165 research outputs found

    The enumeration of transitive self-complementary digraphs

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    Self-complementary graphs and Ramsey numbers Part I: the decomposition and construction of self-complementary graphs

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    AbstractA new method of studying self-complementary graphs, called the decomposition method, is proposed in this paper. Let G be a simple graph. The complement of G, denoted by Ḡ, is the graph in which V(Ḡ)=V(G); and for each pair of vertices u,v in Ḡ,uv∈E(Ḡ) if and only if uv∉E(G). G is called a self-complementary graph if G and Ḡ are isomorphic. Let G be a self-complementary graph with the vertex set V(G)={v1,v2,…,v4n}, where dG(v1)⩽dG(v2)⩽⋯⩽dG(v4n). Let H=G[v1,v2,…,v2n],H′=G[v2n+1,v2n+2,…,v4n] and H∗=G−E(H)−E(H′). Then G=H+H′+H∗ is called the decomposition of the self-complementary graph G.In part I of this paper, the fundamental properties of the three subgraphs H,H′ and H∗ of the self-complementary graph G are considered in detail at first. Then the method and steps of constructing self-complementary graphs are given. In part II these results will be used to study certain Ramsey number problems (see (II))

    Improved Cardinality Bounds for Rectangle Packing Representations

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    Axis-aligned rectangle packings can be characterized by the set of spatial relations that hold for pairs of rectangles (west, south, east, north). A representation of a packing consists of one satisfied spatial relation for each pair. We call a set of representations complete for n ∈ ℕ if it contains a representation of every packing of any n rectangles. Both in theory and practice, fastest known algorithms for a large class of rectangle packing problems enumerate a complete set R of representations. The running time of these algorithms is dominated by the (exponential) size of R. In this thesis, we improve the best known lower and upper bounds on the minimum cardinality of complete sets of representations. The new upper bound implies theoretically faster algorithms for many rectangle packing problems, for example in chip design, while the new lower bound imposes a limit on the running time that can be achieved by any algorithm following this approach. The proofs of both results are based on pattern-avoiding permutations. Finally, we empirically compute the minimum cardinality of complete sets of representations for small n. Our computations directly suggest two conjectures, connecting well-known Baxter permutations with the set of permutations avoiding an apparently new pattern, which in turn seem to generate complete sets of representations of minimum cardinality

    Master index of volumes 61–70

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    Transitivity correlation:A descriptive measure of network transitivity

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    This paper proposes that common measures for network transitivity, based on the enumeration of transitive triples, do not reflect the theoretical statements about transitivity they aim to describe. These statements are often formulated as comparative conditional probabilities, but these are not directly reflected by simple functions of enumerations. We think that a better approach is obtained by considering the probability of a tie between two randomly drawn nodes, conditional on selected features of the network. Two measures of transitivity based on correlation coefficients between the existence of a tie and the existence, or the number, of two-paths between the nodes are developed, and called "Transitivity Phi" and "Transitivity Correlation." Some desirable properties for these measures are studied and compared to existing clustering coefficients, in both random (Erdos-Renyi) and in stylized networks (windmills). Furthermore, it is shown that in a directed graph, under the condition of zero Transitivity Correlation, the total number of transitive triples is determined by four underlying features: size, density, reciprocity, and the covariance between in- and outdegrees. Also, it is demonstrated that plotting conditional probability of ties, given the number of two-paths, provides valuable insights into empirical regularities and irregularities of transitivity patterns

    Some identities for enumerators of circulant graphs

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    We establish analytically several new identities connecting enumerators of different types of circulant graphs of prime, twice prime and prime-squared orders. In particular, it is shown that the semi-sum of the number of undirected circulants and the number of undirected self-complementary circulants of prime order is equal to the number of directed self-complementary circulants of the same order. Keywords: circulant graph; cycle index; cyclic group; nearly doubled primes; Cunningham chain; self-complementary graph; tournament; mixed graphComment: 17 pages, 3 tables Categories: CO Combinatorics (NT Number Theory) Math Subject Class: 05C30; 05A19; 11A4
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