Paths in r-partite self-complementary graphs

AbstractThis paper aims at finding best possible paths in r-partite self-complementary (r-p.s c.) graphs G(r). It is shown that, every connected bi-p.s.c. graphs G(2) of order p. with a bi-partite complementing permutation (bi-p.c.p) σ having mixed cycles, has a (p-3)-path and this result is best possible. Further, if the graph induced on each cycle of bi-p.c.p. of G(2) is connected then G(2) has a hamiltonian path. Lastly the fact that every r-p.s.c graph with an r-partite of σ has non-empty intersection with at least four partitions of G(r), has a hamiltonian path, is established. The graph obtained from G(r) by adding a vertex u constituting (r + 1)-st partition of G(r), which is the fixed point of σ∗ = (u)σ also has a hamiltonian path The last two results generalize the result that every self-complementary graph has a hamiltonian path

Random graph states, maximal flow and Fuss-Catalan distributions

For any graph consisting of $k$ vertices and $m$ edges we construct an ensemble of random pure quantum states which describe a system composed of $2m$ subsystems. Each edge of the graph represents a bi-partite, maximally entangled state. Each vertex represents a random unitary matrix generated according to the Haar measure, which describes the coupling between subsystems. Dividing all subsystems into two parts, one may study entanglement with respect to this partition. A general technique to derive an expression for the average entanglement entropy of random pure states associated to a given graph is presented. Our technique relies on Weingarten calculus and flow problems. We analyze statistical properties of spectra of such random density matrices and show for which cases they are described by the free Poissonian (Marchenko-Pastur) distribution. We derive a discrete family of generalized, Fuss-Catalan distributions and explicitly construct graphs which lead to ensembles of random states characterized by these novel distributions of eigenvalues.Comment: 37 pages, 24 figure

Hipergráfok = Hypergraphs

A projekt célkitűzéseit sikerült megvalósítani. A négy év során több mint száz kiváló eredmény született, amiből eddig 84 dolgozat jelent meg a téma legkiválóbb folyóirataiban, mint Combinatorica, Journal of Combinatorial Theory, Journal of Graph Theory, Random Graphs and Structures, stb. Számos régóta fennálló sejtést bebizonyítottunk, egész régi nyitott problémát megoldottunk hipergráfokkal kapcsolatban illetve kapcsolódó területeken. A problémák némelyike sok éve, olykor több évtizede nyitott volt. Nem egy közvetlen kutatási eredmény, de szintén bizonyos értékmérő, hogy a résztvevők egyike a Norvég Királyi Akadémia tagja lett és elnyerte a Steele díjat. | We managed to reach the goals of the project. We achieved more than one hundred excellent results, 84 of them appeared already in the most prestigious journals of the subject, like Combinatorica, Journal of Combinatorial Theory, Journal of Graph Theory, Random Graphs and Structures, etc. We proved several long standing conjectures, solved quite old open problems in the area of hypergraphs and related subjects. Some of the problems were open for many years, sometimes for decades. It is not a direct research result but kind of an evaluation too that a member of the team became a member of the Norvegian Royal Academy and won Steele Prize

Independent Complementary Distance Pattern Uniform Graphs

AgraphG =(V,E) is called to be Smarandachely uniform k-graph for an integer k ≥ 1ifthereexistsM1,M2, ·· ·,Mk ⊂ V (G) such that fMi (u) ={d(u, v):v ∈ Mi} for ∀u ∈ V (G)−Mi is independent of the choice of u ∈ V (G)−Mi and integer i, 1 ≤ i ≤ k. Each such set Mi, 1 ≤ i ≤ k is called a CDPU set [6, 7]. Particularly, for k = 1, a Smarandachely uniform 1-graph is abbreviated to a complementary distance pattern uniform graph, i.e., CDPU graphs. This paper studies independent CDPU graphs