33,217 research outputs found

    Regularity of Edge Ideals and Their Powers

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    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  reg I(G)\text{ reg } I(G) and the asymptotic linear function  reg I(G)q\text{ reg } I(G)^q, for q1,q \geq 1, in terms of combinatorial data of the given graph G.G.Comment: 31 pages, 15 figure

    Upper bounds on the k-forcing number of a graph

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    Given a simple undirected graph GG and a positive integer kk, the kk-forcing number of GG, denoted Fk(G)F_k(G), 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 kk non-colored neighbors, then each of its non-colored neighbors becomes colored. When k=1k=1, this is equivalent to the zero forcing number, usually denoted with Z(G)Z(G), 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 kk-forcing number. Notable among these, we show that if GG is a graph with order n2n \ge 2 and maximum degree Δk\Delta \ge k, then Fk(G)(Δk+1)nΔk+1+min{δ,k}F_k(G) \le \frac{(\Delta-k+1)n}{\Delta - k + 1 +\min{\{\delta,k\}}}. This simplifies to, for the zero forcing number case of k=1k=1, Z(G)=F1(G)ΔnΔ+1Z(G)=F_1(G) \le \frac{\Delta n}{\Delta+1}. Moreover, when Δ2\Delta \ge 2 and the graph is kk-connected, we prove that Fk(G)(Δ2)n+2Δ+k2F_k(G) \leq \frac{(\Delta-2)n+2}{\Delta+k-2}, which is an improvement when k2k\leq 2, and specializes to, for the zero forcing number case, Z(G)=F1(G)(Δ2)n+2Δ1Z(G)= F_1(G) \le \frac{(\Delta -2)n+2}{\Delta -1}. These results resolve a problem posed by Meyer about regular bipartite circulant graphs. Finally, we present a relationship between the kk-forcing number and the connected kk-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

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    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
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