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ISOMORPHIC GRAPHS An isomorphism between two graphs G = (V, E) and G = (V , E ) is a function f that assigns to each vertex x in V some vertex y = f(x) in V so that the following three conditions are satisfied: 1. f is one-to-one: Each x in V gets assigned a different y = f(x) in V 2. f is onto: Every y in V gets assigned to some x in V. 3. f preserves adjacency: If {x 1, x 2} is an edge in E, then {f(x 1), f(x 2)} is an edge in E . Two graphs are said to be isomorphic if there is an isomorphism from one to the other. The word isomorphic means same form. When applied to graphs, it means that they have the same topological structure. Graphically, two graphs are isomorphic if one can be twisted around to the same shape as the other without breaking any of the edge connections. EXAMPLE 15.8 Isomorphic Graphs

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.

The parameter N is the order of the filter (number of poles in the system function), and Q,. is the 3-dB cutoff frequency. The magnitude of the frequency response may also be written as

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The two graphs in Figure 15.7 on page 289 are isomorphic. The isomorphism is indicated by the corresponding vertex labels. It can be verified that if vertex x 1 is adjacent to vertex x 2 in one graph, then the corresponding vertices are adjacent in the other graph. For example, vertex a is adjacent to vertices b, d, e, and f (but not c, g, or h) in both graphs.

where The frequency response of the Butterworth filter decreases monotonically with increasing 0, and as the filter order increases, the transition band becomes narrower. These properties are illustrated in Fig. 9-6, which shows IH,(jQ)l for Butterworth filters of orders N = 2 , 4 , 8 , and 12. Because

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.

Fig. 9-6. The magnitude of the frequency response for Butterworth filters of orders N = 2.4, 8.

To prove that two graphs are isomorphic (by definition), it is necessary to find an isomorphism between them. This is equivalent to labeling both graphs with the same set of labels so that adjacency applies equally to both labelings. Finding such an isomorphism by chance is unlikely

Therefore, the poles of G,(s) are located at 2N equally spaced points around a circle of radius Q,.,

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

= ( - I ) ~ / ' ~ ( ~ R , Q, expIj =)

CHAP. 15]

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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