Báo cáo toán học: "Integral Cayley graphs defined by greatest common divisors" pot

... m) : a ∈ H}. the electronic journal of combinatorics 18 (2011), #P94 3 Integral Cayley graphs defined by greatest common divisors Walter Klotz Institut f¨ur Mathematik Tech nische Universit¨at ... at every S ∈ B(Γ) can be characterized by greatest common divisors. 1 Introduction The greatest common divisor of nonnegative integers a and b is denoted by gcd(a, b). Let us...

Ngày tải lên: 08/08/2014, 14:23

... 17 (2010), #R81 6 Denote by Z n k the n-fold direct pr oduct o f Z k with itself. It was already noticed by Lov´asz [12] that all Cayley graphs over Z n 2 (“cubelike graphs ) are integral. The ... we defined a finite additive group Γ to be Cayley integral, if for every S ⊆ Γ, 0 ∈ S, − S = S, the Cayley gra ph Cay(Γ, S) is integral. Observe that for this definition Γ may be nonabe...

Ngày tải lên: 08/08/2014, 12:22

... underlying group, we give some infinite families of integral Cayley graphs. The study of Cayley graphs of the symmetric group generated by tra nspositions is interest- ing (See [11]). In this pa ... 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 c...

Ngày tải lên: 08/08/2014, 01:20

... 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 uniformly ... representation theory, can be given by writing C[G]=T ⊕ N, where T is the one-dimensional eigenspace spanned by the uniform vector  g g and N is the orthogonal complement of T. B...

Ngày tải lên: 07/08/2014, 08:20

... circuit constructed by applying the mapping M to the vertices of the circuit C. Then by the above theorem, C ∗ is a circuit in the graph G(∞, u/n) for N (Γ 0 (m)). Since n > 1, by Theorem3.1, C ∗ is ... integer. Thus, by Theorem2.1, Γ + 0 (m) acts transitively on ˆ Q, and it follows that N (Γ 0 (N)) acts transitively on ˆ Q by Theorem2.2. So the proof is completed. 3. Suborbital...

Ngày tải lên: 08/08/2014, 01:20

... finitely many graphs with vertex set [n], unlabeled graphs are finite sets of (labeled) graphs. Indeed, if the automorphism group of G has cardinality a, then [G] is a set of n!/a graphs. We typically ... introduction to this class of graphs. If we choose the weights for the vertices at random, we induce a probability measure on the set of threshold graphs and thereby create a notion...

Ngày tải lên: 08/08/2014, 11:20

... Γ 36 and Γ 37 are Cayley graphs and others are not (See table 4). By table 8, |S| ∈ { 3, 4, 5, 6, 7, 8, 9}. By Theorem 2.11, the graphs Γ 38 , Γ 39 and Γ 40 are not Cayley graphs. It is clear ... Γ 29 is not a Cayley graph of D 10 . Therefore there a re exactly five connected, integral Cayley graphs with 10 vertices.  Lemma 2.23 There are exactly three connected, integral...

Ngày tải lên: 08/08/2014, 14:23

Ngày tải lên: 05/08/2014, 15:20

... points. By the above observation and Lemma 3.1, Fulker- sonian hypergraphs are generalizations of Mengerian hypergraphs. Unlike the Mengerian property, the Fulkersonian property is preserved by passing ... hypergraph, which is defined by a min-max equation in Integer Linear Programming. A nat- ural generalization of the Mengerian hypergraph is the Fulkersonian hypergraph which is defin...

Ngày tải lên: 06/08/2014, 05:20

... 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 ... triangle-free graphs with small independence numbers cannot yield asymptotically better constructions. This is because the independence num- ber is bounded here by bounding the θ-number w...

Ngày tải lên: 07/08/2014, 06:20