By B. Bollobás (Eds.)

ISBN-10: 0720408431

ISBN-13: 9780720408430

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**Spectra of graphs. Theory and application - download pdf or read online**

The speculation of graph spectra can, in a manner, be regarded as an try to make the most of linear algebra together with, specifically, the well-developed concept of matrices for the needs of graph conception and its purposes. besides the fact that, that doesn't suggest that the speculation of graph spectra could be decreased to the idea of matrices; to the contrary, it has its personal attribute beneficial properties and particular methods of reasoning absolutely justifying it to be handled as a thought in its personal correct.

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**Extra info for Advances in Graph Theory**

**Example text**

Fi ,, c,, a,, where c, E C, f, E A n B and Q, E A -B. There are s such intervals and so the vertices of D also form S intervals, some of which may be empty. Let I, be the length of the ith interval. Then CS=,,lz= ID\ = 2 k - r - s. The set D - D,, does not contain adjacent vertices since otherwise G contains a longer cycle than L, as shown in Fig. 4. Consequently s ID,,[2 1[ + I z ] 51 ' 3 1=l (I, - 1)= k - s - ir. ;I Fig. 4. d, E D - D,, and f,E A nB. The cycle in the thick line is longer than L 46 B.

In [7] we conjectured that K,” can be decomposed into hamiltonian cycles if and only if (9/n is an integer and proved this conjecture for n a prime. Here I want to prove two theorems concerning the case h = 3. 2. Theorem. If K: can be decomposed into hamiltonian cycles then K:, also can be decomposed into hamiltonian cycles. (This proof was obtained with D. ) In order to shorten the writing, I will write a hamiltonian cycle as E,,. . E,, . . ,En where EL= (xE,y,, IC,+,). Let the vertex set of K:, be X U X’ with = [X’l=n.

This was to be proved. Part I1 (First version). (A) We start with a graph G" for which h ( G " )3 cn2, fix a c"< c and then a C ' E (c", c) and a p > 0, which is much smaller than c". Using the first part we may assume that the edges are regularly distributed. We may repeat part (A) of the proof of Theorem 7 replacing e( ) by h( ) and c by c'. Then we may assume that G' = G", but have to decrease c': replace the original condition by condition h ( G " ) s c"n2. How we define the graph R k as in the beginning of (B) of the proof of Theorem 7.

### Advances in Graph Theory by B. Bollobás (Eds.)

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