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Page 5749 of Mathematical Reviews Vol. , Issue 2003h [page]

2003 Mathematical Reviews  
of p-point, ¢-line graphs with prescribed minimum degree and line connectivity.  ...  We define a (p,q,7,0) graph as a graph having p points, q lines, line connectivity 2 and minimum degree 0.  ... 

THE MAXIMUM CONNECTIVITY OF A GRAPH

F. Harary
1962 Proceedings of the National Academy of Sciences of the United States of America  
For brevity, we say that a p, q graph is one with p points and q lines.  ...  In addition to finding the maximum connectivity of any graph with a given number of points and lines, we also obtain the minimum connectivity, the maximum diameter, and the minimum diameter.  ...  For brevity, we say that a p, q graph is one with p points and q lines.  ... 
doi:10.1073/pnas.48.7.1142 pmid:16590970 pmcid:PMC220921 fatcat:uu46hrk6lrgmbn7iz5wytcbcy4

Page 772 of Mathematical Reviews Vol. , Issue 2001B [page]

2001 Mathematical Reviews  
Summary: “We define a (p,g,4,A) graph as a graph having p points, q lines, line connectivity 4, and maximum degree A.  ...  degree and line connectivity or minimum degree.  ... 

Page 1303 of Mathematical Reviews Vol. , Issue 84d [page]

1984 Mathematical Reviews  
L. 84d:05114 Realizability of p-point, q-line graphs with prescribed point connectivity, line connectivity, or minimum degree. Networks 12 (1982), no. 3, 341-350.  ...  The lower bounds are expressed in terms of the point connectivity «x, line connectivity A and minimum degree 5 of G.  ... 

Planar and Plane Slope Number of Partial 2-Trees [chapter]

William Lenhart, Giuseppe Liotta, Debajyoti Mondal, Rahnuma Islam Nishat
2013 Lecture Notes in Computer Science  
As a byproduct of our techniques, we answer a long standing question by Garg and Tamassia about the angular resolution of the planar straight-line drawings of series-parallel graphs of bounded degree.  ...  We prove tight bounds (up to a small multiplicative or additive constant) for the plane and the planar slope numbers of partial 2-trees of bounded degree.  ...  A plane graph is a planar graph together with a combinatorial embedding, i.e. a prescribed set of faces including a prescribed outer face.  ... 
doi:10.1007/978-3-319-03841-4_36 fatcat:aizqfijgnjfajgqz6b2irafuaa

A random growth model for power grids and other spatially embedded infrastructure networks

Paul Schultz, Jobst Heitzig, Jürgen Kurths
2014 The European Physical Journal Special Topics  
It consists of an initialization phase with the network extending tree-like for minimum cost and a growth phase with an attachment rule giving a trade-off between cost-optimization and redundancy.  ...  In particular, the mean degree and the slope of the exponential decay can be controlled in partial independence.  ...  ) andγ (dashed) for realizations with different mean degree k and (d) same as in (c) for realizations with different q.  ... 
doi:10.1140/epjst/e2014-02279-6 fatcat:zlogcl3kmngfli7nreo7yva6cm

On the Area-Universality of Triangulations [article]

Linda Kleist
2018 arXiv   pre-print
We study straight-line drawings of planar graphs with prescribed face areas.  ...  A plane graph is 'area-universal' if for every area assignment on the inner faces, there exists a straight-line drawing realizing the prescribed areas.  ...  Acknowledgements I thank Udo Hoffmann and Sven Jäger for helpful comments. Area-universal triangulations  ... 
arXiv:1808.10864v2 fatcat:w4twuzrecvaxrmidkveh75dqzu

A SURVEY OF PROGRESS IN GRAPH THEORY IN THE SOVIET UNION

James Turner, William H. Kautz
1970 Annals of the New York Academy of Sciences  
Very littlv Soviet work has been reported on connectivity propertieb of graphs, matroid theory, the exact enumeration of graphs having prescribed properties, isomorphism testing, graph coloring, and the  ...  The best Sovic work has been concerned with bounds on numerical indices associated with graphs, properties of algebraic stvuctures associated with graphs, and operations on graphs.  ...  Each step of the decomposition 1) locates a point p of degree 1 whose adjacent point is, say, q and (2) removes the points p and q and all edges incident at point q.  ... 
doi:10.1111/j.1749-6632.1970.tb56496.x fatcat:cbb5x22w2farrnw5sijyovn66i

Inverse problems for dynamical systems

Ronald Sverdlove
1981 Journal of Differential Equations  
Formula (11) then tells us that P* and Q* are of degree 8.  ...  Here, we may pose a converse type of question: For a given (local or global) topological structure, what is the minimum degree of polynomial system which realizes it?  ...  This has been established by Kaplan [ 161 for flows with no critical points, and extended by Boothby [7] to the case in which all critical points are of purely hyperbolic type.  ... 
doi:10.1016/0022-0396(81)90034-6 fatcat:wywjfdmhovhinkozhrjp2uw4sa

Planar and Poly-arc Lombardi Drawings [chapter]

Christian A. Duncan, David Eppstein, Michael T. Goodrich, Stephen G. Kobourov, Maarten Löffler
2012 Lecture Notes in Computer Science  
We show that every planar graph has a smooth planar 3-Lombardi drawing and further investigate topics connecting planarity and Lombardi drawings.  ...  In Lombardi drawings of graphs, edges are represented as circular arcs, and the edges incident on vertices have perfect angular resolution.  ...  This research was supported in part by the National Science Foundation under grants CCF-0830403, CCF-0545743, and CCF-1115971, by the Office of Naval Research under MURI grant N00014-08-1-1015, and by  ... 
doi:10.1007/978-3-642-25878-7_30 fatcat:bqduoxtwsbfrvb4twkpozwdu5a

Analytic solution of the two-star model with correlated degrees [article]

Maíra Bolfe, Fernando L. Metz, Edgar Guzmán-González, Isaac Pérez Castillo
2021 arXiv   pre-print
Exponential random graphs are important to model the structure of real-world complex networks. Here we solve the two-star model with degree-degree correlations in the sparse regime.  ...  The model constraints the average correlation between the degrees of adjacent nodes (nearest neighbors) and between the degrees at the end-points of two-stars (next nearest neighbors).  ...  The model allows to generate random graphs with prescribed degree correlations between adjacent nodes and between nodes at the end-points of two-stars.  ... 
arXiv:2102.09629v1 fatcat:t4pvhq6klva2dkfgnxd6tylbsu

Quantifying Loopy Network Architectures

Eleni Katifori, Marcelo O. Magnasco, Jérémie Bourdon
2012 PLoS ONE  
This algorithmic framework decouples the geometric information (exact location of edges and nodes) from the metric topology (connectivity and edge weight) and it ultimately allows us to perform a quantitative  ...  statistical comparison between predictions of theoretical models and naturally occurring loopy graphs.  ...  Daly from the New York Botanical Garden for kindly providing us with cleared leaf images, and P.  ... 
doi:10.1371/journal.pone.0037994 pmid:22701593 pmcid:PMC3368948 fatcat:tlqnosqooneklkifs2ylozj54i

Topology for Distributed Inference on Graphs

Soummya Kar, Saeed Aldosari, JosÉ M. F. Moura
2008 IEEE Transactions on Signal Processing  
We show that, under appropriate conditions, the topology given by the nonbipartite Ramanujan graphs optimizes the convergence rate of this distributed algorithm.  ...  We consider iterative distributed inference with local intersensor communication, which, under simplifying assumptions, is equivalent to distributed average consensus.  ...  The LPS-II graphs are Cayley graphs over the group P 1 (F q ) = f0; 1; . . . ; q 01; 1g, called the Projective line over F q , 4 and which is basically the set of integers modulo q, with an additional  ... 
doi:10.1109/tsp.2008.923536 fatcat:dgkw6yyovzabxfq4os6kno4uc4

The Complexity of the Partial Order Dimension Problem: Closing the Gap

Stefan Felsner, Irina Mustaţă, Martin Pergel
2017 SIAM Journal on Discrete Mathematics  
The dimension of a partial order P is the minimum number of linear orders whose intersection is P . There are efficient algorithms to test if a partial order has dimension at most 2.  ...  The height of a partial order P is the maximum size of a chain in P . Yannakakis also showed that for k ≥ 4 to test if a partial order of height 2 has dimension ≤ k is NP-complete.  ...  The line (p) = p + λ1 shares a point p e with the segmentsê = [û,v ] and a point p f with the segmentf = [x,ŷ ]. Without loss of generality we may assume that p f separates p e and p on (p).  ... 
doi:10.1137/15m1007720 fatcat:yz6lbrji7rduraugowinvau3um

The Complexity of the Partial Order Dimension Problem - Closing the Gap [article]

Stefan Felsner and Irina Mustata and Martin Pergel
2016 arXiv   pre-print
The dimension of a partial order P is the minimum number of linear orders whose intersection is P. There are efficient algorithms to test if a partial order has dimension at most 2.  ...  The height of a partial order P is the maximum size of a chain in P. Yannakakis also showed that for k≥ 4 to test if a partial order of height 2 has dimension ≤ k is NP-complete.  ...  The line ℓ(p) = p + λ1 shares a point p e with the segmentsê = [û,v ] and a point p f with the segmentf = [x ,ŷ ]. Without loss of generality we may assume that p f separates p e and p on ℓ(p).  ... 
arXiv:1501.01147v2 fatcat:jecoz3ejmjgz5fsi4zo3zdqwaq
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