4,209 research outputs found

    A complete solution to the spectrum problem for graphs with six vertices and up to nine edges

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    Let GG be a graph. A GG-design of order nn is a decomposition of the complete graph KnK_n into disjoint copies of GG. The existence problem of graph designs has been completely solved for all graphs with up to five vertices, and all graphs with six vertices and up to seven edges; and almost completely solved for all graphs with six vertices and eight edges leaving two cases of order 32 unsettled. Up to isomorphism there are 20 graphs with six vertices and nine edges (and no isolated vertex). The spectrum problem has been solved completely for 11 of these graphs, and partially for 2 of these graphs. In this article, the two missing graph designs for the six-vertex eight-edge graphs are constructed, and a complete solution to the spectrum problem for the six-vertex nine-edge graphs is given; completing the spectrum problem for all graphs with six vertices and up to nine edges

    Coloring decompositions of complete geometric graphs

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    A decomposition of a non-empty simple graph GG is a pair [G,P][G,P], such that PP is a set of non-empty induced subgraphs of GG, and every edge of GG belongs to exactly one subgraph in PP. The chromatic index χ′([G,P])\chi'([G,P]) of a decomposition [G,P][G,P] is the smallest number kk for which there exists a kk-coloring of the elements of PP in such a way that: for every element of PP all of its edges have the same color, and if two members of PP share at least one vertex, then they have different colors. A long standing conjecture of Erd\H{o}s-Faber-Lov\'asz states that every decomposition [Kn,P][K_n,P] of the complete graph KnK_n satisfies χ′([Kn,P])≤n\chi'([K_n,P])\leq n. In this paper we work with geometric graphs, and inspired by this formulation of the conjecture, we introduce the concept of chromatic index of a decomposition of the complete geometric graph. We present bounds for the chromatic index of several types of decompositions when the vertices of the graph are in general position. We also consider the particular case in which the vertices are in convex position and present bounds for the chromatic index of a few types of decompositions.Comment: 18 pages, 5 figure

    On the Pauli graphs of N-qudits

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    A comprehensive graph theoretical and finite geometrical study of the commutation relations between the generalized Pauli operators of N-qudits is performed in which vertices/points correspond to the operators and edges/lines join commuting pairs of them. As per two-qubits, all basic properties and partitionings of the corresponding Pauli graph are embodied in the geometry of the generalized quadrangle of order two. Here, one identifies the operators with the points of the quadrangle and groups of maximally commuting subsets of the operators with the lines of the quadrangle. The three basic partitionings are (a) a pencil of lines and a cube, (b) a Mermin's array and a bipartite-part and (c) a maximum independent set and the Petersen graph. These factorizations stem naturally from the existence of three distinct geometric hyperplanes of the quadrangle, namely a set of points collinear with a given point, a grid and an ovoid, which answer to three distinguished subsets of the Pauli graph, namely a set of six operators commuting with a given one, a Mermin's square, and set of five mutually non-commuting operators, respectively. The generalized Pauli graph for multiple qubits is found to follow from symplectic polar spaces of order two, where maximal totally isotropic subspaces stand for maximal subsets of mutually commuting operators. The substructure of the (strongly regular) N-qubit Pauli graph is shown to be pseudo-geometric, i. e., isomorphic to a graph of a partial geometry. Finally, the (not strongly regular) Pauli graph of a two-qutrit system is introduced; here it turns out more convenient to deal with its dual in order to see all the parallels with the two-qubit case and its surmised relation with the generalized quadrangle Q(4, 3), the dual ofW(3).Comment: 17 pages. Expanded section on two-qutrits, Quantum Information and Computation (2007) accept\'

    Counterbalancing for serial order carryover effects in experimental condition orders

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    Reactions of neural, psychological, and social systems are rarely, if ever, independent of previous inputs and states. The potential for serial order carryover effects from one condition to the next in a sequence of experimental trials makes counterbalancing of condition order an essential part of experimental design. Here, a method is proposed for generating counterbalanced sequences for repeated-measures designs including those with multiple observations of each condition on one participant and self-adjacencies of conditions. Condition ordering is reframed as a graph theory problem. Experimental conditions are represented as vertices in a graph and directed edges between them represent temporal relationships between conditions. A counterbalanced trial order results from traversing an Euler circuit through such a graph in which each edge is traversed exactly once. This method can be generalized to counterbalance for higher order serial order carryover effects as well as to create intentional serial order biases. Modern graph theory provides tools for finding other types of paths through such graph representations, providing a tool for generating experimental condition sequences with useful properties

    On the existence of small strictly Neumaier graphs

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    A Neumaier graph is a non-complete edge-regular graph containing a regular clique. In this work, we prove several results on the existence of small strictly Neumaier graphs. In particular, we present a theoretical proof of the uniqueness of the smallest Neumaier graphs with parameters (16,9,4;2,4)(16,9,4;2,4), we establish the existence of (25,12,5;2,5)(25,12,5;2,5), and we disprove the existence of Neumaier graphs with parameters (25,16,9;3,5)(25,16,9;3,5), (28,18,11;4,7)(28,18,11;4,7), (33,24,17;6,9)(33,24,17;6,9), (35,22,12;3,5)(35,22,12;3,5) and (55,34,18;3,5)(55,34,18;3,5). Our proofs use combinatorial techniques and a novel application of integer programming methods
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