28,817 research outputs found
On the extremal properties of the average eccentricity
The eccentricity of a vertex is the maximum distance from it to another
vertex and the average eccentricity of a graph is the mean value
of eccentricities of all vertices of . The average eccentricity is deeply
connected with a topological descriptor called the eccentric connectivity
index, defined as a sum of products of vertex degrees and eccentricities. In
this paper we analyze extremal properties of the average eccentricity,
introducing two graph transformations that increase or decrease .
Furthermore, we resolve four conjectures, obtained by the system AutoGraphiX,
about the average eccentricity and other graph parameters (the clique number,
the Randi\' c index and the independence number), refute one AutoGraphiX
conjecture about the average eccentricity and the minimum vertex degree and
correct one AutoGraphiX conjecture about the domination number.Comment: 15 pages, 3 figure
Colouring random graphs and maximising local diversity
We study a variation of the graph colouring problem on random graphs of
finite average connectivity. Given the number of colours, we aim to maximise
the number of different colours at neighbouring vertices (i.e. one edge
distance) of any vertex. Two efficient algorithms, belief propagation and
Walksat are adapted to carry out this task. We present experimental results
based on two types of random graphs for different system sizes and identify the
critical value of the connectivity for the algorithms to find a perfect
solution. The problem and the suggested algorithms have practical relevance
since various applications, such as distributed storage, can be mapped onto
this problem.Comment: 10 pages, 10 figure
On the distance spectrum and distance-based topological indices of central vertex-edge join of three graphs
Topological indices are molecular descriptors that describe the properties of
chemical compounds. These topological indices correlate specific
physico-chemical properties like boiling point, enthalpy of vaporization,
strain energy, and stability of chemical compounds. This article introduces a
new graph operation based on central graph called central vertex-edge join and
provides its results related to graph invariants like eccentric-connectivity
index, connective eccentricity index, total-eccentricity index, average
eccentricity index, Zagreb eccentricity indices, eccentric geometric-arithmetic
index, eccentric atom-bond connectivity index, and Wiener index. Also, we
discuss the distance spectrum of the central vertex-edge join of three regular
graphs. Furthermore, we obtain new families of -equienergetic graphs, which
are non -cospectral
Using Network Analysis to Contrast Three Models of Student Forum Discussions
There is much research about how actors and events in social networks affect each other. In this research, three network models were created for discussion forums in three semesters of undergraduate general physics courses. This study seeks to understand what social network measures are most telling of a online forum classroom dynamic. That is, I wanted to understand more about things like what students are most central to the networks and whether this is consistent across different network models. I also wanted to better understand how students may or may not group together. What relationships (student to student, student to instructor, etc.) are formed, centralization, various clustering and correlation coefficients, and how participation in a forum unfolds were all things that were examined in this data set. Network model construction and measuring how these constructions may affect student interactions was another focus of this study. These attributes are analyzed among individual semesters, but also compared/contrasted across all three, to see if they maintain across different network models. It was found that in general as models increase in connectivity, a rise in network measures like centralization and average degree was observed. A drop in network measure values such as average vertex-vertex distance and diameter was also seen. Finally, it was discovered that changing a model from undirected to directed made an appreciable change in average degree outcomes. Overall, this research gave an appreciation of different network model construction and how different network measures may help describe social networks. It was discovered that centralization metrics may be more telling of social networks than what was anticipated. Average degree, average vertex to vertex distance and diameter followed trends we would expect to see. Other measures looked into were transitivity, average Barrat coefficient and degree correlation coefficient
Graph measures and network robustness
Network robustness research aims at finding a measure to quantify network
robustness. Once such a measure has been established, we will be able to
compare networks, to improve existing networks and to design new networks that
are able to continue to perform well when it is subject to failures or attacks.
In this paper we survey a large amount of robustness measures on simple,
undirected and unweighted graphs, in order to offer a tool for network
administrators to evaluate and improve the robustness of their network. The
measures discussed in this paper are based on the concepts of connectivity
(including reliability polynomials), distance, betweenness and clustering. Some
other measures are notions from spectral graph theory, more precisely, they are
functions of the Laplacian eigenvalues. In addition to surveying these graph
measures, the paper also contains a discussion of their functionality as a
measure for topological network robustness
Small-worlds: How and why
We investigate small-world networks from the point of view of their origin.
While the characteristics of small-world networks are now fairly well
understood, there is as yet no work on what drives the emergence of such a
network architecture. In situations such as neural or transportation networks,
where a physical distance between the nodes of the network exists, we study
whether the small-world topology arises as a consequence of a tradeoff between
maximal connectivity and minimal wiring. Using simulated annealing, we study
the properties of a randomly rewired network as the relative tradeoff between
wiring and connectivity is varied. When the network seeks to minimize wiring, a
regular graph results. At the other extreme, when connectivity is maximized, a
near random network is obtained. In the intermediate regime, a small-world
network is formed. However, unlike the model of Watts and Strogatz (Nature {\bf
393}, 440 (1998)), we find an alternate route to small-world behaviour through
the formation of hubs, small clusters where one vertex is connected to a large
number of neighbours.Comment: 20 pages, latex, 9 figure
The topological structure of 2D disordered cellular systems
We analyze the structure of two dimensional disordered cellular systems
generated by extensive computer simulations. These cellular structures are
studied as topological trees rooted on a central cell or as closed shells
arranged concentrically around a germ cell. We single out the most significant
parameters that characterize statistically the organization of these patterns.
Universality and specificity in disordered cellular structures are discussed.Comment: 18 Pages LaTeX, 16 Postscript figure
Minimizing Unsatisfaction in Colourful Neighbourhoods
Colouring sparse graphs under various restrictions is a theoretical problem
of significant practical relevance. Here we consider the problem of maximizing
the number of different colours available at the nodes and their
neighbourhoods, given a predetermined number of colours. In the analytical
framework of a tree approximation, carried out at both zero and finite
temperatures, solutions obtained by population dynamics give rise to estimates
of the threshold connectivity for the incomplete to complete transition, which
are consistent with those of existing algorithms. The nature of the transition
as well as the validity of the tree approximation are investigated.Comment: 28 pages, 12 figures, substantially revised with additional
explanatio
- …