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

Robert F. Bailey
2013 arXiv   pre-print
The metric 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.  ... 
arXiv:1312.4973v1 fatcat:6njfd6p5tjhkrizn77jtgzhtbi

On embedding of graphs into euclidean spaces of small dimension

J Reiterman, V Rödl, E S̆in̆ajová
1992 Journal of combinatorial theory. Series B (Print)  
APPLICATIONS We apply Theorem 1 to obtain upper bounds for d(G) for classes 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.  ... 
doi:10.1016/0095-8956(92)90002-f fatcat:s5kb6hyt6ba5tojo3ubwjmbs6u

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

J. E. Boillat
1993 Czechoslovak Mathematical Journal  
Nous montrons que l'on peut appliquer cette méthode aux 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.  ... 
doi:10.21136/cmj.1993.128408 fatcat:5pg6s6jnr5b55kfrvxhcit7jzm

The metric dimension of the incidence graphs of projective planes of small order

Tamás Héger, Péter Szilárd, Marcella Takáts
2020 The Australasian Journal of Combinatorics  
We also determine the metric 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.  ... 
dblp:journals/ajc/HegerST20 fatcat:wgpicvc6pre6raoozwqd3pjeeq

The Fractal Dimension of SAT Formulas [chapter]

Carlos Ansótegui, Maria Luisa Bonet, Jesús Giráldez-Cru, Jordi Levy
2014 Lecture Notes in Computer Science  
We study the fractal 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.  ... 
doi:10.1007/978-3-319-08587-6_8 fatcat:buzy72ifgjdpnf5mna3qkcyedq

Flow dimension and capacity for structuring urban street networks

Bin Jiang
2008 Physica A: Statistical Mechanics and its Applications  
To our surprise for the topologies 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).  ... 
doi:10.1016/j.physa.2008.02.047 fatcat:kcenfnhppzevhman3gv65t4i34

Reliable and Efficient Inference of Bayesian Networks from Sparse Data by Statistical Learning Theory [article]

Dominik Janzing, Daniel Herrmann
2003 arXiv   pre-print
This is shown by calculating bounds on the VC 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  ... 
arXiv:cs/0309015v1 fatcat:sj7dr5lyuvfnhk5o63ukwmb3ay

Relative, local and global dimension in complex networks [article]

Robert L. Peach, Alexis Arnaudon, Mauricio Barahona
2022 arXiv   pre-print
In simple models 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  ... 
arXiv:2106.05368v3 fatcat:opqxujlugfg5jeq5iierqpiegy

Constant-Time Reachability in DAGs Using Multidimensional Dominance Drawings

Panagiotis Lionakis, Giacomo Ortali, Ioannis G. Tollis
2021 SN Computer Science  
We also present experimental results that show that the number 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.  ... 
doi:10.1007/s42979-021-00713-6 fatcat:gbg66tmevzdctj45ipb7u7p6wy

Can Life Exist in 2 + 1 Dimensions? [article]

J. H. C. Scargill
2019 arXiv   pre-print
I will examine these arguments and show how a purely scalar theory 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.  ... 
arXiv:1906.05336v1 fatcat:fhmeuvcrfbdgbcw7zxvaggldwi

Existence of life in 2 + 1 dimensions

J. H. C. Scargill
2020 Physical Review Research  
I will examine these arguments and show how a purely scalar theory 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.  ... 
doi:10.1103/physrevresearch.2.013217 fatcat:dhkenrxfezfsfkhymxgohb526u

Towards Optimal Load Balancing Topologies [chapter]

Thomas Decker, Burkhard Monien, Robert Preis
2000 Lecture Notes in Computer Science  
A 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.  ... 
doi:10.1007/3-540-44520-x_37 fatcat:753nmvt4qvgfhcjtxputwn4oqi

Topological graph dimension

Michael B. Smyth, Rueiher Tsaur, Iain Stewart
2010 Discrete Mathematics  
A fairly direct rendering 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.  ... 
doi:10.1016/j.disc.2008.10.003 fatcat:5telt37bzna6helhucn4rnri7m

Unsupervised Dimension Selection using a Blue Noise Spectrum [article]

Jayaraman J. Thiagarajan, Rushil Anirudh, Rahul Sridhar, Peer-Timo Bremer
2018 arXiv   pre-print
, using only a 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.  ... 
arXiv:1810.13427v1 fatcat:tfg3ahj5qjfple3fnihiwbo5ym

Separation dimension of bounded degree graphs [article]

Noga Alon, Manu Basavaraju, L. Sunil Chandran, Rogers Mathew, Deepak Rajendraprasad
2014 arXiv   pre-print
In general, the maximum separation 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.  ... 
arXiv:1407.5075v1 fatcat:phhhanr4v5h5lfpmrtvjgog46m
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