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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

1962
*
Proceedings of the National Academy of Sciences of the United States of America
*

For brevity, we say that a

doi:10.1073/pnas.48.7.1142
pmid:16590970
pmcid:PMC220921
fatcat:uu46hrk6lrgmbn7iz5wytcbcy4
*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*. ...##
###
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]

2013
*
Lecture Notes in Computer Science
*

As a byproduct

doi:10.1007/978-3-319-03841-4_36
fatcat:aizqfijgnjfajgqz6b2irafuaa
*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. ...##
###
A random growth model for power grids and other spatially embedded infrastructure networks

2014
*
The European Physical Journal Special Topics
*

It consists

doi:10.1140/epjst/e2014-02279-6
fatcat:zlogcl3kmngfli7nreo7yva6cm
*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. ... )*and*γ (dashed) for*realizations**with*different mean*degree*k*and*(d) same as in (c) for*realizations**with*different*q*. ...##
###
On the Area-Universality of Triangulations
[article]

2018
*
arXiv
*
pre-print

We study straight-

arXiv:1808.10864v2
fatcat:w4twuzrecvaxrmidkveh75dqzu
*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 ...##
###
A SURVEY OF PROGRESS IN GRAPH THEORY IN THE SOVIET UNION

1970
*
Annals of the New York Academy of Sciences
*

Very littlv Soviet work has been reported on

doi:10.1111/j.1749-6632.1970.tb56496.x
fatcat:cbb5x22w2farrnw5sijyovn66i
*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*. ...##
###
Inverse problems for dynamical systems

1981
*
Journal of Differential Equations
*

Formula (11) then tells us that

doi:10.1016/0022-0396(81)90034-6
fatcat:wywjfdmhovhinkozhrjp2uw4sa
*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. ...##
###
Planar and Poly-arc Lombardi Drawings
[chapter]

2012
*
Lecture Notes in Computer Science
*

We show that every planar

doi:10.1007/978-3-642-25878-7_30
fatcat:bqduoxtwsbfrvb4twkpozwdu5a
*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 ...##
###
Analytic solution of the two-star model with correlated degrees
[article]

2021
*
arXiv
*
pre-print

Exponential random

arXiv:2102.09629v1
fatcat:t4pvhq6klva2dkfgnxd6tylbsu
*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. ...##
###
Quantifying Loopy Network Architectures

2012
*
PLoS ONE
*

This algorithmic framework decouples the geometric information (exact location

doi:10.1371/journal.pone.0037994
pmid:22701593
pmcid:PMC3368948
fatcat:tlqnosqooneklkifs2ylozj54i
*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*. ...##
###
Topology for Distributed Inference on Graphs

2008
*
IEEE Transactions on Signal Processing
*

We show that, under appropriate conditions, the topology given by the nonbipartite Ramanujan

doi:10.1109/tsp.2008.923536
fatcat:dgkw6yyovzabxfq4os6kno4uc4
*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 ...##
###
The Complexity of the Partial Order Dimension Problem: Closing the Gap

2017
*
SIAM Journal on Discrete Mathematics
*

The dimension

doi:10.1137/15m1007720
fatcat:yz6lbrji7rduraugowinvau3um
*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*). ...##
###
The Complexity of the Partial Order Dimension Problem - Closing the Gap
[article]

2016
*
arXiv
*
pre-print

The dimension

arXiv:1501.01147v2
fatcat:jecoz3ejmjgz5fsi4zo3zdqwaq
*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*). ...
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