By Bela Bollobas

ISBN-10: 0444864490

ISBN-13: 9780444864499

From the reviews: "Béla Bollobás introductory path on graph thought merits to be regarded as a watershed within the improvement of this conception as a significant educational topic. ... The e-book has chapters on electric networks, flows, connectivity and matchings, extremal difficulties, colouring, Ramsey conception, random graphs, and graphs and teams. each one bankruptcy starts off at a measured and delicate speed. Classical effects are proved and new perception is supplied, with the examples on the finish of every bankruptcy absolutely supplementing the text... then again this enables an advent not just to a couple of the deeper effects yet, extra vitally, presents outlines of, and company insights into, their proofs. hence in an effortless textual content e-book, we achieve an total realizing of famous common effects, and but even as consistent tricks of, and instructions into, the better degrees of the topic. it's this element of the e-book which may still warrantly it an everlasting position within the literature." #Bulletin of the London Mathematical Society#1

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C. Berrnond, C. Delorme and G. Farhi 24 n(2,D) s 2 D + 1 and f o r A > 2 , n(A,D) 5; A ( AA-2 -1)D-2 s a t i s f y i n g t h e e q u a l i t y a r e c a l l e d Moohe g h a p h b . 231) t h a t Moore g r a p h s can e x i s t o n l y i f A = 2 ( t h e g r a p h s b e i n g t h e ( 2 D + 1 ) c y c l e s ) o r if D = 2 and A = 3 , 7 , 5 1 ( f o r A = 3 and A = I t h e r e e x i s t s a unique Moore graph r e s p e c t i v e l y P e t e r s e n ' s g r a p h on 10 v e r t i c e s , and Hoffman and S i n g l e t o n ' s graph on 50 v e r t i c e s , f o r i s n o t known).

F u r t h e r , t h e o t h e r v e r t i c e s , n o t i n W = V ( F ) must be independent f o r o t h e r w i s e t h e r e is a 2-step p a t h t o a non-cutvertex of F r e s u l t i n g i n an 8 - t r a i l . Also, no v e r t e x v { W c a n be a d j a c e n t t o two v e r t i c e s i n W a s t h i s would y i e l d e i t h e r an 8 - t r a i l or two d i s j o i n t c y c l e s one of which i s C 4 .

Our o b j e c t i s t o i n v e s t i g a t e t h e maximum and minimum v a l u e s of t r ( G ) f o r G(n,m). W e f i r s t n o t e t h a t t h e maximum can be determined w i t h o u t any e f f o r t . 2. THE MAXIMUM TRAIL NUMBER O F A GRAPH T h i s q u e s t i o n i s s u f f i c i e n t l y t r i v i a l t h a t one can d e t e r m i n e t h e answer q u i c k l y and e x a c t l y . W e w i l l see t h a t t h i s is n o t so f o r t h e minimum t r a i l number. Theorem 1. The maximum t r a i l number of a g r a p h G(n,m) n i s odd o r m 2 + 1; o t h e r w i s e it i s );( - (i)- is %+ m if 1.

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Graph Theory: Conference Proceedings by Bela Bollobas

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