241 GraphsExercises1.3.1a) Show that the graph in Figure 1.18 is isomorphic to the Heawood graph (Figure1.16).Fig. 1.18. Another drawing of the Heawood graphb) Deduce that the Heawood graph is vertex-transitive.1.3.2 Show that the fllowing three graphs are isomorphic:D the intersection graph of the Desargues configuration,the line graph of Ks the complement of the Petersen graph.1.3.3 Showthat the line graph of K3.3 is self-complement:a1.3.4 Show that neither of the graphs displayed in Figure 1.19 is a line graph1.3.5 Let H := (V, F) be a hypergraph. The number of edges incident with aex u of H is its degree, denoted d(u). A degree sequence of H is a vectorord :=(d(u) :we V). Let M be the incidence matrix of H and d the correspondingdegree sequence of H. Show that the sum of the columns of M is equal to d.1.3.6 Let H :-- (V, F) be a hypergraph. For u e V, let F, denote the set of edgesof H incident to u. The dual of H is the hypergraph H* whose vertex set is F andwhose edges are the sets Fu, u e V.Fig. 1.19. Two graphs that are not line graphs24 1 Graphs Exercises 1.3.1 a) Show that the graph in Figure 1.18 is isomorphic to the Heawood graph (Fig￾ure 1.16). Fig. 1.18. Another drawing of the Heawood graph b) Deduce that the Heawood graph is vertex-transitive. 1.3.2 Show that the following three graphs are isomorphic:  the intersection graph of the Desargues configuration,  the line graph of K5,  the complement of the Petersen graph. 1.3.3 Show that the line graph of K3,3 is self-complementary. 1.3.4 Show that neither of the graphs displayed in Figure 1.19 is a line graph. 1.3.5 Let H := (V, F) be a hypergraph. The number of edges incident with a vertex v of H is its degree, denoted d(v). A degree sequence of H is a vector d := (d(v) : v ∈ V ). Let M be the incidence matrix of H and d the corresponding degree sequence of H. Show that the sum of the columns of M is equal to d. 1.3.6 Let H := (V, F) be a hypergraph. For v ∈ V , let Fv denote the set of edges of H incident to v. The dual of H is the hypergraph H∗ whose vertex set is F and whose edges are the sets Fv, v ∈ V . Fig. 1.19. Two graphs that are not line graphs