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A new graph parameter related to bounded rank positive semidefinite matrix completions
[article]

2012
*
arXiv
*
pre-print

*to*edges of G, can be

*completed*

*to*

*a*

*positive*

*semidefinite*

*matrix*of

*rank*at most k (assuming

*a*

*positive*

*semidefinite*

*completion*exists). ... We also show some close connections

*to*Euclidean realizations of

*graphs*and

*to*the

*graph*

*parameter*ν^=(G) of H03. ... -Nagy for useful discussions and

*A*. Schrijver for his suggestions for the proof of Theorem 5. ...

##
###
A new graph parameter related to bounded rank positive semidefinite matrix completions

2013
*
Mathematical programming
*

*to*edges of G, can be

*completed*

*to*

*a*

*positive*

*semidefinite*

*matrix*of

*rank*at most k (assuming

*a*

*positive*

*semidefinite*

*completion*exists). ... We also show some close connections

*to*Euclidean realizations of

*graphs*and

*to*the

*graph*

*parameter*ν = (G) of [21] . ... -Nagy for useful discussions and

*A*. Schrijver for his suggestions for the proof of Theorem 5. ...

##
###
Low-rank solutions of matrix inequalities with applications to polynomial optimization and matrix completion problems

2014
*
53rd IEEE Conference on Decision and Control
*

We develop

doi:10.1109/cdc.2014.7040064
dblp:conf/cdc/MadaniFSL14
fatcat:muahfrpfijcb5aasa7rzghcn2a
*a*theory*relating*the*rank*of the minimum-*rank*solution of the LMI problem*to*the sparsity of its underlying*graph*. ... This paper is concerned with the problem of finding*a*low-*rank*solution of an arbitrary sparse linear*matrix*inequality (LMI).*To*this end, we map the sparsity of the LMI problem into*a**graph*. ... Low-*rank**Positive**Semidefinite**Matrix**Completion*The LMI problem (1) encapsulates the low-*rank**positive**semidefinite**matrix**completion*problem, which is described below. ...##
###
On tropical and nonnegative factorization ranks of band matrices
[article]

2017
*
arXiv
*
pre-print

We give

arXiv:1710.02072v1
fatcat:j44tc2ycvbazjmbci26lq4phui
*a*survey of what is known on the algorithmic complexity of Boolean, fuzzy, tropical, nonnegative, and*positive**semidefinite*factorizations, and we examine the behavior of the corresponding*rank*... We show that the Boolean, fuzzy, and tropical versions of*matrix*factorization become polynomial time solvable when restricted*to*this class of matrices, and we also show that the nonnegative*rank*of*a*... Problem 14 can be*related**to*the question asked in [11] . Namely, is it NP-hard*to*decide if the*positive**semidefinite**rank*of*a*n × n*matrix*equals n? ...##
###
The Gram dimension of a graph
[article]

2012
*
arXiv
*
pre-print

The Gram dimension (G) of

arXiv:1112.5960v2
fatcat:i72t263klbeg5jrlqdcqahiwru
*a**graph*is the smallest integer k > 1 such that, for every assignment of unit vectors*to*the nodes of the*graph*, there exists another assignment of unit vectors lying in ^k, having ... We show that*a**graph*has Gram dimension at most 4 if and only if it does not have K_5 and K_2,2,2 as minors. We also show some close connections*to*the notion of d-realizability of*graphs*. ... -Nagy for useful discussions and*A*. Schrijver for his suggestions for the proof of Theorem 5. ...##
###
Maximum Likelihood Threshold and Generic Completion Rank of Graphs
[article]

2017
*
arXiv
*
pre-print

On the other hand we determine both invariants for

arXiv:1703.07849v2
fatcat:36ym77ebgjgadibh3y5gv5jude
*complete*bipartite*graphs*K_m,n and show that for some choices of m and n the two*parameters*may be quite far apart. ... In particular, this gives the first examples of*graphs*on which the maximum likelihood threshold and the generic*completion**rank*do not agree. ... In the end of this paper, we*relate*the generic*completion**rank*of bipartite*graphs**to*the generic*completion**rank*for the general*matrix**completion*problem: Given*a*partially specified m × n*matrix*, we ...##
###
Using Algebraic Properties of Minimal Idempotents for Exhaustive Computer Generation of Association Schemes

2008
*
Electronic Journal of Combinatorics
*

Central

doi:10.37236/754
fatcat:ddilpk2ej5dybel2q7z4gcnbzm
*to*our success is the use of two algebraic constraints based on properties of the minimal idempotents $E_{i}$ of these association schemes : the fact that they are*positive**semidefinite*and that ... $(126,50,13,24)$*graph*and some*new*examples of antipodal distance regular*graphs*. ... Indeed, any principal submatrix of*a**positive**semidefinite**matrix*must again be*positive**semidefinite*, and any principal submatrix of*a**matrix*must have*a**rank*which is at most the*rank*of the original ...##
###
On the graph complement conjecture for minimum rank

2012
*
Linear Algebra and its Applications
*

*Related*results for the

*graph*complement conjecture (and the

*positive*

*semidefinite*version) for joins of

*graphs*are also established. ... This conjecture asks for an upper

*bound*on the sum of the minimum

*rank*of

*a*

*graph*and the minimum

*rank*of its complement, and may be classified as

*a*Nordhaus-Gaddum type problem involving the

*graph*

*parameter*... Acknowledgments The authors thank Sebastian Cioaba for suggesting we examine connections between the

*Graph*Complement Conjecture and other types of Nordhaus-Gaddum

*graph*

*parameter*problems. ...

##
###
Positive semidefinite maximum nullity and zero forcing number

2012
*
The Electronic Journal of Linear Algebra
*

Since any

doi:10.13001/1081-3810.1559
fatcat:yxecfsv6ube2lmnhzddsadt4lq
*positive**semidefinite**matrix**A*can be written as X T X for some X ∈ M n (R) with*rank**A*=*rank*X, every*positive**semidefinite**matrix*is*a*Gram*Matrix*. ... We use this result*to*determine the vertex connectivity of the Cartesian product of*a**complete**graph*and*a*path, thereby providing*a*lower*bound*for the maximum*positive**semidefinite*nullity. ...##
###
Matrices with high completely positive semidefinite rank

2017
*
Linear Algebra and its Applications
*

We also exhibit

doi:10.1016/j.laa.2016.10.015
fatcat:h6mf372z6needfeaznss2ffcqm
*a*class of*completely**positive*matrices with quadratic (in terms of the*matrix*size)*completely**positive**rank*, but with linear*completely**positive**semidefinite**rank*, and we make*a*connection ... The smallest such d is called the (complex)*completely**positive**semidefinite**rank*of M, and it is an open question whether there exists an upper*bound*on this number as*a*function of the*matrix*size. ... We are grateful*to*an anonymous referee for his/her careful reading and helpful comments, and for bringing the works [15, 27]*to*our attention. ...##
###
Finding Low-rank Solutions of Sparse Linear Matrix Inequalities using Convex Optimization

2017
*
SIAM Journal on Optimization
*

We develop

doi:10.1137/14099379x
fatcat:u2b7c2chvvfk7kuemvdy5fnpva
*a*theory*relating*the*rank*of the minimum-*rank*solution of the LMI problem*to*the sparsity of its underlying*graph*. ... This paper is concerned with the problem of finding*a*low-*rank*solution of an arbitrary sparse linear*matrix*inequality (LMI).*To*this end, we map the sparsity of the LMI problem into*a**graph*. ... The authors would like*to*thank Professors Daniel Bienstock and Vineet Goyal of Columbia University for several fruitful discussions. ...##
###
Conic approach to quantum graph parameters using linear optimization over the completely positive semidefinite cone
[article]

2015
*
arXiv
*
pre-print

We investigate the

arXiv:1312.6643v6
fatcat:vcpn7hmhqrcgpae3zz3ox2ytaa
*completely**positive**semidefinite*cone CS_+^n,*a**new**matrix*cone consisting of all n× n matrices that admit*a*Gram representation by*positive**semidefinite*matrices (of any size). ... We use this*new*cone*to*model quantum analogues of the classical independence and chromatic*graph**parameters*α(G) and χ(G), which are roughly obtained by allowing variables*to*be*positive**semidefinite*... In particular, we are grateful*to*one referee for suggesting the result of Lemma 4.4 which permitted*to*correct an error in the proof of Proposition 4.5. ...##
###
Conic Approach to Quantum Graph Parameters Using Linear Optimization Over the Completely Positive Semidefinite Cone

2015
*
SIAM Journal on Optimization
*

We investigate the

doi:10.1137/14097865x
fatcat:c4j45vljn5eg5btyxqxtgj33me
*completely**positive**semidefinite*cone CS n + ,*a**new**matrix*cone consisting of all n × n matrices that admit*a*Gram representation by*positive**semidefinite*matrices (of any size). ... We use this*new*cone*to*model quantum analogues of the classical independence and chromatic*graph**parameters*α(G) and χ(G), which are roughly obtained by allowing variables*to*be*positive**semidefinite*... In particular, we are grateful*to*one referee for suggesting the result of Lemma 4.4 which permitted*to*correct an error in the proof of Proposition 4.5. ...##
###
CONNECTION MATRICES
[chapter]

2007
*
Combinatorics, Complexity, and Chance
*

Acknowledgement I am grateful

doi:10.1093/acprof:oso/9780198571278.003.0012
fatcat:h33lwmporjdwhkfkcdt4ztsq7e
*to*the anonymous referee for pointing out the construction in Example 2.11. ... The property that M (f, k) is*positive**semidefinite*is*related**to*the "reflection*positivity*" property in statistical physics, and we'll call*a**graph**parameter*reflection*positive*if M (f, k) is*positive*... For the chromatic polynomial chr(G, q), it is easy*to*get an upper*bound*on the*rank*of the connection*matrix*M plan (chr(., q), k). ...##
###
Reflection positivity, rank connectivity, and homomorphism of graphs

2006
*
Journal of The American Mathematical Society
*

Acknowledgement We are indebted

doi:10.1090/s0894-0347-06-00529-7
fatcat:yalpe4lumrgxbcvn3wdhrc7lz4
*to*Christian Borgs, Jennifer Chayes, Monique Laurent, Miki Simonovits, Vera T. ... Sós, Balázs Szegedy, Gábor Tardos and Kati Vesztergombi for many valuable discussions and suggestions on the topic of*graph*homomorphisms. ...*matrix*M (f, k) is*positive**semidefinite*and has*rank*1. ...
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