By R. Balakrishnan, K. Ranganathan
This moment variation comprises 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 colorations has been enlarged, protecting extra subject matters resembling homomorphisms and colours and the individuality of the Mycielskian as much as isomorphism.
This e-book additionally introduces a number of attention-grabbing issues resembling 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 evidence of the nonhamiltonicity of the Tutte graph on forty six vertices, and a concrete software of triangulated graphs.
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Additional resources for A Textbook of Graph Theory
1=) where (d) , dz, .. , dn ) is the degree sequence of G, and m = mt G). 4), it follows that m(L(G» = ~2 [t 1] d m. 1 is a path. ~ C« . n ::: 3. The line graph ofa simple graph G is a path if, and only if, G Proof Let G be the path P, on n vertices. Then clearly L(G) is the path Pn - I on n - I vertices. Conversely, let L(G) be a path . Then no vertex of G can have degree greater than 2 because, if G has a vertex v of degree greater than 2, the edges incident to v would form a complete subgraph of L(G) with at least three vertices.
In terms of graphs, isomers are two nonisomorphic graphs having the same degree sequence. 33 represent two isomers of the molecule C3H 7 0 H (propanol). 33. 34. Graph OfC3H7NO ami noacetone C 3H7NO. This has a multiple bond represented by a pair of m ultiple edges between C and O. T he paraffins have the molecular form ula C kH2k+ 2 . They have 3k + 2 atoms (vertices) of which k are carbon atoms and the remaining 2k + 2 are hydrogen atoms. They all have 3k + I bonds (edges). Cayley used enumeration techniques of graph theory (see reference ) to count the number of isomers of CkH2k +2 • His formula shows that for the paraffin C 13H28 , there are 802 different isomers.
Is also onto since, for v' in G' , ¢; J(S( v' » = S(v ) for some v E V (G ), and by the definition of ¢ , ¢ (v ) = v'. Finally, if u v is an edge of G , then ¢I (uv) belon gs to both S(u ' ) and S( v' ), where ¢I (S(U» = S(u ' ) and ¢I (S(V» = S( v' ). This means that u' v' is an edge of G ' . But u' = ¢ (u ) and v' = ¢ (v). Con sequently, ¢ (u )¢ (v) is an edge of G'. If u and v are nonadjacent in G, ¢ (u )¢(v) must be nonadjacent in G ' . Otherwise, ¢(u) ¢(v) belong s to both S(¢ (u » and S(¢ (v» , and hence ¢;I (¢(u )¢ ( v» = u v E E (G ), a contradiction.
A Textbook of Graph Theory by R. Balakrishnan, K. Ranganathan