183 research outputs found
On finite groups all of whose cubic Cayley graphs are integral
For any positive integer , let denote the set of finite
groups such that all Cayley graphs are integral whenever
. Estlyi and Kovcs \cite{EK14}
classified for each . In this paper, we characterize
the finite groups each of whose cubic Cayley graphs is integral. Moreover, the
class is characterized. As an application, the classification
of is obtained again, where .Comment: 11 pages, accepted by Journal of Algebra and its Applications on June
201
Horizontal isogeny graphs of ordinary abelian varieties and the discrete logarithm problem
Fix an ordinary abelian variety defined over a finite field. The ideal class
group of its endomorphism ring acts freely on the set of isogenous varieties
with same endomorphism ring, by complex multiplication. Any subgroup of the
class group, and generating set thereof, induces an isogeny graph on the orbit
of the variety for this subgroup. We compute (under the Generalized Riemann
Hypothesis) some bounds on the norms of prime ideals generating it, such that
the associated graph has good expansion properties.
We use these graphs, together with a recent algorithm of Dudeanu, Jetchev and
Robert for computing explicit isogenies in genus 2, to prove random
self-reducibility of the discrete logarithm problem within the subclasses of
principally polarizable ordinary abelian surfaces with fixed endomorphism ring.
In addition, we remove the heuristics in the complexity analysis of an
algorithm of Galbraith for explicitly computing isogenies between two elliptic
curves in the same isogeny class, and extend it to a more general setting
including genus 2.Comment: 18 page
Groups all of whose undirected Cayley graphs are integral
Let be a finite group, be a set such that if
, then , where denotes the identity element of .
The undirected Cayley graph of over the set is the graph
whose vertex set is and two vertices and are adjacent whenever
. The adjacency spectrum of a graph is the multiset of all
eigenvalues of the adjacency matrix of the graph. A graph is called integral
whenever all adjacency spectrum elements are integers. Following Klotz and
Sander, we call a group Cayley integral whenever all undirected Cayley
graphs over are integral. Finite abelian Cayley integral groups are
classified by Klotz and Sander as finite abelian groups of exponent dividing
or . Klotz and Sander have proposed the determination of all non-abelian
Cayley integral groups. In this paper we complete the classification of finite
Cayley integral groups by proving that finite non-abelian Cayley integral
groups are the symmetric group of degree , and
for some integer , where is the
quaternion group of order .Comment: Title is change
Isogeny graphs of ordinary abelian varieties
Fix a prime number . Graphs of isogenies of degree a power of
are well-understood for elliptic curves, but not for higher-dimensional abelian
varieties. We study the case of absolutely simple ordinary abelian varieties
over a finite field. We analyse graphs of so-called -isogenies,
resolving that they are (almost) volcanoes in any dimension. Specializing to
the case of principally polarizable abelian surfaces, we then exploit this
structure to describe graphs of a particular class of isogenies known as
-isogenies: those whose kernels are maximal isotropic subgroups
of the -torsion for the Weil pairing. We use these two results to write
an algorithm giving a path of computable isogenies from an arbitrary absolutely
simple ordinary abelian surface towards one with maximal endomorphism ring,
which has immediate consequences for the CM-method in genus 2, for computing
explicit isogenies, and for the random self-reducibility of the discrete
logarithm problem in genus 2 cryptography.Comment: 36 pages, 4 figure
On prisms, M\"obius ladders and the cycle space of dense graphs
For a graph X, let f_0(X) denote its number of vertices, d(X) its minimum
degree and Z_1(X;Z/2) its cycle space in the standard graph-theoretical sense
(i.e. 1-dimensional cycle group in the sense of simplicial homology theory with
Z/2-coefficients). Call a graph Hamilton-generated if and only if the set of
all Hamilton circuits is a Z/2-generating system for Z_1(X;Z/2). The main
purpose of this paper is to prove the following: for every s > 0 there exists
n_0 such that for every graph X with f_0(X) >= n_0 vertices, (1) if d(X) >=
(1/2 + s) f_0(X) and f_0(X) is odd, then X is Hamilton-generated, (2) if d(X)
>= (1/2 + s) f_0(X) and f_0(X) is even, then the set of all Hamilton circuits
of X generates a codimension-one subspace of Z_1(X;Z/2), and the set of all
circuits of X having length either f_0(X)-1 or f_0(X) generates all of
Z_1(X;Z/2), (3) if d(X) >= (1/4 + s) f_0(X) and X is square bipartite, then X
is Hamilton-generated. All these degree-conditions are essentially
best-possible. The implications in (1) and (2) give an asymptotic affirmative
answer to a special case of an open conjecture which according to [European J.
Combin. 4 (1983), no. 3, p. 246] originates with A. Bondy.Comment: 33 pages; 5 figure
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