- 10.7.1: Find all possible spanning trees for each of the graphs in 1 and 2.
- 10.7.2: Find all possible spanning trees for each of the graphs in 1 and 2.
- 10.7.3: Find a spanning tree for each of the graphs in 3 and 4.
- 10.7.4: Find a spanning tree for each of the graphs in 3 and 4.
- 10.7.5: Use Kruskals algorithm to find a minimum spanning tree for each of ...
- 10.7.6: Use Kruskals algorithm to find a minimum spanning tree for each of ...
- 10.7.7: Use Prims algorithm starting with vertex a or v0 to find a minimum ...
- 10.7.8: Use Prims algorithm starting with vertex a or v0 to find a minimum ...
- 10.7.9: For each of the graphs in 9 and 10, find all minimum spanning trees...
- 10.7.10: For each of the graphs in 9 and 10, find all minimum spanning trees...
- 10.7.11: A pipeline is to be built that will link six cities. The cost (in h...
- 10.7.12: Use Dijkstras algorithm for the airline route system of Figure 10.7...
- 10.7.13: Use Dijkstras algorithm to find the shortest path from a to z for e...
- 10.7.14: Use Dijkstras algorithm to find the shortest path from a to z for e...
- 10.7.15: Use Dijkstras algorithm to find the shortest path from a to z for e...
- 10.7.16: Use Dijkstras algorithm to find the shortest path from a to z for e...
- 10.7.17: The graph of exercise 9 with a = a and z = f
- 10.7.18: The graph of exercise 10 with a = u and z = w
- 10.7.19: Prove part (2) of Proposition 10.7.1: Any two spanning trees for a ...
- 10.7.20: Given any two distinct vertices of a tree, there exists a unique pa...
- 10.7.21: a. Suppose T1 and T2 are two different spanning trees for a graph G...
- 10.7.22: Prove that an edge e is contained in every spanning tree for a conn...
- 10.7.23: Consider the spanning trees T1 and T2 in the proof of Theorem 10.7....
- 10.7.24: Suppose that T is a minimum spanning tree for a connected, weighted...
- 10.7.25: Prove that if G is a connected, weighted graph and e is an edge of ...
- 10.7.26: If G is a connected, weighted graph and no two edges of G have the ...
- 10.7.27: Prove that if G is a connected, weighted graph and e is an edge of ...
- 10.7.28: Suppose a disconnected graph is input to Kruskals algorithm. What w...
- 10.7.29: Suppose a disconnected graph is input to Prims algorithm. What will...
- 10.7.30: Prove that if a connected, weighted graph G is input to Algorithm 1...
- 10.7.31: Modify Algorithm 10.7.3 so that the output consists of the sequence...
Solutions for Chapter 10.7: Spanning Trees and Shortest Paths
Full solutions for Discrete Mathematics with Applications | 4th Edition
Upper triangular systems are solved in reverse order Xn to Xl.
peA) = det(A - AI) has peA) = zero matrix.
Remove row i and column j; multiply the determinant by (-I)i + j •
Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra -eij in the i, j entry (i #- j). Then Eij A subtracts eij times row j of A from row i.
Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.
A symmetric matrix with eigenvalues of both signs (+ and - ).
Inverse matrix A-I.
Square matrix with A-I A = I and AA-l = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B-1 A-I and (A-I)T. Cofactor formula (A-l)ij = Cji! detA.
Left inverse A+.
If A has full column rank n, then A+ = (AT A)-I AT has A+ A = In.
Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.
Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .
Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A - AI) if no eigenvalues are repeated; always meA) divides peA).
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.
Every v in V is orthogonal to every w in W.
Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.
Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.
Solvable system Ax = b.
The right side b is in the column space of A.
Symmetric matrix A.
The transpose is AT = A, and aU = a ji. A-I is also symmetric.
Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.
Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.
Stretch and shift the time axis to create Wjk(t) = woo(2j t - k).