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vb.net code to generate barcode Figure 15.53 Incidence matrices in Java
Figure 15.53 Incidence matrices UPC  13 Scanner In Java Using Barcode Control SDK for Java Control to generate, create, read, scan barcode image in Java applications. EAN13 Creation In Java Using Barcode encoder for Java Control to generate, create GS1  13 image in Java applications. Figure 15.54 Digraph
EAN13 Supplement 5 Decoder In Java Using Barcode reader for Java Control to read, scan read, scan image in Java applications. Bar Code Creation In Java Using Barcode maker for Java Control to generate, create barcode image in Java applications. CHAP. 15] Bar Code Recognizer In Java Using Barcode decoder for Java Control to read, scan read, scan image in Java applications. Print GTIN  13 In Visual C#.NET Using Barcode encoder for .NET Control to generate, create EAN13 image in Visual Studio .NET applications. GRAPHS
EAN / UCC  13 Creator In .NET Framework Using Barcode creator for ASP.NET Control to generate, create GS1  13 image in ASP.NET applications. GS1  13 Creation In VS .NET Using Barcode maker for .NET framework Control to generate, create GS1  13 image in .NET applications. Figure 15.55 Incidence matrix
EAN 13 Generation In VB.NET Using Barcode generator for .NET framework Control to generate, create UPC  13 image in .NET framework applications. Generating Barcode In Java Using Barcode maker for Java Control to generate, create bar code image in Java applications. Figure 15.56 Digraph
EAN13 Creator In Java Using Barcode encoder for Java Control to generate, create European Article Number 13 image in Java applications. Paint UCC128 In Java Using Barcode generator for Java Control to generate, create USS128 image in Java applications. a. b. c. d.
Print RM4SCC In Java Using Barcode generator for Java Control to generate, create British Royal Mail 4State Customer Code image in Java applications. Make Matrix Barcode In VB.NET Using Barcode creator for .NET Control to generate, create 2D Barcode image in Visual Studio .NET applications. The adjacency matrix is shown in Figure 15.57. The adjacency list is shown in Figure 15.57. The graph is not connected because there is no path from B to A. The graph is not acyclic because it contains the cycle BECDB. Scanning UPCA In .NET Framework Using Barcode recognizer for .NET framework Control to read, scan read, scan image in VS .NET applications. Decoding Barcode In Visual Basic .NET Using Barcode Control SDK for Visual Studio .NET Control to generate, create, read, scan barcode image in VS .NET applications. Figure 15.57 Adjacency matrix and adjacency list
Making UPCA Supplement 5 In Visual Basic .NET Using Barcode generation for .NET Control to generate, create UPC Symbol image in Visual Studio .NET applications. Drawing Bar Code In Visual Basic .NET Using Barcode creation for VS .NET Control to generate, create barcode image in VS .NET applications. a. The adjacency matrix for a wheel graph looks like matrix A shown in Figure 15.58 on page 316. b. The incidence matrix for a wheel graph looks like matrix B shown in Figure 15.58 (for the case n = 4). In general, it will have n 1s followed by n 0s on the first row. Below that will lie the identity matrix (all 1s on the diagonal and 0s elsewhere) followed by the square matrix with 1s on the diagonal and the subdiagonal. Compare this with the recursive solution to Problem 15.9 on page 308. c. The adjacency list for a wheel graph looks like the list shown in Figure 15.58 on page 316. The edge list for the first vertex (the central vertex) has n edge nodes, one for every other vertex. Every other edge list has three edge nodes: one pointing to the central vertex (labeled a in Figure 15.58) and one to each of its neighbors. a. The two graphs are isomorphic. The bijection is defined by the vertex labels shown in Figure 15.59 on page 316. b. An euler cycle for G2 is ABCDEBFCADFEA. c. A hamiltonian cycle for G2 is ABCDFEA. The trace of Dijkstra s algorithm is shown in Figure 15.60 on page 316. The trace of Dijkstra s algorithm is shown in Figure 15.61 on page 316. a. If the depthfirst search is applied to a tree, it does a preorder traversal. b. If the breadthfirst search is applied to a tree, it does a levelorder traversal. Read Code128 In VB.NET Using Barcode reader for VS .NET Control to read, scan read, scan image in .NET applications. Scanning Bar Code In Java Using Barcode decoder for Java Control to read, scan read, scan image in Java applications. 15.15 15.16 15.17 GRAPHS
[CHAP. 15
Figure 15.58 Adjacency matrix and adjacency list
Figure 15.59 Isomorphic graphs
Figure 15.60 Dijkstra s algorithm
Figure 15.61 Dijkstra s algorithm
CHAP. 15] GRAPHS
The seven graphs are labeled in Figure 15.62. Among them: G1 is isomorphic to G2 : The isomorphism is shown by the vertex labels a j. G3 is isomorphic to G4 : The isomorphism is shown by the vertex labels p y. G6 cannot be isomorphic to any of the other graphs because it has 25 edges and all the others have 20. G3 (and thus also G4) cannot be isomorphic to any of the other graphs because it has a pyramid of four adjacent 3cycles (pqr, prs, pst, and ptq) and none of the other graphs (except G6) does. G6 cannot be isomorphic to any of the other graphs because it has a chain of three adjacent 4cycles (ABCD, CEFG, and FHIJ) and none of the other graphs (except G6) does. Similarly, G7 cannot be isomorphic to any of the other graphs because it has a chain of four adjacent 3cycles (PQS, QSR, SRT, and RTU) and none of the other graphs (except G6) does. Figure 15.62 Graph isomorphisms
15.19 15.20 The adjacency matrix and the adjacency list are shown in Figure 15.63 on page 318. a. The breadthfirst search visits ABDECHFIGKLJMONPQ; its spanning tree is shown on the left in Figure 15.64 on page 318. b. The depthfirst search visits ABCFEIHDKLMJGNPQO; its spanning tree is shown on the right in Figure 15.64 on page 318. GRAPHS
[CHAP. 15
Figure 15.63 Adjacency matrix and adjacency list
Figure 15.64 Breadthfirst search and depthfirst search
APPENDIX
Essential Mathematics
This appendix summarizes mathematical topics used in the study of data structures. THE FLOOR AND CEILING FUNCTIONS The floor and ceiling functions return the two nearest integers of a given real number. The floor of x, denoted by x , is the greatest integer that is not greater than x. The ceiling of x, denoted by x , is the smallest integer that is not smaller than x. Here are the main properties of these two functions. (The symbol stands for the set of all integers.) Theorem A.1 Properties of the Floor and Ceiling Functions 1. x = max{m  m x}, and x = min{n  n x}. 2. x x < x + 1, and x 1 < x x . 3. x 1 < x x x < x + 1. 4. If n and n x < n + 1, then n = x . If n and n 1 < x < n, then n = x . 5. If x , then x = x = x . 6. If x , then x < x < x . 7. x x and x = x 8. x + 1 = x and x + 1 x LOGARITHMS The logarithm with base b of a positive number x is the exponent y on b for which b y = x. For example, the logarithm of 1000 base 10 is 3 because 103 = 1000. This is written log10 1000 = 3. The logarithm with base 2 is called the binary logarithm and is written lg x = log 2 x. For example, lg 8 = 3. As a mathematical function, the logarithm is the inverse of the exponential function with the same base: y = log b x by = x For example, 3 = lg 8 because 23 = 8.

