8. If we remove an edge e from a graph G and G- e is still connected, then show that e lies along some cycle of G.

Determine whether the following graph is strongly connected

7. a) How many different paths of length 2 are there in the undirected graph G in Fig. 11.43? b) Let G = (V, E) be a loop-free undirected graph, where V= {U, v2. ..., v) and deg(v,) = d,, for all I sisn. How many different paths of length 2 are there in G? Figure 11.43

(iii) If f(x + p) = f(x) for all r in D, where p is a positive constant, then f is called a periodic function and the smallest such number p is called the period. For instance, y = sin x has period 2m and y = tan x has period . If we know what the graph looks like in an interval of length p, then we can use translation to sketch the entire graph (see Figure 4). period p FIGURE 4 Periodic function: а-р a+p a+ 2p a translational symmetry D. Asymptotes (i) Horizontal Asymptotes. Recall from Section 2.6 that if either lim,-»» f(x) = L or lim,- f(x) = L, then the line y = L is a horizontal asymptote of the curve y =f(x). If it turns out that lim,»z f(x) = ∞ (or -x), then we do not have an asymptote to the right, but this fact is still useful information for sketching the curve. (ii) Vertical Asymptotes. Recall from Section 2.2 that the asymptote if at least one of the following statements is true: Guidelines for Sketching a Curve The following checklist is intended as a guide to sketching a…