 Chapter 23.23.1: Sketch the graph consisting of the vertices and edges of a square. ...
 Chapter 23.23.2: Find a shortest path P: s > t and its length by Moore's BFS algor...
 Chapter 23.23.3: The net of roads in Fig. 487 connecting four villages is to be redu...
 Chapter 23.23.4: Find a shortest spanning tree by Kruskal' s algorithm.
 Chapter 23.23.5: Find a sh0l1est spanning tree by Prim's algorithm. Sketch it. For t...
 Chapter 23.23.6: Find flow augmenting paths:
 Chapter 23.23.7: Do the computations indicated near the end of Example 1 in detail.
 Chapter 23.23.8: Are the following graphs bipartite? If you answer is yes, find S an...
 Chapter 23.1: What is a graph? A digraph? A tree? A cycle? A path?
 Chapter 23.23.1: Worker WI can do jobs 11 and 13 , worker W2 job 14 , worker W3 jobs...
 Chapter 23.23.2: Find a shortest path P: s > t and its length by Moore's BFS algor...
 Chapter 23.23.3: Find shortest paths for the following graphs.
 Chapter 23.23.4: Find a shortest spanning tree by Kruskal' s algorithm.
 Chapter 23.23.5: Find a sh0l1est spanning tree by Prim's algorithm. Sketch it. For t...
 Chapter 23.23.6: Find flow augmenting paths:
 Chapter 23.23.7: Solve Example 1 by FordFulkerson with initial now O. Is it more wo...
 Chapter 23.23.8: Are the following graphs bipartite? If you answer is yes, find S an...
 Chapter 23.2: State from memory how you can handle graphs and digraphs on computers.
 Chapter 23.23.1: Explain how the following may be regarded as graphs or digraphs: fl...
 Chapter 23.23.2: Find a shortest path P: s > t and its length by Moore's BFS algor...
 Chapter 23.23.3: Find shortest paths for the following graphs.
 Chapter 23.23.4: Find a shortest spanning tree by Kruskal' s algorithm.
 Chapter 23.23.5: Find a sh0l1est spanning tree by Prim's algorithm. Sketch it. For t...
 Chapter 23.23.6: Find flow augmenting paths:
 Chapter 23.23.7: Which are the "bottleneck" edges by which the flow in Example 1 is ...
 Chapter 23.23.8: Are the following graphs bipartite? If you answer is yes, find S an...
 Chapter 23.3: Describe situations and problems that can be modeled using graphs o...
 Chapter 23.23.1: How would you represent a net of oneway and twoway streets by a d...
 Chapter 23.23.2: Find a shortest path P: s > t and its length by Moore's BFS algor...
 Chapter 23.23.3: Find shortest paths for the following graphs.
 Chapter 23.23.4: Find a shortest spanning tree by Kruskal' s algorithm.
 Chapter 23.23.5: Find a sh0l1est spanning tree by Prim's algorithm. Sketch it.
 Chapter 23.23.6: Find flow augmenting paths:
 Chapter 23.23.7: Find the maximum How by FordFulkerson:
 Chapter 23.23.8: Are the following graphs bipartite? If you answer is yes, find S an...
 Chapter 23.4: What is a shortest path problem? Give applications.
 Chapter 23.23.1: Give further examples of situations that could be represented by a ...
 Chapter 23.23.2: Find a shortest path P: s > t and its length by Moore's BFS algor...
 Chapter 23.23.3: Find shortest paths for the following graphs.
 Chapter 23.23.4: Find a shortest spanning tree by Kruskal' s algorithm.
 Chapter 23.23.5: Find a sh0l1est spanning tree by Prim's algorithm. Sketch it.
 Chapter 23.23.6: Find the maximum flow by inspection: In Prob. 1.
 Chapter 23.23.7: Find the maximum How by FordFulkerson:
 Chapter 23.23.8: Are the following graphs bipartite? If you answer is yes, find S an...
 Chapter 23.5: What is BFS? DFS? In what connection did these concepts occur?
 Chapter 23.23.1: Find the adjacency matrix of the graph in Fig. 476.
 Chapter 23.23.2: Find a shortest path P: s > t and its length by Moore's BFS algor...
 Chapter 23.23.3: Find shortest paths for the following graphs.
 Chapter 23.23.4: Find a shortest spanning tree by Kruskal' s algorithm.
 Chapter 23.23.5: Find a sh0l1est spanning tree by Prim's algorithm. Sketch it.
 Chapter 23.23.6: Find the maximum flow by inspection:In Prob. 2.
 Chapter 23.23.7: Find the maximum How by FordFulkerson:
 Chapter 23.23.8: Are the following graphs bipartite? If you answer is yes, find S an...
 Chapter 23.6: Give some applications in which spanning trees playa role.
 Chapter 23.23.1: When will the adjacency matrix of a graph be symmetric? Of a digraph?
 Chapter 23.23.2: (Nonuniqueness) A shonest path s > t for given sand t need not be...
 Chapter 23.23.3: Find shortest paths for the following graphs.
 Chapter 23.23.4: CAS PROBLEM. Kruskal's Algorithm. Write a corresponding program. (S...
 Chapter 23.23.5: Find a sh0l1est spanning tree by Prim's algorithm. Sketch it.
 Chapter 23.23.6: Find the maximum flow by inspection:In Prob. 3.
 Chapter 23.23.7: Find the maximum How by FordFulkerson:
 Chapter 23.23.8: Can you obtain the answer to Prob. 3 from that to Prob. I?
 Chapter 23.7: What are bipartite graphs? What applications motivate this concept?
 Chapter 23.23.1: Find the adjacency matrix of the graph or digraph.
 Chapter 23.23.2: (Maximum length) If P is a shortest path between any two vertices i...
 Chapter 23.23.3: Show that in Dijkstra's algorithm, for L" there is a path P: I ~ k ...
 Chapter 23.23.4: Design an algorithm for obtaining longest spanning trees.
 Chapter 23.23.5: (Complexity) Show that Prim's algorithm has complexity 0(n2).
 Chapter 23.23.6: Find the maximum flow by inspection:In Prob. 4.
 Chapter 23.23.7: What is the (simple) reason that Kirchhoffs law is preserved in aug...
 Chapter 23.23.8: Find an augmenting path:
 Chapter 23.8: What is the traveling salesman problem?
 Chapter 23.23.1: Find the adjacency matrix of the graph or digraph.
 Chapter 23.23.2: (Moore's algorithm) Show that if a vertex v has label .A(v) = k, th...
 Chapter 23.23.3: Show that in Dijkstra's algorithm. at each instant the demand on st...
 Chapter 23.23.4: Apply the algorithm in Prob. 8 to the graph in Example I. Compare w...
 Chapter 23.23.5: How does Prim's algorithm prevent the formation of cycles as one gr...
 Chapter 23.23.6: In Fig. 495 find T and cap (5. T) if 5 equals[1,2.31
 Chapter 23.23.7: How does FordFulkerson prevent the fOimation of cycles?
 Chapter 23.23.8: Find an augmenting path:
 Chapter 23.9: What is a network? What optimization problems are connected with it?
 Chapter 23.23.1: Find the adjacency matrix of the graph or digraph.
 Chapter 23.23.2: Call the length of a shortest path s ~ v the distance of v from s. ...
 Chapter 23.23.3: CAS PROBLEM. Dijkstra's Algorithm. Write a program and apply it to ...
 Chapter 23.23.4: To get a minimum spanning tree, instead of adding shortest edges, o...
 Chapter 23.23.5: For a complete graph (or one that is almost complete), if our dara ...
 Chapter 23.23.6: In Fig. 495 find T and cap (5. T) if 5 equals[I. 2.4.51
 Chapter 23.23.7: How can you see that FordFulkerson follows a BFS technique?
 Chapter 23.23.8: Find an augmenting path:
 Chapter 23.10: Can a forward edge in one path be a backward edge in another path? ...
 Chapter 23.23.1: Find the adjacency matrix of the graph or digraph.
 Chapter 23.23.2: (Hamiltonian cycle) Find and sketch a Hamiltonian cycle in the grap...
 Chapter 23.23.4: Apply the method suggested in Prob. IO to the graph in Example 1. D...
 Chapter 23.23.5: In what case will Prim's algorithm give S = E as the final result?
 Chapter 23.23.6: In Fig. 495 find T and cap (5. T) if 5 equals [1, 3, 51
 Chapter 23.23.7: Are the consecutive How augmenting paths produced by FordFulkerson...
 Chapter 23.23.8: Augmenting the given matching, find a maximum cardinality matching:...
 Chapter 23.11: There is a famous theorem on cut sets. Can you remember and explain...
 Chapter 23.23.1: Find the adjacency matrix of the graph or digraph.
 Chapter 23.23.2: Find and sketch a Hamiltonian cycle in the graph of a dodecahedron....
 Chapter 23.23.4: Find a shortest spanning tree in the complete graph of all possible...
 Chapter 23.23.5: TEAM PROJECT. Center of a Graph and Related Concepts. (a) Distance,...
 Chapter 23.23.6: Find a minimum cut set in Fig. 495 and verify that its capacity equ...
 Chapter 23.23.7: (Integer flow theorem) Prove that if the capacities in a network G ...
 Chapter 23.23.8: Augmenting the given matching, find a maximum cardinality matching:...
 Chapter 23.12: Find the adjacency matrix of:
 Chapter 23.23.1: Find the adjacency matrix of the graph or digraph.
 Chapter 23.23.2: Find and sketch a Hamiltonian cycle In Fig. 479. Sec. 23.1.
 Chapter 23.23.4: (Forest) A (not necessarily connected) graph without cycles is call...
 Chapter 23.23.5: What would the result be if you applied Prim's algorithm to a graph...
 Chapter 23.23.6: Find examples of flow augmenting paths and the maximum flow in the ...
 Chapter 23.23.7: CAS PROBLEM. FordFulkerson. Write a program and apply it to Probs....
 Chapter 23.23.8: Augmenting the given matching, find a maximum cardinality matching:...
 Chapter 23.13: Find the adjacency matrix of:
 Chapter 23.23.1: Sketch the graph whose adjacency matrix is:
 Chapter 23.23.2: (Euler graph) An EllIeI' graph G is a graph that has a clo:.ed Eul...
 Chapter 23.23.4: (Uniqueness) The path connecting any two vertice~ 1I and u in a tre...
 Chapter 23.23.5: CAS PROBLEM. Prim's Algorithm. Write a program and apply it to Prob...
 Chapter 23.23.6: In Fig. 498 find T and cap (5. T) if 5 equals
 Chapter 23.23.7: If the FordFulkerson algorithm stops without reaching t. sho~ that...
 Chapter 23.23.8: (Scheduling and matching) Three teachers Xl, X2' '3 teach four cla...
 Chapter 23.14: Find the adjacency matrix of:
 Chapter 23.23.1: Sketch the graph whose adjacency matrix is:
 Chapter 23.23.2: Is the graph in Fig. 483 an Euler graph? (Give a reason.)
 Chapter 23.23.4: If in a graph any two vertices are connected by a unique path, the ...
 Chapter 23.23.6: In Fig. 498 find T and cap (5. T) if 5 equals
 Chapter 23.23.7: (Several sources and sinks) If a network has several sources Sl' .....
 Chapter 23.23.8: (Vertex coloring and exam scheduling) What is the smallest number o...
 Chapter 23.15: Find the adjacency matrix of:
 Chapter 23.23.1: Sketch the graph whose adjacency matrix is:
 Chapter 23.23.2: Find 4 different closed Euler trails in Fig. 484.
 Chapter 23.23.4: If a graph has no cycles, it must have at least 2 vertices of degre...
 Chapter 23.23.6: In Fig. 498 find T and cap (5. T) if 5 equals
 Chapter 23.23.7: Find the maximum flow in the network in Fig. 50 I with two sources ...
 Chapter 23.23.8: How many colors do you need in vertex coloring the graph in Prob. 5?
 Chapter 23.16: Find the adjacency matrix of:
 Chapter 23.23.1: The matrix in Prob. 14.
 Chapter 23.23.2: The postman problem is the problem of finding a closed walk W: s ~ ...
 Chapter 23.23.4: A tree with exactly two vertices of degree 1 must be a path.
 Chapter 23.23.6: In Fig. 498 find a minimum cut set and its capacity.
 Chapter 23.23.7: Find a minimum cut set in Fig. 499 and its capacity.
 Chapter 23.23.8: Show that all trees can be vertex colored with two colors.
 Chapter 23.17: Find the adjacency matrix of:
 Chapter 23.23.1: The matrix in Prob. 16.
 Chapter 23.23.2: Show that the length of a shortest postman trail is the same for ev...
 Chapter 23.23.4: A tree with 11 vertices has 11  I edges. (Proof by induction.)
 Chapter 23.23.6: Why are backward edge~ not considered III the definition of the cap...
 Chapter 23.23.7: Show that in a network G with all Cij = I, the maximum flow equals ...
 Chapter 23.23.8: (Harbor management) How many piers does a harbor master need for ac...
 Chapter 23.18:
 Chapter 23.23.1: (Complete graph) Show that a graph G with 11 vertices can have at m...
 Chapter 23.23.2: (Order) Show that 0(1113 ) + 0(1113 ) = 0(1113 ) and kO(111P) = O(m...
 Chapter 23.23.4: If two vertices in a tree are joined by a new edge. a cycle is formed.
 Chapter 23.23.6: In which case can an edge U, j) be used as a forward as well as a b...
 Chapter 23.23.7: In Prob. 17, the cut set contains precisely all forward edges used ...
 Chapter 23.23.8: What would be the answer to Prob. 18 if only the five 987 ship~ 51>...
 Chapter 23.19:
 Chapter 23.23.1: In what case are all the offdiagonal entries of the adjacency matr...
 Chapter 23.23.2: Show that ~ = 0(111), O.02em + 100m2 = O(em ).
 Chapter 23.23.4: A graph with 11 vertices is a tree if and only if it has 11  1 edg...
 Chapter 23.23.6: (Incremental network) Sketch the network in Fig. 498, and on each e...
 Chapter 23.23.7: Show that in a network G with capacities all equal to I, the capaci...
 Chapter 23.23.8: (Complete bipartite graphs) A bipartite graph G = (5, T: E) is call...
 Chapter 23.20:
 Chapter 23.23.1: The graph in Prob. 9.
 Chapter 23.23.2: If we switch from one computer to another that is 100 times as fast...
 Chapter 23.23.8: (Planar graph) A planar graph is a graph that can be drawn on a she...
 Chapter 23.21: Make a vertex incidence list of the digraph in Prob. 13.
 Chapter 23.23.1: The graph in Prob. 8.
 Chapter 23.23.2: CAS PROBLEM. Moore's Algorithm. Write a computer program for the al...
 Chapter 23.23.8: (Bipartite graph K3,3 not planar) Three factories 1, 2, 3 are each ...
 Chapter 23.22: Make a vertex incidence list of the digraph in Prob. 14.
 Chapter 23.23.1: Find the incidence matrix of: The digraph in Prob. II.
 Chapter 23.23.8: (Four (vertex) color theorem) The famous Jourcolor theorem states...
 Chapter 23.23: Find a shortest path and its length by Moore's BFS algorithm. assum...
 Chapter 23.23.1: Find the incidence matrix of: The digraph in Prob. 13
 Chapter 23.23.8: (Edge coloring) The edge chromatic number xeCG) of a graph G is the...
 Chapter 23.24: Find a shortest path and its length by Moore's BFS algorithm. assum...
 Chapter 23.23.1: Find the incidence matrix of: Make a vertex incidence list ofthe di...
 Chapter 23.23.8: Vizing's theorem states that for any graph G (without multiple edge...
 Chapter 23.25: Find a shortest path and its length by Moore's BFS algorithm. assum...
 Chapter 23.26: Find a shortest path and its length by Moore's BFS algorithm. assum...
 Chapter 23.27: Find a shortest path and its length by Moore's BFS algorithm. assum...
 Chapter 23.28: Find a shortest path and its length by Moore's BFS algorithm. assum...
 Chapter 23.29: (Shortest spanning tree) Find a shortest spanning tree for the grap...
 Chapter 23.30: Find a shortest ~panning tree in Prob. 27.
 Chapter 23.31: Cayley's theorem states that the number of spanning trees in a comp...
 Chapter 23.32: Show that 0(1Il3) + 0(1112) = 0(11/3).
 Chapter 23.33: MAXIMUM FLOW. Find the maximum flow. where the given numbers are ca...
 Chapter 23.34: MAXIMUM FLOW. Find the maximum flow. where the given numbers are ca...
 Chapter 23.35: Company A has offices in Chicago. Los Angeles. and New York. Compan...
 Chapter 23.36: (Maximum cal'dinality matching). Augmenting the given matching. fin...
Solutions for Chapter Chapter 23: Advanced Engineering Mathematics 9th Edition
Full solutions for Advanced Engineering Mathematics  9th Edition
ISBN: 9780471488859
Solutions for Chapter Chapter 23
Get Full SolutionsSince 192 problems in chapter Chapter 23 have been answered, more than 49908 students have viewed full stepbystep solutions from this chapter. Chapter Chapter 23 includes 192 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Advanced Engineering Mathematics was written by and is associated to the ISBN: 9780471488859. This textbook survival guide was created for the textbook: Advanced Engineering Mathematics, edition: 9.

Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

Column space C (A) =
space of all combinations of the columns of A.

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

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.

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

Kirchhoff's Laws.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

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).

Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A  AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

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.

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·

Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn 1 with P(Xi) = bi. Vij = (Xi)jI and det V = product of (Xk  Xi) for k > i.

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