A Textbook of Graph Theory (2nd Edition) (Universitext) by R. Balakrishnan, K. Ranganathan PDF
By R. Balakrishnan, K. Ranganathan
Graph conception skilled a massive progress within the twentieth century. one of many major purposes for this phenomenon is the applicability of graph idea in different disciplines similar to physics, chemistry, psychology, sociology, and theoretical computing device technology. This textbook offers an outstanding historical past within the simple issues of graph thought, and is meant for a complicated undergraduate or starting graduate direction in graph theory.
This moment variation contains new chapters: one on domination in graphs and the opposite at the spectral homes of graphs, the latter together with a dialogue on graph power. The bankruptcy on graph colors has been enlarged, masking extra subject matters comparable to homomorphisms and colors and the individuality of the Mycielskian as much as isomorphism. This booklet additionally introduces a number of attention-grabbing themes 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 evidence of Kuratowski's theorem on planar graphs, the evidence of the nonhamiltonicity of the Tutte graph on forty six vertices, and a concrete program of triangulated graphs.
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Additional info for A Textbook of Graph Theory (2nd Edition) (Universitext)
2. 3. 4. 5. v// D N. 7 Line Graphs Let G be a loopless graph. G/, and hence we assume in this section that G has no isolated vertices. We also assume that G has no loops. G/ of a graph G follow: 1. G/ is connected. 2. G/: 3. G/: 4. v/ 2: 5. G/ v1 e1 v2 e2 v4 e4 e5 e3 e7 v7 v3 v6 e6 v5 G Fig. 1. 2. 1. The line graph of a simple graph G is a path if and only if G is a path. Proof. Let G be the path Pn on n vertices. G/ is the path Pn 1 on n 1 vertices. G/ be a path. G/ with at least three vertices.
G/ D 2; then K2 is a spanning subgraph of G; and so no vertex of G is a cut vertex of G: This completes the proof of the theorem. 11. , 3-regular) connected graph G has a cut vertex if and only if it has a cut edge. Proof. Let G have a cut vertex v0 : Let v1 ; v2 ; v3 be the vertices of G that are adjacent to v0 in G: Consider G v0 ; which has either two or three components. If G v0 has three components, no two of v1 ; v2 ; and v3 can belong to the same component of G v0 : In this case, each of v0 v1 ; v0 v2 ; and v0 v3 is a cut edge of G: (See Fig.
12. If G is simple and ı least k: k; then G contains a path of length at Proof. 1. An automorphism of a graph G is an isomorphism of G onto itself. G/ ! G/ of automorphisms of G is a group. 2. G/ of all automorphisms of a simple graph G is a group with respect to the composition ı of mappings as the group operation. Proof. G/ preserving adjacency and nonadjacency. v/ are adjacent in G: But . u/ D 1 . u// and . v/ D 1 . v//: Hence, . u/ and . v/ are adjacent in GI that is, 1 ı 2 preserves adjacency.
A Textbook of Graph Theory (2nd Edition) (Universitext) by R. Balakrishnan, K. Ranganathan