List of Figures 1.1 The Petersen graph. 3 1.2 Isomorphic graphs. 4 1.3 A graph G with v(G) = a(G) = 3 and X (G) = 4 . 8 1.4 A directed path and cycle. 8 2.1 A sparse and a dense graph. 21 2.2 The graph G and a plot of its degree distribution. 22 2.3 A power law graph with 400 vertices. 23 2.4 The log-log plot of the degree distribution of a power law graph. 24 2.5 A graph G with L(G) = 1.4. 26 2.6 A graph H with c(H) = 0.75. 27 2.7 A bipartite core. 28 3.1 A graph with 25 vertices, where edges were drawn with probability ^. 35 3.2 The 1-e.c. graphs of order 4. 38 3.3 The graph K3DK3. 38 4.1 A graph with 100 vertices and edges drawn with probability ^. 63 4.2 A graph generated by the preferential attachment model with 100 vertices and average degree close to the average degree of the graph in Figure 4.1. 64 vn

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