19,500 research outputs found
Ihara zeta functions for periodic simple graphs
The definition and main properties of the Ihara zeta function for graphs are
reviewed, focusing mainly on the case of periodic simple graphs. Moreover, we
give a new proof of the associated determinant formula, based on the treatment
developed by Stark and Terras for finite graphs.Comment: 17 pages, 7 figures. V3: minor correction
Ihara's zeta function for periodic graphs and its approximation in the amenable case
In this paper, we give a more direct proof of the results by Clair and
Mokhtari-Sharghi on the zeta functions of periodic graphs. In particular, using
appropriate operator-algebraic techniques, we establish a determinant formula
in this context and examine its consequences for the Ihara zeta function.
Moreover, we answer in the affirmative one of the questions raised by
Grigorchuk and Zuk. Accordingly, we show that the zeta function of a periodic
graph with an amenable group action is the limit of the zeta functions of a
suitable sequence of finite subgraphs.Comment: 21 pages, 4 figure
Zeta functions for infinite graphs and functional equations
The definitions and main properties of the Ihara and Bartholdi zeta functions
for infinite graphs are reviewed. The general question of the validity of a
functional equation is discussed, and various possible solutions are proposed.Comment: 23 pages, 3 figures. Accepted for publication in "Fractals in Applied
Mathematics", Contemporary Mathematics, Editors Carfi, Lapidus, Pearse, van
Frankenhuijse
Comparability in the graph monoid
Let be the infinite cyclic group on a generator To avoid
confusion when working with -modules which also have an additional
-action, we consider the -action to be a -action
instead.
Starting from a directed graph , one can define a cancellative commutative
monoid with a -action which agrees with the monoid
structure and a natural order. The order and the action enable one to label
each nonzero element as being exactly one of the following: comparable
(periodic or aperiodic) or incomparable. We comprehensively pair up these
element features with the graph-theoretic properties of the generators of the
element. We also characterize graphs such that every element of is
comparable, periodic, graphs such that every nonzero element of is
aperiodic, incomparable, graphs such that no nonzero element of is
periodic, and graphs such that no element of is aperiodic.
The Graded Classification Conjecture can be formulated to state that
is a complete invariant of the Leavitt path algebra of
over a field Our characterizations indicate that the Graded
Classification Conjecture may have a positive answer since the properties of
are well reflected by the structure of Our work also implies
that some results of [R. Hazrat, H. Li, The talented monoid of a Leavitt path
algebra, J. Algebra, 547 (2020) 430-455] hold without requiring the graph to be
row-finite.Comment: This version contains some modifications based on the input of a
referee for the New York Journal of Mathematic
Random subshifts of finite type
Let be an irreducible shift of finite type (SFT) of positive entropy, and
let be its set of words of length . Define a random subset
of by independently choosing each word from with some
probability . Let be the (random) SFT built from the set
. For each and tending to infinity, we compute
the limit of the likelihood that is empty, as well as the limiting
distribution of entropy for . For near 1 and tending
to infinity, we show that the likelihood that contains a unique
irreducible component of positive entropy converges exponentially to 1. These
results are obtained by studying certain sequences of random directed graphs.
This version of "random SFT" differs significantly from a previous notion by
the same name, which has appeared in the context of random dynamical systems
and bundled dynamical systems.Comment: Published in at http://dx.doi.org/10.1214/10-AOP636 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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