63,511 research outputs found
Determination of the prime bound of a graph
Given a graph , a subset of is a module of if for each
, is adjacent to all the elements of or to none
of them. For instance, , and () are
modules of called trivial. Given a graph , (respectively
) denotes the largest integer such that there is a module
of which is a clique (respectively a stable) set in with . A
graph is prime if and if all its modules are trivial. The
prime bound of is the smallest integer such that there is a prime
graph with , and . We establish the following. For every graph such that
and
is not an integer,
. Then, we prove that
for every graph such that where , or . Moreover if and only if or its complement
admits isolated vertices. Lastly, we show that for every non
prime graph such that and .Comment: arXiv admin note: text overlap with arXiv:1110.293
Polynomials with prescribed bad primes
We tabulate polynomials in Z[t] with a given factorization partition, bad
reduction entirely within a given set of primes, and satisfying auxiliary
conditions associated to 0, 1, and infinity. We explain how these sets of
polynomials are of particular interest because of their role in the
construction of nonsolvable number fields of arbitrarily large degree and
bounded ramification. Finally we discuss the similar but technically more
complicated tabulation problem corresponding to removing the auxiliary
conditions.Comment: 26 pages, 3 figure
James' Conjecture for Hecke algebras of exceptional type, I
In this paper, and a second part to follow, we complete the programme
(initiated more than 15 years ago) of determining the decomposition numbers and
verifying James' Conjecture for Iwahori--Hecke algebras of exceptional type.
The new ingredients which allow us to achieve this aim are:
- the fact, recently proved by the first author, that all Hecke algebras of
finite type are cellular in the sense of Graham--Lehrer, and
- the explicit determination of -graphs for the irreducible (generic)
representations of Hecke algebras of type and by Howlett and Yin.
Thus, we can reduce the problem of computing decomposition numbers to a
manageable size where standard techniques, e.g., Parker's {\sf MeatAxe} and its
variations, can be applied. In this part, we describe the theoretical
foundations for this procedure.Comment: 24 pages; corrected some misprints, added Remark 4.1
Bipartite entangled stabilizer mutually unbiased bases as maximum cliques of Cayley graphs
We examine the existence and structure of particular sets of mutually
unbiased bases (MUBs) in bipartite qudit systems. In contrast to well-known
power-of-prime MUB constructions, we restrict ourselves to using maximally
entangled stabilizer states as MUB vectors. Consequently, these bipartite
entangled stabilizer MUBs (BES MUBs) provide no local information, but are
sufficient and minimal for decomposing a wide variety of interesting operators
including (mixtures of) Jamiolkowski states, entanglement witnesses and more.
The problem of finding such BES MUBs can be mapped, in a natural way, to that
of finding maximum cliques in a family of Cayley graphs. Some relationships
with known power-of-prime MUB constructions are discussed, and observables for
BES MUBs are given explicitly in terms of Pauli operators.Comment: 8 pages, 1 figur
Drawing a Graph in a Hypercube
A -dimensional hypercube drawing of a graph represents the vertices by
distinct points in , such that the line-segments representing the
edges do not cross. We study lower and upper bounds on the minimum number of
dimensions in hypercube drawing of a given graph. This parameter turns out to
be related to Sidon sets and antimagic injections.Comment: Submitte
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