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Note that by definition an edge is a set with exactly two elements. This prevents the possibility of a loop being an edge because a loop involves only one vertex. So the definition of simple graphs excludes the possibility of loops. Also note that since E is a set, an edge cannot be listed more than once. (Sets do not allow repeated members.) So the definition of simple graphs excludes the possibility of multiple edges. In general, graphs may include loops and multiple edges; simple graphs do not. GRAPH TERMINOLOGY If G = (V, E) is a graph and G = (V , E ) where V V and E of G. If V = V, then G is called a spanning subgraph of G. Every graph is a spanning subgraph of itself. E, then G is called a subgraph

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14.25 14.26

where T&) is a kth-order Chebyshev polynomial [see Eq. (9.9)]. Therefore, Eq. (9.4) may be written as

Thus, A(ejw)is an Lth-order polynomial in coso. With Ad(ei") a desired amplitude, and W(eiw)a positive weighting function, let

The graph G1 = (V1, E1) in Figure 15.2 with vertex set V1 = {a, b, d} and edge set E1 = {ad, bd} is a nonspanning subgraph of the graph in Example 15.1. This subgraph has size 3. The graph G2 = (V2, E2) in Figure 15.2 with vertex set V2 = {a, b, c, d} and edge set E2 = {ab, ac, cd} is a spanning subgraph of the graph in Example 15.1. This subgraph has size 4.

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partition() method: private static int partition(int[] a, int p, int q) { int m = (p + q)/2; m = indexOfMedian(a, p, m, q-1); swap(a, p, m); // The rest is the same as lines 20-32 in Example 14.9 on page 264 }

be a weighted approximation error. The equiripple filter design problem thus involves finding the coefficients a ( k ) that minimize the maximum absolute value of E(ejU)over a set of frequencies, F,

For example, to design a low-pass filter, the set F will be the frequencies in the passband. [O, w,,], and the n], illustrated in Fig. 9-4. The transition band, (w,, o,), is a don't care region, and it is not as stopband,

The degree (or valence) of a vertex is the number of edges that are incident upon it. For example, in the graph in Figure 15.1 on page 285, vertex a has degree 3 and vertex b has degree 2. But in the subgraph G1 in Figure 15.2, vertices a and b both have degree 1. An isolated point is a vertex of degree 0.

This requires a method for locating the index of three array elements:

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Don't Care Fig. 9-4. The set R in the equiripple filter design problem, consisting of the passband [0, w,] and the stopband [op. , ] . The transition band ( o , , w,) is a don't care region. o

Theorem 15.1 The sum of the degrees of the vertices of a graph with m edges is 2m. Each edge contributes 1 to the degree of each of the two vertices that determine it. So the total contribution if m edges is 2m. A complete graph is a simple graph in which every pair of vertices is connected by an edge. For a given number n of vertices, there is only one complete graph of that size, so we refer to the complete graph on a given vertex set. EXAMPLE 15.3 The Complete Graph on a Set of Four Vertices

considered in the minimization of the weighted error. The solution to this optimization problem is given in the alternation theorem. which is as follows:

private static int indexOfMedian(int[] a, int i, int j, int k) { // Returns the index of the median of {a[i], a[j], a[k]} if (a[i] <= a[j] && a[j] <= a[k]) return j; if (a[i] <= a[k] && a[k] <= a[j]) return k; if (a[j] <= a[i] && a[i] <= a[k]) return i; if (a[j] <= a[k] && a[k] <= a[i]) return k; if (a[k] <= a[i] && a[i] <= a[j]) return i; return j; }

to be the unique function that minimizes the maximum value of the weighted error I E(eJW)I over the set 3is that the E(eJW) have at least L 2 alternations. That is to say, there must be at least L 2 extremalfiequencies.

Figure 15.3 shows the complete graph on the set V = {a, b, c, d}. Its edge set E is E = {ab, ac, ad, bc, bd, cd}. Note that the graphs in the previous examples are subgraphs of this one.

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