53 research outputs found

### Extending Utility Representations of Partial Orders

The problem is considered as to whether a monotone function defined on a
subset P of a Euclidean space can be strictly monotonically extended to the
whole space. It is proved that this is the case if and only if the function is
{\em separably increasing}. Explicit formulas are given for a class of
extensions which involves an arbitrary bounded increasing function. Similar
results are obtained for monotone functions that represent strict partial
orders on arbitrary abstract sets X. The special case where P is a Pareto
subset is considered.Comment: 15 page

### The graph bottleneck identity

A matrix $S=(s_{ij})\in{\mathbb R}^{n\times n}$ is said to determine a
\emph{transitional measure} for a digraph $G$ on $n$ vertices if for all
$i,j,k\in\{1,\...,n\},$ the \emph{transition inequality} $s_{ij} s_{jk}\le
s_{ik} s_{jj}$ holds and reduces to the equality (called the \emph{graph
bottleneck identity}) if and only if every path in $G$ from $i$ to $k$ contains
$j$. We show that every positive transitional measure produces a distance by
means of a logarithmic transformation. Moreover, the resulting distance
$d(\cdot,\cdot)$ is \emph{graph-geodetic}, that is, $d(i,j)+d(j,k)=d(i,k)$
holds if and only if every path in $G$ connecting $i$ and $k$ contains $j$.
Five types of matrices that determine transitional measures for a digraph are
considered, namely, the matrices of path weights, connection reliabilities,
route weights, and the weights of in-forests and out-forests. The results
obtained have undirected counterparts. In [P. Chebotarev, A class of
graph-geodetic distances generalizing the shortest-path and the resistance
distances, Discrete Appl. Math., URL
http://dx.doi.org/10.1016/j.dam.2010.11.017] the present approach is used to
fill the gap between the shortest path distance and the resistance distance.Comment: 12 pages, 18 references. Advances in Applied Mathematic

### Which Digraphs with Ring Structure are Essentially Cyclic?

We say that a digraph is essentially cyclic if its Laplacian spectrum is not
completely real. The essential cyclicity implies the presence of directed
cycles, but not vice versa. The problem of characterizing essential cyclicity
in terms of graph topology is difficult and yet unsolved. Its solution is
important for some applications of graph theory, including that in
decentralized control. In the present paper, this problem is solved with
respect to the class of digraphs with ring structure, which models some typical
communication networks. It is shown that the digraphs in this class are
essentially cyclic, except for certain specified digraphs. The main technical
tool we employ is the Chebyshev polynomials of the second kind. A by-product of
this study is a theorem on the zeros of polynomials that differ by one from the
products of Chebyshev polynomials of the second kind. We also consider the
problem of essential cyclicity for weighted digraphs and enumerate the spanning
trees in some digraphs with ring structure.Comment: 19 pages, 8 figures, Advances in Applied Mathematics: accepted for
publication (2010) http://dx.doi.org/10.1016/j.aam.2010.01.00

### 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

### Second-Order Agents on Ring Digraphs

The paper addresses the problem of consensus seeking among second-order
linear agents interconnected in a specific ring topology. Unlike the existing
results in the field dealing with one-directional digraphs arising in various
cyclic pursuit algorithms or two-directional graphs, we focus on the case where
some arcs in a two-directional ring graph are dropped in a regular fashion. The
derived condition for achieving consensus turns out to be independent of the
number of agents in a network.Comment: 6 pages, 10 figure

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