Integral Cayley Graphs Defined by Greatest Common Divisors
نویسندگان
چکیده
منابع مشابه
Integral Cayley Graphs Defined by Greatest Common Divisors
An undirected graph is called integral, if all of its eigenvalues are integers. Let Γ = Zm1 ⊗· · ·⊗Zmr be an abelian group represented as the direct product of cyclic groups Zmi of order mi such that all greatest common divisors gcd(mi,mj) ≤ 2 for i 6= j. We prove that a Cayley graph Cay(Γ, S) over Γ is integral, if and only if S ⊆ Γ belongs to the the Boolean algebra B(Γ) generated by the subg...
متن کاملDivisibility and Greatest Common Divisors
Definition 2.1. When a and b are integers, we say a divides b if b = ak for some k ∈ Z. We then write a | b (read as “a divides b”). Example 2.2. We have 2 | 6 (because 6 = 2 · 3), 4 | (−12), and 5 | 0. We have ±1 | b for every b ∈ Z. However, 6 does not divide 2 and 0 does not divide 5. Divisibility is a relation, much like inequalities. In particular, the relation 2 | 6 is not the number 3, e...
متن کاملOrdered Groups with Greatest Common Divisors Theory
An embedding (called a GCD theory) of partly ordered abelian group G into abelian l-group Γ is investigated such that any element of Γ is an infimum of a subset (possible non-finite) from G. It is proved that a GCD theory need not be unique. A complete GCD theory is introduced and it is proved that G admits a complete GCD theory if and only if it admits a GCD theory G Γ such that Γ is an Archim...
متن کاملApproximate Greatest Common Divisors and Polynomials Roots
This lecture will show by example some of the problems that occur when the roots of a polynomial are computed using a standard polynomial root solver. In particular, polynomials of high degree with a large number of multiple roots will be considered, and it will be shown that even roundoff error due to floating point arithmetic, in the absence of data errors, is sufficient to cause totally inco...
متن کاملComputing Greatest Common Divisors and Factorizations
In a quadratic number field Q(V~D), D a squarefree integer, with class number 1, any algebraic integer can be decomposed uniquely into primes, but for only 21 domains Euclidean algorithms are known. It was shown by Cohn [5] that for D < —19 even remainder sequences with possibly nondecreasing norms cannot determine the GCD of arbitrary inputs. We extend this result by showing that there does no...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: The Electronic Journal of Combinatorics
سال: 2011
ISSN: 1077-8926
DOI: 10.37236/581