142 research outputs found
On (2-d)-kernels in the cartesian product of graphs
In this paper we study the problem of the existence of (2-d)-kernels in the cartesian product of graphs. We give sufficient conditions for the existence of (2-d)-kernels in the cartesian product and also we consider the number of (2-d)-kernels
A hierarchical structure of transformation semigroups with applications to probability limit measures
The structure of transformation semigroups on a finite set is analyzed by
introducing a hierarchy of functions mapping subsets to subsets. The resulting
hierarchy of semigroups has a corresponding hierarchy of minimal ideals, or
kernels. This kernel hierarchy produces a set of tools that provides direct
access to computations of interest in probability limit theorems; in
particular, finding certain factors of idempotent limit measures. In addition,
when considering transformation semigroups that arise naturally from edge
colorings of directed graphs, as in the road-coloring problem, the hierarchy
produces simple techniques to determine the rank of the kernel and to decide
when a given kernel is a right group. In particular, it is shown that all
kernels of rank one less than the number of vertices must be right groups and
their structure for the case of two generators is described.Comment: 35 pages, 4 figure
Spectral analysis for adjacency operators on graphs
We put into evidence graphs with adjacency operator whose singular subspace
is prescribed by the kernel of an auxiliary operator. In particular, for a
family of graphs called admissible, the singular continuous spectrum is absent
and there is at most an eigenvalue located at the origin. Among other examples,
the one-dimensional XY model of solid-state physics is covered. The proofs rely
on commutators methods.Comment: 16 pages, 9 figure
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