178,539 research outputs found
About Vertex Mappings
In [6] partial graph mappings were formalized in the Mizar system [3]. Such mappings map some vertices and edges of a graph to another while preserving adjacency. While this general approach is appropriate for the general form of (multidi)graphs as introduced in [7], a more specialized version for graphs without parallel edges seems convenient. As such, partial vertex mappings preserving adjacency between the mapped verticed are formalized [email protected] Gutenberg University, Mainz, GermanyIan Anderson. A first course in discrete mathematics. Springer Undergraduate Mathematics Series. Springer, London, 2001. ISBN 1-85233-236-0.Grzegorz Bancerek, Czesław Bylinski, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, and Karol Pak. The role of the Mizar Mathematical Library for interactive proof development in Mizar. Journal of Automated Reasoning, 61(1):9–32, 2018. doi:10.1007/s10817-017-9440-6.Adam Grabowski, Artur Korniłowicz, and Adam Naumowicz. Four decades of Mizar. Journal of Automated Reasoning, 55(3):191–198, 2015. doi:10.1007/s10817-015-9345-1.Pavol Hell and Jaroslav Nesetril. Graphs and homomorphisms. Oxford Lecture Series in Mathematics and Its Applications; 28. Oxford University Press, Oxford, 2004. ISBN 0-19-852817-5.Ulrich Knauer. Algebraic graph theory: morphisms, monoids and matrices, volume 41 of De Gruyter Studies in Mathematics. Walter de Gruyter, 2011.Sebastian Koch. About graph mappings. Formalized Mathematics, 27(3):261–301, 2019. doi:10.2478/forma-2019-0024.Gilbert Lee and Piotr Rudnicki. Alternative graph structures. Formalized Mathematics, 13(2):235–252, 2005.Robin James Wilson. Introduction to Graph Theory. Oliver & Boyd, Edinburgh, 1972. ISBN 0-05-002534-1.27330331
On the algorithmic complexity of twelve covering and independence parameters of graphs
The definitions of four previously studied parameters related to total coverings and total matchings of graphs can be restricted, thereby obtaining eight parameters related to covering and independence, each of which has been studied previously in some form. Here we survey briefly results concerning total coverings and total matchings of graphs, and consider the aforementioned 12 covering and independence parameters with regard to algorithmic complexity. We survey briefly known results for several graph classes, and obtain new NP-completeness results for the minimum total cover and maximum minimal total cover problems in planar graphs, the minimum maximal total matching problem in bipartite and chordal graphs, and the minimum independent dominating set problem in planar cubic graphs
A graph-based mathematical morphology reader
This survey paper aims at providing a "literary" anthology of mathematical
morphology on graphs. It describes in the English language many ideas stemming
from a large number of different papers, hence providing a unified view of an
active and diverse field of research
A survey of statistical network models
Networks are ubiquitous in science and have become a focal point for
discussion in everyday life. Formal statistical models for the analysis of
network data have emerged as a major topic of interest in diverse areas of
study, and most of these involve a form of graphical representation.
Probability models on graphs date back to 1959. Along with empirical studies in
social psychology and sociology from the 1960s, these early works generated an
active network community and a substantial literature in the 1970s. This effort
moved into the statistical literature in the late 1970s and 1980s, and the past
decade has seen a burgeoning network literature in statistical physics and
computer science. The growth of the World Wide Web and the emergence of online
networking communities such as Facebook, MySpace, and LinkedIn, and a host of
more specialized professional network communities has intensified interest in
the study of networks and network data. Our goal in this review is to provide
the reader with an entry point to this burgeoning literature. We begin with an
overview of the historical development of statistical network modeling and then
we introduce a number of examples that have been studied in the network
literature. Our subsequent discussion focuses on a number of prominent static
and dynamic network models and their interconnections. We emphasize formal
model descriptions, and pay special attention to the interpretation of
parameters and their estimation. We end with a description of some open
problems and challenges for machine learning and statistics.Comment: 96 pages, 14 figures, 333 reference
EEG Resting-State Brain Topological Reorganization as a Function of Age
Resting state connectivity has been increasingly studied to investigate the effects of aging on the brain. A reduced organization
in the communication between brain areas was demonstrated b
y combining a variety of different imaging technologies (fMRI,
EEG, and MEG) and graph theory. In this paper, we propose a methodology to get new insights into resting state connectivity and
its variations with age, by combining advanced techniques of effective connectivity estimation, graph theoretical approach, and
classification by SVM method. We analyzed high density EEG signal
srecordedatrestfrom71healthysubjects(age:20–63years).
Weighted and directed connectivity was computed by means of Partial Directed Coherence based on a General Linear Kalman filter
approach. To keep the information collected by the estimator, weighted and directed graph indices were extracted from the resulting
networks. A relation between brain network properties and age of the subject was found, indicating a tendency of the network to
randomly organize increasing with age. This result is also confirmed dividing the whole population into two subgroups according
to the age (young and middle-aged adults): significant differences exist in terms of network organization measures. Classification
of the subjects by means of such indices returns an accuracy greater than 80
- …