A Textbook of Graph Theory (2nd Edition) (Universitext) by R. Balakrishnan, K. Ranganathan

By R. Balakrishnan, K. Ranganathan

Graph concept skilled an important progress within the twentieth century. one of many major purposes for this phenomenon is the applicability of graph thought in different disciplines akin to physics, chemistry, psychology, sociology, and theoretical machine technological know-how. This textbook presents an exceptional heritage within the uncomplicated themes of graph thought, and is meant for a complicated undergraduate or starting graduate direction in graph theory.

This moment variation comprises new chapters: one on domination in graphs and the opposite at the spectral houses of graphs, the latter together with a dialogue on graph power. The bankruptcy on graph shades has been enlarged, overlaying extra themes akin to homomorphisms and hues and the distinctiveness of the Mycielskian as much as isomorphism. This publication additionally introduces a number of fascinating subject matters corresponding 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 facts 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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Denote a general graph product of two simple graphs by G H: We define the product in such a way that G H is also simple. v1 ; v2 / of G1 G2 ; consider the following possibilities: (i) u1 adjacent to v1 in G1 or u1 nonadjacent to v1 in G1 . (ii) u2 adjacent to v2 in G2 or u2 nonadjacent to v2 in G2 . (iii) u1 D v1 and/or u2 D v2 : We use, with respect to any graph, the symbols E; N; and D to denote adjacency (edge), nonadjacency (no edge), and equality of vertices, respectively. v1 ; v2 / of G1 G2 : Since each ı can take two options, there are in all 28 D 256 graph products G1 G2 that can be defined using G1 and G2 : 2 3 a11 a12 a13 If S D 4a21 D a23 5 ; then the edge-nonedge entry of S will correspond to a31 a32 a33 the nonedge-edge entry of the structure matrix of G2 G1 : Hence, the product is commutative, that is, G1 and G2 commute under if and only if the double array S is symmetric.

V2 ; E2 / be two simple graphs. 1. 8 Operations on Graphs G1 25 G2 G3 G4 G7 G5 G8 G6 G9 Fig. 2. 3. Join of two graphs: Let G1 and G2 be two vertex-disjoint graphs. 26 illustrates the graph G1 _ G2 : If G1 D K1 and G2 D Cn ; then G1 _ G2 is called the wheel Wn : W5 is shown in Fig. 27. G2 /: 26 1 Basic Results Fig. 26 G1 _ G2 v1 u1 G1 : G2 : v2 v4 u2 v3 v1 u1 G1 ∨ G2: v2 v4 u2 v3 Fig. 9 Graph Products We now define graph products. Denote a general graph product of two simple graphs by G H: We define the product in such a way that G H is also simple.

Determine the connectivity and edge connectivity of the Petersen graph P: (See graph P of Fig. 7. 5. The connectivity and edge connectivity of a simple cubic graph G are equal.

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