Download A Textbook of Graph Theory by R. Balakrishnan, K. Ranganathan PDF

By R. Balakrishnan, K. Ranganathan

 This moment version contains new chapters: one on domination in graphs and the opposite at the spectral homes of graphs, the latter including a dialogue on graph energy.  The bankruptcy on graph hues has been enlarged, masking extra themes reminiscent of homomorphisms and colors and the distinctiveness of the Mycielskian as much as isomorphism. 

This publication additionally introduces numerous attention-grabbing subject matters reminiscent of 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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Give an example of a graph G such that both G and GC have exactly two pendant vertices. 14 Show that r (en) is Dn , the dihedral group of order 2n. 15 Determine r(K p •q ) , p =1= q . 16 Determine the order of the automorphism group of (a) K4 - e; (b) Pn . 17 Show that the line graph of K; is regularof degree 2n -4. Draw the line graph of K4. q is regular of degree p + q - 2. 3. 19 Show that a connected graph G is complete bipartite if, and only if, no induced subgraph of G is a K3 or P4. 20 IfG is connected and diam(G) ~ 3, wherediam(G) then show that GC is connected and diam(G C ) ::: 3.

If (u , z) E A(T) , then C" = (C - [x , y }) + (u , z) is a directed cycle containing u and having length shorter than that of C . Hence (u , z) doe s not belong to A(T). Therefore, (z . u) E A(T ), and (u , x, y , Z, u) is a directed 4-cycle containing u . 2 states that eve ry vertex of a d iconnected tournament lies on a k-cycle for every k, 3 ::: k ::: n . Ho wever, this property is not true for a diconnected k-partite tournament. 8 is a counterexample. T is a 3-partite tournament with {x , w}, {u} and {v} as partite sets.

The number of directed 3-cycle s in a tournament T of order n is obtained by subtracting the total number of transitive triples of vertices of T from the total number of triples of vertice s of T . Thus the total number of directed 3-cycles in T . ,vEV(T) S V S v - 1). Th. ,VEV(T) 2 the score vector of a tourn ament T determ ines the number of directed 3-cycles in T. But, in a general k-partite tournament , the score vector need not determine the number of directed 3-cycles. 9. Both have the same score vector (2, 2, 2, 2, 2, 2).

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