Matrices of forests, analysis of networks, and ranking problems

The matrices of spanning rooted forests are studied as a tool for analysing the structure of networks and measuring their properties. The problems of revealing the basic bicomponents, measuring vertex proximity, and ranking from preference relations / sports competitions are considered. It is shown that the vertex accessibility measure based on spanning forests has a number of desirable properties. An interpretation for the stochastic matrix of out-forests in terms of information dissemination is given.Comment: 8 pages. This article draws heavily from arXiv:math/0508171. Published in Proceedings of the First International Conference on Information Technology and Quantitative Management (ITQM 2013). This version contains some corrections and addition

On packing dijoins in digraphs and weighted digraphs

In this paper, we make some progress in addressing Woodall's Conjecture, and the refuted Edmonds-Giles Conjecture on packing dijoins in unweighted and weighted digraphs. Let $D=(V,A)$ be a digraph, and let $w\in \mathbb{Z}^A_{\geq 0}$. Suppose every dicut has weight at least $\tau$, for some integer $\tau\geq 2$. Let $\rho(\tau,D,w):=\frac{1}{\tau}\sum_{v\in V} m_v$, where each $m_v$ is the integer in $\{0,1,\ldots,\tau-1\}$ equal to $w(\delta^+(v))-w(\delta^-(v))$ mod $\tau$. In this paper, we prove the following results, amongst others: (1) If $w={\bf 1}$, then $A$ can be partitioned into a dijoin and a $(\tau-1)$-dijoin. (2) If $\rho(\tau,D,w)\in \{0,1\}$, then there is an equitable $w$-weighted packing of dijoins of size $\tau$. (3) If $\rho(\tau,D,w)= 2$, then there is a $w$-weighted packing of dijoins of size $\tau$. (4) If $w={\bf 1}$, $\tau=3$, and $\rho(\tau,D,w)=3$, then $A$ can be partitioned into three dijoins. Each result is best possible: (1) and (4) do not hold for general $w$, (2) does not hold for $\rho(\tau,D,w)=2$ even if $w={\bf 1}$, and (3) does not hold for $\rho(\tau,D,w)=3$. The results are rendered possible by a \emph{Decompose, Lift, and Reduce procedure}, which turns $(D,w)$ into a set of \emph{sink-regular weighted $(\tau,\tau+1)$-bipartite digraphs}, each of which is a weighted digraph where every vertex is a sink of weighted degree $\tau$ or a source of weighted degree $\tau,\tau+1$, and every dicut has weight at least $\tau$. Our results give rise to a number of approaches for resolving Woodall's Conjecture, fixing the refuted Edmonds-Giles Conjecture, and the $\tau=2$ Conjecture for the clutter of minimal dijoins. They also show an intriguing connection to Barnette's Conjecture.Comment: 71 page

Effective Generation of Subjectively Random Binary Sequences

We present an algorithm for effectively generating binary sequences which would be rated by people as highly likely to have been generated by a random process, such as flipping a fair coin.Comment: Introduction and Section 6 revise

An extensive English language bibliography on graph theory and its applications, supplement 1

Graph theory and its applications - bibliography, supplement

Directed graphs, 2D state models, and characteristic polynomials of irreducible matrix pairs

AbstractThe definition and main properties of a 2D digraph, namely a directed graph with two kinds of arcs, are introduced. Under the assumption of strong connectedness, the analysis of its paths and cycles is performed, based on an integer matrix whose rows represent the compositions of all circuits, and on the corresponding row module. Natural constraints on the composition of the paths connecting each pair of vertices lead to the definition of a 2D strongly connected digraph. For a 2D digraph of this kind the set of vertices can be partitioned into disjoint 2D-imprimitivity classes, whose number and composition are strictly related to the structure of the row module. Irreducible matrix pairs, i.e. pairs endowed with a 2D strongly connected digraph, are subsequently discussed. Equivalent descriptions of irreducibility, naturally extending those available for a single irreducible matrix, are obtained. These refer to the free evolution of the 2D state models described by the pairs and to their characteristic polynomials. Finally, primitivity is viewed as a special case of irreducibility, and completely characterized in terms of 2D-digraphs, characteristic polynomials, and 2D system dynamics