 13.3.6: Use Prims algorithm to find a minimum spanning tree in the connecte...
 13.3.7: Let G be a connected weighted graph. If every spanning tree of G is...
 13.3.8: Let G be a connected weighted graph and let w1 < w2 < < wk be the k...
 13.3.9: Let G be a connected weighted graph. Describe, with justification, ...
 13.3.10: The weights of the edges of a complete graph of order 5 are 1, 2, 2...
 13.3.11: For each of the weighted graphs G1 and G2 in Figure 13.34, determin...
 13.3.12: Let G is a connected weighted graph of order n 4 all of whose edges...
 13.3.13: Let G be a connected weighted graph and T a minimum spanning tree o...
 13.3.14: Show, for each integer k 2, that there exists a connected weighted ...
Solutions for Chapter 13.3: The Minimum Spanning Tree Problem
Full solutions for Discrete Mathematics  1st Edition
ISBN: 9781577667308
Solutions for Chapter 13.3: The Minimum Spanning Tree Problem
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Since 14 problems in chapter 13.3: The Minimum Spanning Tree Problem have been answered, more than 13617 students have viewed full stepbystep solutions from this chapter. Discrete Mathematics was written by and is associated to the ISBN: 9781577667308. This textbook survival guide was created for the textbook: Discrete Mathematics, edition: 1. Chapter 13.3: The Minimum Spanning Tree Problem includes 14 full stepbystep solutions.

Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

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.

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

Fundamental Theorem.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n  r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

lAII = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n  1, volume of box = I det( A) I.

Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

Outer product uv T
= column times row = rank one matrix.

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

Solvable system Ax = b.
The right side b is in the column space of A.

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.