By R. Balakrishnan, K. Ranganathan

This moment version contains new chapters: one on domination in graphs and the opposite at the spectral houses of graphs, the latter including a dialogue on graph energy. The bankruptcy on graph shades has been enlarged, overlaying extra themes equivalent to homomorphisms and colorations and the distinctiveness of the Mycielskian as much as isomorphism.

This booklet additionally introduces numerous attention-grabbing issues similar to Dirac's theorem on k-connected graphs, Harary-Nashwilliam's theorem at the hamiltonicity of line graphs, Toida-McKee's characterization of Eulerian graphs, the Tutte matrix of a graph, Fournier's evidence of Kuratowski's theorem on planar graphs, the facts of the nonhamiltonicity of the Tutte graph on forty six vertices, and a concrete software of triangulated graphs.

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2. 4, A ::': K = 2. Hence A = 2. Finally, the above arguments show that if A = 3, then K = 3, and if A = 2, then K = 2. 2 Connectivityand Edge-Connectivity .. 51 e - - t - -... v U~---t (a) v U v U U . - - - - - -.. 7. 6 Give examples of cubic graphs G I , G 2 , and G3 with = 2, and K(G3) = 3. 6 A family of two or more paths in a graph G is said to be internally disjoint if no vertex of G is an internal vertex of more than one path in the family . We now state and prove Whitney's theorem on 2-connected graphs.

3 I. 3 Prove that G 1 0 G 2 is isomorphic to G 2 0 G I . 4 Prove that (a) (b) (c) (d) n (GdG 2]) m(G l [G 2]) m (G I m eG ] = n (G 2[Gd) = n (G l 0 G 2) = n (G dn (G 2) = n (G I )m (G 2) + n (G 2)2m (G d 0 G 2 ) = n (G dm (G 2) + m (G t )n (G 2 ) + 2m(Gdm(G 2) ® G 2) = 2m(Gdm(G 2) . 8 An Application to Chemistry 29 We now introduce the powers of a simple graph G. 11 ThekthpowerG k ofG has V(G k ) = V(G) , where u and v are adjacent in G k whenever dG(u, v) ~ k. 32. By definition of o', G is a spanning subgraph of c-, k :::: 1.

Proof Let C t = UIUZ . UkUI and C z = VI Vz ' " VkVI be two longest cycles in G. 8). U; and Ut divide C I into two subpaths. Let L I be the longer of these subpaths. ) Let L z be defined in a similar manner in c-. Then L I U PI U Lz U P z is a cycle of length greater than that of C I (or C z)· Hence C I and Cz cannot be disjoint. ) Suppose that C I and C; have exactly one vertex , say UI = VJ, in common. Since G is 2-connected, UI is not a cut vertex of G, and so there exists a path P with one end vertex u, in CI-UI and the other end vertex Vj in Cz-vJ, which is internally disjoint from C I U Cz- Let PI denote the longer ofthe two UI-U i sections of CJ, and QI denote the longer of the two VI-Vj sections of C z.