Symbolic powers of monomial ideals and Cohen-Macaulay vertex-weighted digraphs

In this paper we study irreducible representations and symbolic Rees algebras of monomial ideals. Then we examine edge ideals associated to vertex-weighted oriented graphs. These are digraphs having no oriented cycles of length two with weights on the vertices. For a monomial ideal with no embedded primes we classify the normality of its symbolic Rees algebra in terms of its primary components. If the primary components of a monomial ideal are normal, we present a simple procedure to compute its symbolic Rees algebra using Hilbert bases, and give necessary and sufficient conditions for the equality between its ordinary and symbolic powers. We give an effective characterization of the Cohen--Macaulay vertex-weighted oriented forests. For edge ideals of transitive weighted oriented graphs we show that Alexander duality holds. It is shown that edge ideals of weighted acyclic tournaments are Cohen--Macaulay and satisfy Alexander dualityComment: Special volume dedicated to Professor Antonio Campillo, Springer, to appea

Exponents of 2-regular digraphs

AbstractA digraph G is called primitive if for some positive integer k, there is a walk of length exactly k from each vertex u to each vertex v (possibly u again). If G is primitive, the smallest such k is called the exponent of G, denoted by exp(G). A digraph G is said to be r-regular if each vertex in G has outdegree and indegree exactly r.It is proved that if G is a primitive 2-regular digraph with n vertices, then exp(G)⩽(n−1)2/4+1. Also all 2-regular digraphs with exponents attaining the bound are characterized. This supports a conjecture made by Shen and Greegory

Enumerating Hamiltonian Cycles in A 2-connected Regular Graph Using Planar Cycle Bases

Planar fundamental cycle basis belong to a 2-connected simple graph is used for enumerating Hamiltonian cycles contained in the graph. This is because a fun- damental cycle basis is easily constructed. Planar basis is chosen since it has a weighted induced graph whose values are limited to 1. Hence making it is possible to be used in the Hamiltonian cycle enumeration procedures efficiently. In this paper a Hamiltonian cycle enumeration scheme is obtained through two stages. Firstly, i cycles out of m bases cycles are determined using an appropriate con- structed constraint. Secondly, to search all Hamiltonian cycles which are formed by the combination of i basis cycles obtained in the first stage efficiently. This ef- ficiency is achieved through the generation of a class of objects consisting of Ill-bit binary strings which is a representation of i cycle combinations between m cycle basis cycle

Toward a Theory of Markov Influence Systems and their Renormalization

Nonlinear Markov chains are probabilistic models commonly used in physics, biology, and the social sciences. In "Markov influence systems" (MIS), the transition probabilities of the chains change as a function of the current state distribution. This work introduces a renormalization framework for analyzing the dynamics of MIS. It comes in two independent parts: first, we generalize the standard classification of Markov chain states to the dynamic case by showing how to "parse" graph sequences. We then use this framework to carry out the bifurcation analysis of a few important MIS families. In particular, we show that irreducible MIS are almost always asymptotically periodic. We also give an example of "hyper-torpid" mixing, where a stationary distribution is reached in super-exponential time, a timescale that cannot be achieved by any Markov chain