We derive closed form expressions and limiting formulae for a variety of functions of a permutation resulting from repeated riffle shuffles. The results allow new formulae and approximations for the number of permutations inS n with given... more
We derive closed form expressions and limiting formulae for a variety of functions of a permutation resulting from repeated riffle shuffles. The results allow new formulae and approximations for the number of permutations inS n with given cycle type and number of descents. The theorems are derived from a bijection discovered by Gessel. A self-contained proof of Gessel's result is given.
A set of vertices $W$ resolves a graph $G$ if every vertex is uniquely determined by its vector of distances to the vertices in $W$. A metric dimension of $G$ is the minimum cardinality of a resolving set of $G$. A bipartite graph G(n,n)... more
A set of vertices $W$ resolves a graph $G$ if every vertex is uniquely determined by its vector of distances to the vertices in $W$. A metric dimension of $G$ is the minimum cardinality of a resolving set of $G$. A bipartite graph G(n,n) is a graph whose vertex set $V$ can be partitioned into two subsets $V_1$ and $V_2,$ with $|V_1|=|V_2|=n,$ such that every edge of $G$ joins $V_1$ and $V_2$. The graph $G$ is called $k$-regular if every vertex of $G$ is adjacent to $k$ other vertices. In this paper, we determine the metric dimension of $k$-regular bipartite graphs G(n,n) where $k=n-1$ or $k=n-2$.
A set S of vertices in a graph G(V, E) is called a dominating set if every vertex v ∈ V is either an element of S or is adjacent to an element of S. A set S of vertices in a graph G(V, E) is called a total dominating set if every vertex v... more
A set S of vertices in a graph G(V, E) is called a dominating set if every vertex v ∈ V is either an element of S or is adjacent to an element of S. A set S of vertices in a graph G(V, E) is called a total dominating set if every vertex v ∈ V is adjacent to an element of S. The domination number of a graph G denoted by γ(G) is the minimum cardinality of a dominating set in G. Respectively the total domination number of a graph G denoted by γt(G) is the minimum cardinality of a total dominating set in G. An upper bound for γt(G) which has been achieved by Cockayne and et al. in [1] is: for any graph G with no isolated vertex and maximum degree Δ(G) and n vertices, γt(G) ≤ n − Δ(G )+1 . Here we characterize bipartite graphs and trees which achieve this upper bound. Further we present some another upper and lower bounds for γt(G). Also, for circular complete graphs, we determine the value of γt(G).
The edge-chromatic number of the complete graph on n vertices, X'(Kn), is well-known and simple to find. This number has applications in round-robin tournaments and what we will call the "efficient handshake" problem: namely, it gives... more
The edge-chromatic number of the complete graph on n vertices, X'(Kn), is well-known and simple
to find. This number has applications in round-robin tournaments and what we will call the "efficient
handshake" problem: namely, it gives the minimum number of periods needed to complete all n choose 2 interactions among a group of n members such that no member may be involved in two interactions at
once. However, algorithms for this most efficient sequence can be time-consuming to execute, particularly
in the case that the interaction is brief and dependent on physical arrangement, as in the case of a
handshake. This paper will describe an alternate method in terms of both handshaking and graph
theory, examine its time costs, and derive a simple means of comparing its efficiency to that of the
optimal algorithm.
We define a biclique to be the complement of a bipartite graph, consisting of two cliques joined by a number of edges. In this paper we study algebraic aspects of the chromatic polynomials of these graphs. We derive a formula for the... more
We define a biclique to be the complement of a bipartite graph, consisting
of two cliques joined by a number of edges. In this paper we study algebraic aspects
of the chromatic polynomials of these graphs. We derive a formula for the chromatic
polynomial of an arbitrary biclique, and use this to give certain conditions under which
two of the graphs have chromatic polynomials with the same splitting field. Finally,
we use a subfamily of bicliques to prove the cubic case of the α + n conjecture, by
showing that for any cubic integer α, there is a natural number n such that α + n is a
chromatic root.
In this paper, we deal with the notion of star coloring of graphs. A star coloring of an undirected graph G is a proper vertex coloring of G (i.e., no two neighbors are assigned the same color) such that any path of length 3 in G is not... more
In this paper, we deal with the notion of star coloring of graphs. A star coloring of an undirected graph G is a proper vertex coloring of G (i.e., no two neighbors are assigned the same color) such that any path of length 3 in G is not bicolored. We give the exact value of the star chromatic number of different families of graphs such as trees, cycles, complete bipartite graphs, outerplanar graphs and 2-dimensional grids. We also study and give bounds for the star chromatic number of other families of graphs, such as hypercubes, tori, d-dimensional grids, graphs with bounded treewidth and planar graphs.
We show that the dimer model on a bipartite graph on a torus gives rise to a quantum integrable system of special type - a cluster integrable system. The phase space of the classical system contains, as an open dense subset, the moduli... more
We show that the dimer model on a bipartite graph on a torus gives rise to a quantum integrable system of special type - a cluster integrable system. The phase space of the classical system contains, as an open dense subset, the moduli space of line bundles with connections on the graph. The sum of Hamiltonians is essentially the partition function of the dimer model. Any graph on a torus gives rise to a bipartite graph on the torus. We show that the phase space of the latter has a Lagrangian subvariety. We identify it with the space parametrizing resistor networks on the original graph.We construct several discrete quantum integrable systems.
A subset of vertices in a graph is called a dissociation set if it induces a subgraph with a vertex degree of at most 1. The maximum dissociation set problem, i.e., the problem of finding a dissociation set of maximum size in a given... more
A subset of vertices in a graph is called a dissociation set if it induces a subgraph with a vertex degree of at most 1. The maximum dissociation set problem, i.e., the problem of finding a dissociation set of maximum size in a given graph is known to be NP-hard for bipartite graphs. We show that the maximum dissociation set
Objectives: The current study is the first to examine the network structure of an encrypted online drug distribution network. It examines 1) the global network structure, 2) the local network structure, and 3) identifies those vendor... more
Objectives: The current study is the first to examine the network structure of an encrypted online drug distribution network. It examines 1) the global network structure, 2) the local network structure, and 3) identifies those vendor characteristics that best explain variation in the network structure. In doing so, it evaluates the role of trust in online drug markets. Methods: The study draws on a unique dataset of transaction level data from an encrypted online drug market. Structural measures and community detection analysis are used to characterize and investigate the network structure. Exponential random graph modeling is used to evaluate which vendor characteristics explain variation in purchasing patterns. Results: Vendors’ trustworthiness explains more variation in the overall network structure than the affordability of vendor products or the diversity of vendor product listings. This results in a highly localized network structure with a few key vendors accounting for most transactions. Conclusions: The results indicate that vendors’ trustworthiness is a better predictor of vendor selection than product diversity or affordability. These results illuminate the internal market dynamics that sustain digital drug markets and highlight the importance of examining how new anonymizing technologies shape global drug distribution networks.
The sum-product algorithm (belief/probability propagation) can be naturally mapped into analog transistor circuits. These circuits enable the construction of analog-VLSI decoders for turbo codes, low-density parity-check codes, and... more
The sum-product algorithm (belief/probability propagation) can be naturally mapped into analog transistor circuits. These circuits enable the construction of analog-VLSI decoders for turbo codes, low-density parity-check codes, and similar codes.
In this paper we explore the biclique structure of a biconvex bipartite graph G. We define two concatenation operators on bicliques of G. According to these operations, we show that G can be decomposed into two chain graphs G L and G R ,... more
In this paper we explore the biclique structure of a biconvex bipartite graph G. We define two concatenation operators on bicliques of G. According to these operations, we show that G can be decomposed into two chain graphs G L and G R , and a bipartite permutation graph G P . Using this representation, we propose linear-time algorithms for the treewidth and pathwidth problems on biconvex bipartite graphs.
