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Why is it that the natural definition of a simple graph prohibits loops while the natural definition of a digraph allows them True or false: a. If a graph has n vertices and n(n 1)/2 edges, then it must be a complete graph. b. The length of a path must be less than the size of the graph. c. The length of a cycle must equal the number of distinct vertices it has. d. If the incidence matrix for a graph has n rows and n(n 1)/2 columns, then the graph must be a complete graph. e. In an incidence matrix for a digraph, the sum of the entries in each row equals the indegree for that vertex. f. The sum of all the entries in an incidence matrix for a graph is 2|E|. g. The sum of all the entries in an incidence matrix for a digraph is always 0. A graph (V, E) is called dense if |E| = (|V|2), and it is called sparse if |E| = O(|V|). a. Which of the three representations (adjacency matrix, incidence matrix, or adjacency list) would be best for a dense graph b. Which representation would be best for a sparse graph Why is it that, in the incidence matrix of a simple graph, there are always exactly two 1s in each column

as a short circuit. Consequently, v can never reach the negative values, and the current i can increase negatively without limit. The corresponding i-v plot is sketched in Fig. 2-39(c).

15.1 Find each of the following properties for the graph shown in Figure 15.32: a. Its size n b. Its vertex set V c. Its edge set E d. The degree d(x) of each vertex x e. A path of length 3 f. A path of length 5 g. A cycle of length 4 h. A spanning tree i. Its adjacency matrix j. Its incidence matrix k. Its adjacency list Find each of the following properties for the digraph shown in Figure 15.33: a. Its size n b. Its vertex set V c. Its edge set E d. The indegree id(x) of each vertex x e. The outdegree od(x) of each vertex x f. A path of length 3 g. A path of length 5 h. A cycle of length 4

At the beginning of Second, Second must execute a P() operation (also called test,.

In the small-signal circuit of Fig. 2-40, the capacitor models the diode di usion capacitance, so that C Cd 0:02 F, and vth is known to be of frequency ! 107 rad/s. Also, rd 2:5 and ZTh RTh 10 . Find the phase angle (a) between id and vd and (b) between vd and vth .

i. Its adjacency matrix j. Its incidence matrix k. Its adjacency list 15.3 15.4 15.5 Draw the complete graph on n vertices for n = 2, 3, 4, 5, and 6. Determine whether the graph G1 in Figure 15.34 is either eulerian or hamiltonian. Determine whether the graph G2 in Figure 15.34 is either eulerian or hamiltonian.

Fig. 2-40 (a) The di usion capacitance produces a reactance 1 1 5 !Cd 107 0:02 10 6 2:5 5j 908 Zd rd k jxd 2:236j 26:578 2 j1 2:5 j5 xd

Figure 15.35 shows 12 subgraphs of the graph in Figure 15.6 on page 288. Determine whether each of these is connected, acyclic, and/or spanning.

Thus, id leads vd by a phase angle of 26.578. (b) Let Zeq be the impedance looking to the right from vth ; then Zeq ZTh Zd 10 2 j1 12 j1 12:04j 4:768 Hence, vth leads vd by an angle of 26.578 4:768 21:818.

wait, acquire, etc) on the semaphore The P() operation will block Second When First completes, the last thing First must do is execute a V() operation (also called increment, release, signal, etc) on the semaphore The V() operation will release the waiting process Second, thus guaranteeing that First will complete before Second 610 Describe how transaction logging might apply to a file system, and give an example of a file system that employs transaction logging Microsoft s NTFS is one file system that uses transaction logging to protect the file system from corruption due to system failure When a change must be made to the file system, such as adding a new file, the file system first writes the changes to a write-ahead log When the system has recorded all the changes in the write-ahead log, the system marks the transaction as committed.

Using ideal diodes, resistors, and batteries, synthesize a function-generator circuit that will yield the i-v characteristic of Fig. 2-41(a).

CHAP. 15]