Báo cáo toán hoc:"Which Cayley graphs are integral" docx

... integral Cayley graphs and introduce some infinite family of connected integ r al Cayley graphs. The main results are the following. Theorem 1.1 There are exactly seven connected cubic integral Cayley ... C m and D 2n are used for the cyclic group of order m and dihedral gro up of order 2n (n > 2). The main question that we are concerned here is the following: Which Cayle...

Ngày tải lên: 08/08/2014, 01:20

... theorem of N. Alon and Y. Roichman [4] asserts that random Cayley graphs are expanders. Specifically, they study the spectrum of the Cayley graphs obtained by selecting k elements, independently and ... is a family of graphs {G i | i>0}, where G i is a (n i ,d,)- expander,  and d are constants independent of n i ,andthen i tend to infinity in i. Graphs with these properties...

Ngày tải lên: 07/08/2014, 08:20

... V. Chv´atal, Almost all graphs with 1.44n edges are 3-colorable, Random Structures & Algorithms, 2 (1991), 11–28. [5] P. Erd˝os, A. R´enyi, On the evolution of random graphs, Publication of ... for non-k-colorability are given for k>3. 1991 Mathematics Subject Classification: Primary 05C80; Secondary 05C15. 1 Introduction Let N(n, m, A) denote the number of graphs with vertices...

Ngày tải lên: 07/08/2014, 06:20

... and describe an explicit construction of triangle free graphs with independence numbers m and Ω(m 3/2 ) vertices. Our graphs are Cayley graphs and their construc- tion is based on some of the ... easy; see [14]) that the independence number of G does not exceed θ(G). The graphs G n we construct here are triangle free graphs on n vertices satisfying θ(G n )=Θ(n 2/3 ), and hence...

Ngày tải lên: 07/08/2014, 06:20

... reduced two -graphs, and let E(F) denote the class of two -graphs containing no member of F as a substructure. If f n and g n denote the numbers of labelled n-element two -graphs and reduced two -graphs ... Y unchanged.) When I speak of the pentagon and hexagon two -graphs below, I mean the two -graphs obtained from the pentagon and hexagon graphs by this procedure. All this material c...

Ngày tải lên: 07/08/2014, 06:20

... non-hamiltonian graphs which barely miss satisfying Dirac’s condition. We are sure this must be known, but we have not found it in the literature. We are grateful to Arthur Hobbs [6], who made us aware of ... every component of G − v. ✷ Summarizing, we now know all extremal graphs, G with f (G)=2δ +1. The connected extremal graphs are those given in Theorem 3 along with the trivial...

Ngày tải lên: 07/08/2014, 06:20

... particular, circulant graphs have been studied intensively in the literature. These graphs are easily seen to be vertex transitive. In the case of a prime vertex number n circulant graphs are known to ... only vertex transitive graphs. Because of their high symmetry, Cayley graphs are ideal models for commu- nication networks. Routing and weight balancing is easily done on su...

Ngày tải lên: 07/08/2014, 06:22

... full cycle. Cayley graphs, and in particular circulants, have been studied intensively in the lit- erature. These graphs are vertex-transitive. In the case of a prime vertex number n, circulants are ... on the choice of the pair (x, y) ∈ γ. The elements of Γ are called basic relations, their graphs basic graphs, and the numbers p γ α,β are called the structure constants of the...

Ngày tải lên: 07/08/2014, 06:22

... up G(k, t) was to obtain graphs with more vertices than n 1 and still having -holes in large subgraphs. Next we consider a random “contraction” of G(k, t) to obtain graphs with fewer than n 1 vertices ... M. Capalbo, Y. Kohayakawa, V. R¨odl, A. Ruci´nski, E. Szemer´edi, Near- optimum universal graphs for graphs with bounded degrees, APPROX-RANDOM 2001, LNCS 3139 (2001) 170-180 the el...

Ngày tải lên: 07/08/2014, 06:23

... properties is sharp (hence the parentheses). Together with the 2 Hoffman-Singleton subgraphs H 1 and H 2 , Lemma 35 produces 252 Hoffman-Singleton subgraphs. A further 252 of them are obtained by applying ... subgraphs. The next two Lemmas show that there are no further Hoffman-Singleton subgraphs: the total number of 5-cycles in G is 443 520 (Lemma 41), each contained in two Hoffman- Singlet...

Ngày tải lên: 07/08/2014, 08:22