In this paper, we study the controllability and stabilizability properties of the Kolmogorov forward equation of a continuous time Markov chain (CTMC) evolving on a finite state space, using the transition rates as the control parameters. Firstly, we prove small-time local and global controllability from and to strictly positive equilibrium configurations when the underlying graph is strongly c...

We s(ate the following conjecture and prove it for the case where q is a proper prime power: Let A be a nonsingular n by n matrix over the finite fieM GF~, q~-4, then there exists a vector x in (GFa) ~ such that both x and ~4x have no zero component. In this note we consider-the following conjecture: Conjecture 1. Let A be a nonsingular n by n matrix over the finite field GFa, q~_4, then there ...

A famous and wide-open problem, going back to at least the early 1970s, concerns classification of chromatic polynomials graphs. Toward this one may ask for necessary inequalities among coefficients a polynomial, we contribute such when polynomial $$\chi _G(n)=\chi ^*_0\left( {\begin{array}{c}n+d\\ d\end{array}}\right) +\chi ^*_1\left( {\begin{array}{c}n+d-1\\ +\dots ^*_d\left( {\begin{array}{c...

let $r$ be a commutative ring with zero-divisor set $z(r)$. the total graph of $r$, denoted by$t(gamma(r))$, is the simple (undirected) graph with vertex set  $r$ where two distinct vertices areadjacent if their sum lies in $z(r)$.this work considers minimum zero-sum $k$-flows for $t(gamma(r))$.both for $vert rvert$ even and the case when $vert rvert$ is odd and $z(g)$ is an ideal of $r$it is s...

It is proved in this paper that every bipartite graphic sequence with the minimum degree δ ≥ 2 has a realization that admits a nowhere-zero 4-flow. This result implies a conjecture originally proposed by Keedwell (1993) and reproposed by Cameron (1999) about simultaneous edgecolorings and critical partial Latin squares. ∗Mathematics Subject Classification (2000): 05C15, 05B15, 05C38, 05C70, 05C07.

For an arbitrary ring $R$, the zero-divisor graph of $R$, denoted by $Gamma (R)$, is an undirected simple graph that its vertices are all nonzero zero-divisors of $R$ in which any two vertices $x$ and $y$ are adjacent if and only if either $xy=0$ or $yx=0$. It is well-known that for any commutative ring $R$, $Gamma (R) cong Gamma (T(R))$ where $T(R)$ is the (total) quotient ring of $R$. In this...

We extend a well known uncrossing technique in linear programs (LPs) to work with partitions. Using this technique, we tie together three previously unrelated papers on Steiner trees, by showing that the following three values are equal: (1) the objective value of a subtour based LP by Polzin and Vahdati Daneshmand; (2) the objective value of a partition based LP by Könemann and Tan; (3) a “max...

In this paper, we unify the Markov theory of a variety of different types of graphs used in graphical Markov models by introducing the class of loopless mixed graphs, and show that all independence models induced by m-separation on such graphs are compositional graphoids. We focus in particular on the subclass of ribbonless graphs which as special cases include undirected graphs, bidirected gra...