A graph G is Nm-locally connected if for every vertex v in G, the vertices not equal to v and with distance at most m to v induce a connected subgraph in G. We show that both connectedN2-locally connected claw-free graph and 3-edge-connected N3-locally connected claw-free graph have connected even [2, 4]-factors, which settle a conjecture by Li in [6].

Subtraction-free computational complexity is the version of arithmetic circuit complexity that allows only three operations: addition, multiplication, and division. We use cluster transformations to design efficient subtraction-free algorithms for computing Schur functions and their skew, double, and supersymmetric analogues, thereby generalizing earlier results by P. Koev. We develop such algo...

We propose a novel a framework for deriving approximations for intractable probabilistic models. This framework is based on a free energy (negative log marginal likelihood) and can be seen as a generalization of adaptive TAP [1, 2, 3] and expectation propagation (EP) [4, 5]. The free energy is constructed from two approximating distributions which encode different aspects of the intractable mod...

For a graph H and an integer k ≥ 2, let σk(H) denote the minimum degree sum of k independent vertices of H . We prove that if a connected claw-free graph G satisfies σk+1(G) ≥ |G| − k, then G has a spanning tree with at most k leaves. We also show that the bound |G| − k is sharp and discuss the maximum degree of the required spanning trees.

– A complete characterization of real networks requires us to understand the consequences of the uneven interaction strengths between a system’s components. Here we use minimum spanning trees (MSTs) to explore the effect of correlations between link weights and network topology on scale-free networks. Solely by changing the nature of the correlations between weights and network topology, the st...

A graph G is Nm-locally connected if for every vertex v in G, the vertices not equal to v and with distance at most m to v induce a connected subgraph in G. We show that both connected N2-locally connected claw-free graph and 3-edge-connected N3-locally connected claw-free graph have connected even [2, 4]-factors, which settle a conjecture by Li in [6].

We present a framework for approximate inference in probabilistic data models which is based on free energies. The free energy is constructed from two approximating distributions which encode different aspects of the intractable model. Consistency between distributions is required on a chosen set of moments. We find good performance using sets of moments which either specify factorized nodes or...

For an integer k with k ≥ 2, a k-tree (resp. a k-forest) is a tree (resp. forest) with maximum degree at most k. In this paper, we show that for any integer k with k ≥ 3, any connected K1,k+1-free graph has a spanning k-tree or a spanning k-forest with only large components.