31,101 research outputs found

    On Middle Cube Graphs

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    We study a family of graphs related to the nn-cube. The middle cube graph of parameter k is the subgraph of Q2k−1Q_{2k-1} induced by the set of vertices whose binary representation has either k−1k-1 or kk number of ones. The middle cube graphs can be obtained from the well-known odd graphs by doubling their vertex set. Here we study some of the properties of the middle cube graphs in the light of the theory of distance-regular graphs. In particular, we completely determine their spectra (eigenvalues and their multiplicities, and associated eigenvectors)

    On middle cube graphs

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    We study a family of graphs related to the nn-cube. The middle cube graph of parameter k is the subgraph of Q2k−1Q_{2k-1} induced by the set of vertices whose binary representation has either k−1k-1 or kk number of ones. The middle cube graphs can be obtained from the well-known odd graphs by doubling their vertex set. Here we study some of the properties of the middle cube graphs in the light of the theory of distance-regular graphs. In particular, we completely determine their spectra (eigenvalues and their multiplicities, and associated eigenvectors).Postprint (author's final draft

    Daisies and Other Turan Problems

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    We make some conjectures about extremal densities of daisy-free families, where a `daisy' is a certain hypergraph. These questions turn out to be related to some Turan problems in the hypercube, but they are also natural in their own right

    Upper bounds on the size of 4- and 6-cycle-free subgraphs of the hypercube

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    In this paper we modify slightly Razborov's flag algebra machinery to be suitable for the hypercube. We use this modified method to show that the maximum number of edges of a 4-cycle-free subgraph of the n-dimensional hypercube is at most 0.6068 times the number of its edges. We also improve the upper bound on the number of edges for 6-cycle-free subgraphs of the n-dimensional hypercube from the square root of 2 - 1 to 0.3755 times the number of its edges. Additionally, we show that if the n-dimensional hypercube is considered as a poset, then the maximum vertex density of three middle layers in an induced subgraph without 4-cycles is at most 2.15121 times n choose n/2.Comment: 14 pages, 9 figure
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