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The metric dimension of small distance-regular and strongly regular graphs
[article]

2013
*
arXiv
*
pre-print

The metric

arXiv:1312.4973v1
fatcat:6njfd6p5tjhkrizn77jtgzhtbi
*dimension**of*Γ is the smallest size*of*a resolving set for Γ. ... In this paper, we present the results*of*computer calculations which have found the metric*dimension**of*all distance-regular*graphs*on up to 34 vertices, low-valency distance transitive*graphs*on up to ... Acknowledgements Most*of*this work was carried out while the author was a postdoctoral fellow at Ryerson University. ...##
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On embedding of graphs into euclidean spaces of small dimension

1992
*
Journal of combinatorial theory. Series B (Print)
*

APPLICATIONS We apply Theorem 1 to obtain upper bounds for d(G) for classes

doi:10.1016/0095-8956(92)90002-f
fatcat:s5kb6hyt6ba5tojo3ubwjmbs6u
*of**graphs*which admit orientations with K, I*small*. ... These results are corollaries*of*Theorem 1, the proof*of*which is based on random representations*of**graphs*. ...##
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Cheeger constant and first nonzero eigenvalue of the Laplacian of a graph of small dimension

Constante de Cheeger et première valeur propre non nulle du Laplacien d'un graphe de petite dimension

1993
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Czechoslovak Mathematical Journal
*

Constante de Cheeger et première valeur propre non nulle du Laplacien d'un graphe de petite dimension

Nous montrons que l'on peut appliquer cette méthode aux

doi:10.21136/cmj.1993.128408
fatcat:5pg6s6jnr5b55kfrvxhcit7jzm
*graphes*de petites*dimensions*et nous déterminons une borne inférieure pour la première valeur propre non nulle du Laplacien d'un*graphe*connexe ... Le résultat précédent montre que l'inégalité du théorème 1.2 est faible en petites*dimensions*. En effet, cette inégalité prévoit un domaine plutôt convexe, alors qu'il est concave en réalité. ...##
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The metric dimension of the incidence graphs of projective planes of small order

2020
*
The Australasian Journal of Combinatorics
*

We also determine the metric

dblp:journals/ajc/HegerST20
fatcat:wgpicvc6pre6raoozwqd3pjeeq
*dimension**of*the incidence*graphs**of*finite affine planes. ... The metric*dimension*μ q*of*the incidence*graph**of*a projective plane*of*order q is known for q ≥ 23 and q ≤ 5. ... Tamás Héger gratefully acknowledges the support by the ÚNKP-18-4 New National Excellence Program*of*the Ministry*of*Human Capacities, grant no. ...##
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The Fractal Dimension of SAT Formulas
[chapter]

2014
*
Lecture Notes in Computer Science
*

We study the fractal

doi:10.1007/978-3-319-08587-6_8
fatcat:buzy72ifgjdpnf5mna3qkcyedq
*dimension**of*SAT formulas, and show that most industrial families*of*formulas are self-similar, with a*small*fractal*dimension*. ... We explore how the*dimension**of*a formula, together with other*graph*properties can be used to characterize SAT instances. ... It would also mean that the diameter d max*of*the*graph*grows as d max ∼ n 1/d , where d is the fractal*dimension**of*the*graph*, and not as d max ∼ log n, as in random*graphs*or*small*-world*graphs*. ...##
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Flow dimension and capacity for structuring urban street networks

2008
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Physica A: Statistical Mechanics and its Applications
*

To our surprise for the topologies

doi:10.1016/j.physa.2008.02.047
fatcat:kcenfnhppzevhman3gv65t4i34
*of*urban street networks, previously confirmed as a form*of**small*world and scale-free networks, we find that (1) the range*of*their flow*dimension*is rather wider than ... The findings confirm that (1) both the wider range*of*flow*dimension*and the higher flow capacity can be a signature*of**small*world networks, and (2) the flow capacity can be an alternative quantity for ... The ratio*of*m to n tends to a very*small*value (as a reminder, we are dealing with large sparse*graphs*). ...##
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Reliable and Efficient Inference of Bayesian Networks from Sparse Data by Statistical Learning Theory
[article]

2003
*
arXiv
*
pre-print

This is shown by calculating bounds on the VC

arXiv:cs/0309015v1
fatcat:sj7dr5lyuvfnhk5o63ukwmb3ay
*dimension**of*the set*of*those probability measures that correspond to simple*graphs*. ... We consider the case that sparse data strongly suggest that the probabilities can be described by a simple Bayesian network, i.e., by a*graph*with*small*in-degree \Delta. ... But we will show that it does make sense from the point*of*view*of*learning theory to consider*graphs*with*small*in-degree since we can derive upper bounds on the VC*dimension**of*the set*of*corresponding ...##
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Relative, local and global dimension in complex networks
[article]

2022
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arXiv
*
pre-print

In simple models

arXiv:2106.05368v3
fatcat:opqxujlugfg5jeq5iierqpiegy
*of*epidemics on networks, the relative*dimension*is predictive*of*the spreading capability*of*nodes, and identifies scales at which the*graph*structure is predictive*of*infectivity. ... We further apply our*dimension*measures to neuronal networks, economic trade, social networks, ocean flows, and to the comparison*of*random*graphs*. ... AA was supported by funding to the Blue Brain Project, a research center*of*the École polytechnique fédérale de Lausanne (EPFL), from the Swiss government's ETH Board*of*the Swiss Federal Institutes*of*...##
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Constant-Time Reachability in DAGs Using Multidimensional Dominance Drawings

2021
*
SN Computer Science
*

We also present experimental results that show that the number

doi:10.1007/s42979-021-00713-6
fatcat:gbg66tmevzdctj45ipb7u7p6wy
*of**dimensions*, k, in the solutions produced by our techniques is low. ... Answering reachability queries in directed acyclic*graphs*is an operation required by many applications. ...*of**dimensions*used is equal to the width*of*the*graph*. ...##
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Can Life Exist in 2 + 1 Dimensions?
[article]

2019
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arXiv
*
pre-print

I will examine these arguments and show how a purely scalar theory

arXiv:1906.05336v1
fatcat:fhmeuvcrfbdgbcw7zxvaggldwi
*of*gravity may evade the first one, before considering certain families*of*planar*graphs*which share properties which are observed in ... There are two main arguments levelled against the possibility*of*life in 2 + 1*dimensions*: the lack*of*a local gravitational force and Newtonian limit in 3D general relativity, and the claim that the restriction ...*of*regular lattices and the*small*average path length*of*random*graphs*. ...##
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Existence of life in 2 + 1 dimensions

2020
*
Physical Review Research
*

I will examine these arguments and show how a purely scalar theory

doi:10.1103/physrevresearch.2.013217
fatcat:dhkenrxfezfsfkhymxgohb526u
*of*gravity may evade the first one, before considering certain families*of*planar*graphs*which share properties which are observed in ... There are two main arguments leveled against the possibility*of*life in 2 + 1*dimensions*: the lack*of*a local gravitational force and Newtonian limit in three-dimensional general relativity, and the claim ...*of*regular lattices and the*small*average path length*of*random*graphs*. ...##
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Towards Optimal Load Balancing Topologies
[chapter]

2000
*
Lecture Notes in Computer Science
*

A

doi:10.1007/3-540-44520-x_37
fatcat:753nmvt4qvgfhcjtxputwn4oqi
*small*flow volume and a*small*diameter*of*the*graph*keeps the time requirement*of*this phase low. We compare and propose several network topologies based on these measurements. ... Its time requirement depends on the maximum node degree and on the number*of*eigenvalues*of*the network. The second phase migrates the load according to this flow. ... We also considered the ¢ torus, the butterfly*graph**of**dimension*4 and the DeBruijn*graph**of**dimension*6. Several*graphs*can be expressed as a cartesian product like e. g. ...##
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Topological graph dimension

2010
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Discrete Mathematics
*

A fairly direct rendering

doi:10.1016/j.disc.2008.10.003
fatcat:5telt37bzna6helhucn4rnri7m
*of*the concept*of**small*inductive*dimension*for*graphs*is given in Definition 1*of*[3] . This approach, however, is not without problems. ... Mukhin, Dimensional properties*of**graphs*and digital spaces, J. Math. Imaging Vision 6 (1996) 109-119]. Our definition is analogous to that*of*(*small*inductive)*dimension*in topology. ...##
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Unsupervised Dimension Selection using a Blue Noise Spectrum
[article]

2018
*
arXiv
*
pre-print

, using only a

arXiv:1810.13427v1
fatcat:tfg3ahj5qjfple3fnihiwbo5ym
*small*subset*of*the original features. ... By analyzing synthetic*graph*signals with a blue noise spectrum, we show that we can measure the importance*of*each*dimension*. ...*small*data scenarios. ...##
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Separation dimension of bounded degree graphs
[article]

2014
*
arXiv
*
pre-print

In general, the maximum separation

arXiv:1407.5075v1
fatcat:phhhanr4v5h5lfpmrtvjgog46m
*dimension**of*a*graph*on n vertices is Θ( n). ... In this article, we focus on bounded degree*graphs*and show that the separation*dimension**of*a*graph*with maximum degree d is at most 2^9log^ d d. ... A critical ingredient*of*our proof is the*small*set expansion property*of*random regular*graphs*. ...
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