33,217 research outputs found
Regularity of Edge Ideals and Their Powers
We survey recent studies on the Castelnuovo-Mumford regularity of edge ideals
of graphs and their powers. Our focus is on bounds and exact values of and the asymptotic linear function , for in terms of combinatorial data of the given graph Comment: 31 pages, 15 figure
Upper bounds on the k-forcing number of a graph
Given a simple undirected graph and a positive integer , the
-forcing number of , denoted , is the minimum number of vertices
that need to be initially colored so that all vertices eventually become
colored during the discrete dynamical process described by the following rule.
Starting from an initial set of colored vertices and stopping when all vertices
are colored: if a colored vertex has at most non-colored neighbors, then
each of its non-colored neighbors becomes colored. When , this is
equivalent to the zero forcing number, usually denoted with , a recently
introduced invariant that gives an upper bound on the maximum nullity of a
graph. In this paper, we give several upper bounds on the -forcing number.
Notable among these, we show that if is a graph with order and
maximum degree , then . This simplifies to, for the zero forcing number case
of , . Moreover, when and the graph is -connected, we prove that , which is an improvement when , and
specializes to, for the zero forcing number case, . These results resolve a problem posed by
Meyer about regular bipartite circulant graphs. Finally, we present a
relationship between the -forcing number and the connected -domination
number. As a corollary, we find that the sum of the zero forcing number and
connected domination number is at most the order for connected graphs.Comment: 15 pages, 0 figure
Unfolding Latent Tree Structures using 4th Order Tensors
Discovering the latent structure from many observed variables is an important
yet challenging learning task. Existing approaches for discovering latent
structures often require the unknown number of hidden states as an input. In
this paper, we propose a quartet based approach which is \emph{agnostic} to
this number. The key contribution is a novel rank characterization of the
tensor associated with the marginal distribution of a quartet. This
characterization allows us to design a \emph{nuclear norm} based test for
resolving quartet relations. We then use the quartet test as a subroutine in a
divide-and-conquer algorithm for recovering the latent tree structure. Under
mild conditions, the algorithm is consistent and its error probability decays
exponentially with increasing sample size. We demonstrate that the proposed
approach compares favorably to alternatives. In a real world stock dataset, it
also discovers meaningful groupings of variables, and produces a model that
fits the data better
- …