 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
ISBN: 9780495391326
Solutions for Chapter 10.7: Spanning Trees and Shortest Paths
Get Full SolutionsThis textbook survival guide was created for the textbook: Discrete Mathematics with Applications , edition: 4th. Since 31 problems in chapter 10.7: Spanning Trees and Shortest Paths have been answered, more than 24596 students have viewed full stepbystep solutions from this chapter. Discrete Mathematics with Applications was written by Sieva Kozinsky and is associated to the ISBN: 9780495391326. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 10.7: Spanning Trees and Shortest Paths includes 31 full stepbystep solutions.

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

Cofactor Cij.
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.

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

Inverse matrix AI.
Square matrix with AI A = I and AAl = 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 B1 AI and (AI)T. Cofactor formula (Al)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 = Ql. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

Orthogonal subspaces.
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. AI 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.

Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t  k).