35 research outputs found
Spanning Forests and the Golden Ratio
For a graph G, let f_{ij} be the number of spanning rooted forests in which
vertex j belongs to a tree rooted at i. In this paper, we show that for a path,
the f_{ij}'s can be expressed as the products of Fibonacci numbers; for a
cycle, they are products of Fibonacci and Lucas numbers. The {\em doubly
stochastic graph matrix} is the matrix F=(f_{ij})/f, where f is the total
number of spanning rooted forests of G and n is the number of vertices in G. F
provides a proximity measure for graph vertices. By the matrix forest theorem,
F^{-1}=I+L, where L is the Laplacian matrix of G. We show that for the paths
and the so-called T-caterpillars, some diagonal entries of F (which provides a
measure of the self-connectivity of vertices) converge to \phi^{-1} or to
1-\phi^{-1}, where \phi is the golden ratio, as the number of vertices goes to
infinity. Thereby, in the asymptotic, the corresponding vertices can be
metaphorically considered as "golden introverts" and "golden extroverts,"
respectively. This metaphor is reinforced by a Markov chain interpretation of
the doubly stochastic graph matrix, according to which F equals the overall
transition matrix of a random walk with a random number of steps on G.Comment: 12 pages, 2 figures, 25 references. As accepted by Disc. Appl. Math.
(2007
Chip-firing may be much faster than you think
A new bound (Theorem \ref{thm:main}) for the duration of the chip-firing game
with chips on a -vertex graph is obtained, by a careful analysis of the
pseudo-inverse of the discrete Laplacian matrix of the graph. This new bound is
expressed in terms of the entries of the pseudo-inverse.
It is shown (Section 5) to be always better than the classic bound due to
Bj{\"o}rner, Lov\'{a}sz and Shor. In some cases the improvement is dramatic.
For instance: for strongly regular graphs the classic and the new bounds
reduce to and , respectively. For dense regular graphs -
- the classic and the new bounds reduce to
and , respectively.
This is a snapshot of a work in progress, so further results in this vein are
in the works
Similarities on Graphs: Kernels versus Proximity Measures
We analytically study proximity and distance properties of various kernels
and similarity measures on graphs. This helps to understand the mathematical
nature of such measures and can potentially be useful for recommending the
adoption of specific similarity measures in data analysis.Comment: 16 page
The graph bottleneck identity
A matrix is said to determine a
\emph{transitional measure} for a digraph on vertices if for all
the \emph{transition inequality} holds and reduces to the equality (called the \emph{graph
bottleneck identity}) if and only if every path in from to contains
. We show that every positive transitional measure produces a distance by
means of a logarithmic transformation. Moreover, the resulting distance
is \emph{graph-geodetic}, that is,
holds if and only if every path in connecting and contains .
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
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
Do logarithmic proximity measures outperform plain ones in graph clustering?
We consider a number of graph kernels and proximity measures including
commute time kernel, regularized Laplacian kernel, heat kernel, exponential
diffusion kernel (also called "communicability"), etc., and the corresponding
distances as applied to clustering nodes in random graphs and several
well-known datasets. The model of generating random graphs involves edge
probabilities for the pairs of nodes that belong to the same class or different
predefined classes of nodes. It turns out that in most cases, logarithmic
measures (i.e., measures resulting after taking logarithm of the proximities)
perform better while distinguishing underlying classes than the "plain"
measures. A comparison in terms of reject curves of inter-class and intra-class
distances confirms this conclusion. A similar conclusion can be made for
several well-known datasets. A possible origin of this effect is that most
kernels have a multiplicative nature, while the nature of distances used in
cluster algorithms is an additive one (cf. the triangle inequality). The
logarithmic transformation is a tool to transform the first nature to the
second one. Moreover, some distances corresponding to the logarithmic measures
possess a meaningful cutpoint additivity property. In our experiments, the
leader is usually the logarithmic Communicability measure. However, we indicate
some more complicated cases in which other measures, typically, Communicability
and plain Walk, can be the winners.Comment: 11 pages, 5 tables, 9 figures. Accepted for publication in the
Proceedings of 6th International Conference on Network Analysis, May 26-28,
2016, Nizhny Novgorod, Russi