We characterize the rank of edge connection matrices of partition functions of real vertex models, as the dimension of the homogeneous components of the algebra of @math -invariant tensors. Here @math is the sub- group of the real orthogonal group that stabilizes the vertex model. This answers a question of Bal 'azs Szegedy from 2007.
Let @math be strictly positive and let @math be symmetric. Following de la Harpe and Jones @cite_6 we call the pair @math a . Let @math be the collection of all graphs, allowing multiple edges but no loops or circles. The of @math is the graph parameter @math defined by for @math .
@cite_6
2070948578 2950966800 2610522381 2110596924
Let @math be a 3-uniform hypergraph of order @math and size @math , and let @math be a subset of vertices of @math . The set @math is a strong transversal in @math if @math contains at least two vertices from every edge of @math . The strong transversal number @math of @math is the minimum size of a strong transversal in @math . We show that @math , and we characterize the hypergraphs that achieve equality in this bound. In particular, we show that the Fano plane is the only connected 3-uniform hypergraph @math of order @math and size @math that achieves equality in this bound. A set @math of vertices in a graph @math is a double total dominating set of @math if every vertex of @math is adjacent to at least two vertices in @math . The minimum cardinality of a double total dominating set of @math is the double total domination number @math of @math . Let @math be a connected graph of order @math with minimum degree at least three. As an application of our hypergraph results, we show that @math with equality if and only if @math is the Heawood graph (equivalently, the incidence bipartite graph of the Fano plane). Further if @math is not the Heawood graph, we show that @math , while if @math is a cubic graph different from the Heawood graph, we show that @math , and this bound is sharp. A graph @math is called @math -full if @math contains a subgraph @math of @math edges such that, for any non-empty @math , @math holds. Here, @math denotes the set of vertices incident to @math . It is known that the family of edge sets of @math -full graphs forms a family of matroid, known as the sparsity matroid of @math . In this paper, we give a constant-time approximation algorithm for the rank of the sparsity matroid of a degree-bounded undirected graph. This leads to a constant-time tester for @math -fullness in the bounded-degree model, (i.e., we can decide with high probability whether an input graph satisfies a property @math or far from @math ). Depending on the values of @math and @math , it can test various properties of a graph such as connectivity, rigidity, and how many spanning trees can be packed. Based on this result, we also propose a constant-time tester for @math -edge-connected-orientability in the bounded-degree model, where an undirected graph @math is called @math -edge-connected-orientable if there exists an orientation @math of @math with a vertex @math such that @math contains @math arc-disjoint dipaths from @math to each vertex @math and @math arc-disjoint dipaths from each vertex @math to @math . A tester is called a one-sided error tester for @math if it always accepts a graph satisfying @math . We show, for @math and (proper) @math , any one-sided error tester for @math -fullness and @math -edge-connected-orientability requires @math queries. For a positive parameter @math , the @math -bounded distance between a pair of vertices @math in a weighted undirected graph @math is the length of the shortest @math path in @math with at most @math edges, aka hops . For @math as above and @math , a @math -hopset of @math is a graph @math on the same vertex set, such that all distances in @math are @math -approximated by @math -bounded distances in @math . Hopsets are a fundamental graph-theoretic and graph-algorithmic construct, and they are widely used for distance-related problems in a variety of computational settings. Currently existing constructions of hopsets produce hopsets either with @math edges, or with a hopbound @math . In this paper we devise a construction of linear-size hopsets with hopbound @math . This improves the previous bound almost exponentially. We also devise efficient implementations of our construction in PRAM and distributed settings. The only existing PRAM algorithm EN16 for computing hopsets with a constant (i.e., independent of @math ) hopbound requires @math time. We devise a PRAM algorithm with polylogarithmic running time for computing hopsets with a constant hopbound, i.e., our running time is exponentially better than the previous one. Moreover, these hopsets are also significantly sparser than their counterparts from EN16 . We use our hopsets to devise a distributed routing scheme that exhibits near-optimal tradeoff between individual memory requirement @math of vertices throughout preprocessing and routing phases of the algorithm, and stretch @math , along with a near-optimal construction time @math , where @math is the hop-diameter of the input graph. Suppose that @math is a sequence of finite connected planar graphs, and in each @math a special vertex, called the root, is chosen randomly-uniformly. We introduce the notion of a distributional limit @math of such graphs. Assume that the vertex degrees of the vertices in @math are bounded, and the bound does not depend on @math . Then after passing to a subsequence, the limit exists, and is a random rooted graph @math . We prove that with probability one @math is recurrent. The proof involves the Circle Packing Theorem. The motivation for this work comes from the theory of random spherical triangulations.