Orthogonal Representations, Projective Rank, and Fractional Minimum Positive Semidefinite Rank: Connections and New Directions

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1 Electonic Jounal of Linea Algeba Volume 32 Volume 32 (2017) Aticle Othogonal Repesentations, Pojective Rank, and Factional Minimum Positive Semidefinite Rank: Connections and New Diections Leslie Hogben Iowa State Univesity, hogben@aimathog Kevin F Palmowski Iowa State Univesity, kpalmow@iastateedu David E Robeson Nanyang Technological Univesity, davideobeson@gmailcom Simone Seveini Univesity College London, simoseve@gmailcom Follow this and additional woks at: Pat of the Discete Mathematics and Combinatoics Commons, and the Quantum Physics Commons Recommended Citation Hogben, Leslie; Palmowski, Kevin F; Robeson, David E; and Seveini, Simone (2017), "Othogonal Repesentations, Pojective Rank, and Factional Minimum Positive Semidefinite Rank: Connections and New Diections", Electonic Jounal of Linea Algeba, Volume 32, pp DOI: This Aticle is bought to you fo fee and open access by Wyoming Scholas Repositoy It has been accepted fo inclusion in Electonic Jounal of Linea Algeba by an authoized edito of Wyoming Scholas Repositoy Fo moe infomation, please contact scholcom@uwyoedu

2 Electonic Jounal of Linea Algeba, ISSN A publication of the Intenational Linea Algeba Society Volume 32, pp , Apil ORTHOGONAL REPRESENTATIONS, PROJECTIVE RANK, AND FRACTIONAL MINIMUM POSITIVE SEMIDEFINITE RANK: CONNECTIONS AND NEW DIRECTIONS LESLIE HOGBEN, KEVIN F PALMOWSKI, DAVID E ROBERSON, AND SIMONE SEVERINI Abstact Factional minimum positive semidefinite ank is defined fom -fold faithful othogonal epesentations and it is shown that the pojective ank of any gaph equals the factional minimum positive semidefinite ank of its complement An -fold vesion of the taditional definition of minimum positive semidefinite ank of a gaph using Hemitian matices that fit the gaph is also pesented This pape also intoduces -fold othogonal epesentations of gaphs and fomalizes the undestanding of pojective ank as factional othogonal ank Connections of these concepts to quantum theoy, including Tsielson s poblem, ae discussed Key wods Pojective ank, Othogonal epesentation, Minimum positive semidefinite ank, Factional, Tsielson s poblem, Gaph, Matix AMS subject classifications 15B10, 05C72, 05C90, 15A03, 15B57, 81P45 1 Intoduction This pape deals with factional vesions of gaph paametes defined by othogonal epesentations, including minimum positive semidefinite ank In Section 2, we extend the existing idea of an othogonal epesentation fo a gaph via a highe-dimensional constuction With this, we intoduce a new paamete, -fold othogonal ank, that is to othogonal ank as b-fold chomatic numbe is to chomatic numbe (see Section 12 fo the definition of b-fold chomatic numbe and othe tems elated to factional chomatic numbe) This allows us to fomally chaacteize pojective ank as factional othogonal ank, a concept that was peviously undestood (eg, in [15, 11]) but not igoously pesented (fomal definitions of pojective ank and othe paametes ae given in Section 13) In Section 3, we apply this factionalization pocess to the minimum positive semidefinite ank poblem (viewed via faithful othogonal epesentations) and develop two new gaph paametes, namely, -fold and factional minimum positive semidefinite ank We also povide an altenate definition of -fold minimum positive semidefinite ank that is based on the minimum ank of a matix that -fits a gaph, allowing us to view the highe-dimensional poblem though eithe of the two viewpoints taditionally associated with the classical minimum positive semidefinite ank poblem Ou final esult, found in Section 35, shows that the factional minimum positive semidefinite ank of a gaph is equal to the pojective ank of the complement of the gaph This esult seves to connect the two Received by the editos on Septembe 2, 2015 Accepted fo publication on Febuay 22, 2017 Handling Edito: Byan L Shade Depatment of Mathematics, Iowa State Univesity, Ames, IA 50011, USA (hogben@iastateedu), and Ameican Institute of Mathematics, 600 E Bokaw Rd, San Jose, CA 95112, USA (hogben@aimathog) Depatment of Mathematics, Iowa State Univesity, Ames, IA 50011, USA (kevinpalmowski@gmailcom) Division of Mathematical Sciences, Nanyang Technological Univesity, SPMS-MAS-03-01, 21 Nanyang Link, Singapoe (davideobeson@gmailcom) Reseach suppoted in pat by the Singapoe National Reseach Foundation unde NRF RF Awad no NRF-NRFF Depatment of Compute Science, Univesity College London, Gowe Steet, London WC1E 6BT, United Kingdom (simoseve@gmailcom) Reseach suppoted by the Royal Society and EPSRC

3 Electonic Jounal of Linea Algeba, ISSN A publication of the Intenational Linea Algeba Society Volume 32, pp , Apil Factional Appoach to Othogonal Repesentations seemingly diffeent poblems; moving fowad, this will allow the extensive existing liteatue on minimum positive semidefinite ank to be used to infom new developments in the moe ecently intoduced aea of pojective ank In the emainde of this intoduction, we discuss applications of the factional paametes discussed (Section 11), give a bief intoduction to the factional appoach to chomatic numbe to motivate ou definitions (Section 12), and povide necessay notation and teminology (Section 13) 11 Applications Linea algebaic stuctues and associated gaph theoetic famewoks have ecently become moe impotant tools to study the fundamental diffeences that chaacteize theoies of natue, like classical mechanics, quantum mechanics, and geneal pobabilistic theoies Matices, gaphs, and thei elated combinatoial optimization techniques tun out to povide a supisingly geneal language with which to appoach questions connected with foundational ideas, such as the analysis of contextual inequalities and non-local games [2, 3], and with concete aspects, such as quantifying vaious capacities of entanglement-assisted channels [6, 10], and the ovehead needed to classically simulate quantum computation [9] A point of stength of such famewoks is thei ability to efomulate mathematical questions in a coase manne that is nonetheless effective, in some cases, to single out specific facts Tsielson s poblem [17] povides a emakable example: deciding whethe the mathematical models of non-elativistic quantum mechanics, whee obseves have linea opeatos acting on a finite dimensional tenso poduct space, and algebaic quantum field theoy, whee obseves have commuting linea opeatos on a single (possibly infinite dimensional) space, poduce the same set of coelations We know that if Tsielson s poblem has a positive answe, then the notoious Connes Embedding conjectue [4, 12], oiginally concened with an appoximation popety fo finite von Neumann algebas, is tue Tsielson s poblem can be seen fom a combinatoial matix point of view by woking with gaphs and thei associated algebaic stuctues [14] Roughly speaking, instead of constucting sets of coelation matices, we can ty looking fo vaious pattens of zeoes in the sets, as in the spiit of combinatoial matix theoy The pojective ank, denoted ξ f, is a ecently intoduced gaph paamete with the potential fo settling the above discussion Indeed, it has been shown that if thee exists a gaph whose pojective ank is iational, then Tsielson s poblem has a negative answe [13] Pojective epesentations and pojective ank wee oiginally defined in [11] as a tool fo studying quantum coloings and quantum homomophisms of gaphs Quantum coloings and the quantum chomatic numbe give quantitative measues of the advantage that quantum entanglement povides in pefoming distibuted tasks and in distinguishing scenaios elated to classical and quantum physics, espectively In fact, the existence of a quantum n-coloing fo a given gaph is equivalent to the existence of a pojective epesentation of value n fo the Catesian poduct of the gaph with a complete gaph on n vetices It was also shown in [11] that pojective ank is monotone with espect to quantum homomophisms, ie, if thee exists a quantum homomophism fom a gaph G to a gaph H, then ξ f (G) ξ f (H) This shows that pojective ank is a lowe bound fo quantum chomatic numbe, and moe geneally povides a method fo fobidding the existence of quantum homomophisms Indeed, this appoach was used to detemine the quantum odd gith of the Knese gaphs in [15] Pojective ank has also been studied fom a puely gaph theoetic point of view, and in [5] it was shown that this paamete is multiplicative with espect to the lexicogaphic and disjunctive gaph poducts Using this fact the authos wee able to

4 Electonic Jounal of Linea Algeba, ISSN A publication of the Intenational Linea Algeba Society Volume 32, pp , Apil L Hogben, KF Palmowski, DE Robeson, and S Seveini 100 find a sepaation between quantum chomatic numbe and a ecently defined semidefinite elaxation of this paamete, answeing a question posed in [14] This pape takes a linea algebaic appoach to these questions, building connections between ecent gaph theoetical appoaches to quantum questions and existing liteatue on othogonal epesentations and minimum positive semidefinite ank 12 A factional appoach To demonstate the factional appoach that we use with othogonal epesentations and minimum positive semidefinite ank, conside the following deivation of the factional chomatic numbe as found in [16] The chomatic numbe χ(g) of a gaph G is the least numbe c such that G can be coloed with c colos; that is, we can assign to each vetex of G one of c colos in such a way that adjacent vetices eceive diffeent colos A coloing with c colos can be genealized to a b-fold coloing with c colos, o a c:b-coloing: fom a palette of c colos, assign b colos to each vetex of G such that adjacent vetices eceive disjoint sets of colos Fo a fixed b, the b-fold chomatic numbe of G, χ b (G), is the smallest c such that G has a c:b-coloing With this, the factional chomatic numbe of G as χ f (G) = inf b χ b (G) b While it is not obvious, it can be shown that χ f (G) is always a ational numbe, as thee is an altenative linea pogamming fomulation fo the paamete fo which stong duality holds Fo futhe infomation on factional coloing, including a time-scheduling intepetation of the poblem, see the discussions in the Peface and Chapte 3 of [16] The pocess of assigning objects to the vetices of a gaph, subject to cetain constaints, is a key element common to the poblems we examine in this wok, and the pocedue of genealizing fom assigning one object to assigning b-many objects (o, in ou case, b-dimensional o ank-b objects) is an undelying theme At each stage of the pocess, we ae inteested in gaph paametes that give infomation about the most efficient set of objects we can use, with the end goal of developing factional vesions of existing paametes (in the spiit of [16]) and connecting the moe ecent wok on pojective ank with existing ideas fom the ealm of minimum positive semidefinite ank Rathe than the colos used fo coloing poblems, the objects that we assign to the vetices of a gaph ae vectos and matices, which adds a distinctly linea algebaic flavo to both the poblems and the constaints: the idea of diffeent colos tanslates to othogonality conditions on ou objects As such, ou esults often see linea algeba and gaph theoy woking hand-in-hand, with stuctue found in one discipline influencing esults that ae based in the othe 13 Backgound, definitions, and notation The natual numbes, N, stat at 1 We use the notation [a : b] to denote the set of integes {a, a + 1,, b 1, b} Thoughout, d and ae used to epesent natual numbes Vectos ae denoted by boldface font, typically x, and matices ae capital lettes, typically A, B, P, o X, depending on context The symbol 0 denotes eithe the scala zeo o a zeo matix, and an identity matix is denoted by I; any of these may be subscipted to claify thei sizes We follow the usual convention of denoting the j th standad basis vecto in C d (fo some d) as e j Rows and columns of matices may be indexed eithe by natual numbes o by vetices of a gaph, depending on context The elements of a matix A ae denoted a ij ; if A is a block matix, then its blocks ae denoted A ij Gaphs ae usually denoted by G o H, vetices by u, v o i, j, and edges by uv o ij

5 Electonic Jounal of Linea Algeba, ISSN A publication of the Intenational Linea Algeba Society Volume 32, pp , Apil Factional Appoach to Othogonal Repesentations If A C p p and B C q q, then the diect sum of A and B, denoted A B, is the block diagonal matix [ A 0 0 B ] C (p+q) (p+q) We denote the conjugate tanspose of A by A A Hemitian matix satisfies A = A A Hemitian matix A C n n is positive semidefinite, denoted A 0, if x Ax 0 fo all x C n, o equivalently, if all of its eigenvalues ae nonnegative Typically, G = (V, E) will denote a simple undiected gaph on n vetices, whee V = V (G) is the set of vetices of G and E = E(G) is the set of edges of G An isolated vetex is a vetex that is not adjacent to any othe vetex of G A subgaph of a gaph G is a gaph H such that V (H) V (G) and E(H) E(G) An induced subgaph of a gaph G, denoted G[W ] fo some set W V (G), is a subgaph with vetex set W such that if u, v W and uv E(G), then uv E(G[W ]) The union of gaphs G and H, denoted G H, is the gaph with vetex set V (G H) = V (G) V (H) and edge set E(G H) = E(G) E(H) If V (G) V (H) =, then this union is disjoint and denoted G H The complement of G, denoted G, is the gaph with V (G) = V (G) and E(G) = {uv : u v, uv / E(G)} An independent set in G is a set W V (G) such that if u, v W, then uv / E(G) The independence numbe of G, denoted α(g), is the lagest possible cadinality of an independent set in G A clique in G is an induced subgaph H that is a complete gaph, ie, uv E(H) fo evey u, v V (H) The clique numbe of G, denoted ω(g), is the lagest possible ode of a clique in G A clique-sum of gaphs G and H on K t, ie, the gaph G H whee G H = K t, is denoted by G K t H; this is also called a t-clique-sum of G and H A chodal gaph is a gaph that does not have any induced cycles of length geate than 3; any chodal gaph can be constucted as clique-sum(s) of complete gaphs A pefect gaph is a gaph G fo which evey induced subgaph H of G satisfies ω(h) = χ(h) A cut-vetex of a connected gaph G is a vetex whose deletion disconnects G A gaph with a cut-vetex can be viewed as a 1-clique-sum We wok in the vecto space C d fo some d N We use S to denote a subspace of a vecto space A basis matix fo an -dimensional subspace S of C d is a matix X C d that has othonomal columns and satisfies S = ange(x) We say that two subspaces S 1 and S 2 of C d ae othogonal, denoted S 1 S 2, if u 1u 2 = 0 fo all u 1 S 1 and all u 2 S 2 ; an equivalent condition is that X 1 X 2 = 0, whee X 1 and X 2 ae basis matices fo S 1 and S 2, espectively Given some gaph G and d N, an othogonal epesentation in C d fo G is a set of unit vectos {x u } u V (G) C d such that x ux v = 0 if uv E(G) It is clea that such a epesentation always exists fo d = V (G) Povided that G has at least one edge, it is clea that such a epesentation cannot be made fo d = 1 We define the othogonal ank of G to be ξ(g) = min { d : G has an othogonal epesentation in C d} Let d, N with d A d/-pojective epesentation, o d/-epesentation, is an assignment of matices {P u } u V (G) to the vetices of G such that fo each u V (G), P u C d d, ankp u =, P u = P u, and P 2 u = P u ; and if uv E(G), then P u P v = 0 In wods, a d/-epesentation is an assignment of ank- (d d) othogonal pojection matices (pojectos) to the vetices of G such that adjacent vetices eceive pojectos that ae othogonal The pojective ank

6 Electonic Jounal of Linea Algeba, ISSN A publication of the Intenational Linea Algeba Society Volume 32, pp , Apil L Hogben, KF Palmowski, DE Robeson, and S Seveini 102 of G is defined as { } d ξ f (G) = inf : G has a d/-epesentation d, Pojective ank was fist intoduced in 2012 by Robeson and Mančinska, whee it is noted that ξ f (G) ξ(g); see [15] and [11] fo additional infomation, popeties, and applications Complementay to the idea of an othogonal epesentation is that of a faithful othogonal epesentation (hee we follow the complementay usage in the minimum ank liteatue) In ode fo the definitions given next to coincide with those in the minimum ank liteatue, we must assume that the gaph G has no isolated vetices A faithful othogonal epesentation in C d fo a gaph G is a set of unit vectos {x u } u V (G) C d such that x ux v = 0 if and only if uv / E(G) We define the minimum positive semidefinite ank of G as (11) m + (G) = min { d : G has a faithful othogonal epesentation in C d} We say that a matix A C n n fits the ode-n gaph G if a ii = 1 fo all i [1 : n], and fo all i j, we have a ij = 0 if and only if ij / E(G) Let H + (G) = {A C n n : A 0 and A fits G} A faithful othogonal epesentation in C d fo G coesponds to a matix A H + (G) with anka d, and a matix A H + (G) with ank d can be factoed as A = B B fo some B C d n Thus, an altenate chaacteization (see, eg, [7]) of m + (G) is m + (G) = min{anka : A H + (G)} (and in fact, this is the customay definition of this paamete) The definitions and explanation given hee coincide with those in the liteatue povided that the gaph G has no isolated vetices The most common definition of H + (G) in the liteatue does not contain the assumption that a ii = 1 If vetex i is adjacent to at least one othe vetex, then popeties of positive semidefinite matices equie a ii > 0, and so A can be scaled by a positive diagonal conguence to a matix of the same ank and nonzeo patten that has all diagonal enties equal to one Howeve, conside the case whee G consists of n isolated vetices (no edges): then as defined in [1, 7], etc, m + (G) = 0, wheeas with ou definition m + (G) = n The two definitions of minimum positive semidefinite ank coincide pecisely when G has no isolated vetices Ou definition facilitates connections to the use of othogonal ank in the study of quantum issues, and the assumption of no isolated vetices is needed only when connecting to the minimum ank liteatue, so we omit it except when discussing connections to such wok (whee we state eithe this assumption o one that implies it, such as the gaph being connected and of ode at least two) We also note that fo any gaph the values of the paametes studied can be computed fom thei values on the connected components of the gaph (see Section 3), which facilitates handling cases with isolated vetices sepaately 2 Othogonal subspace epesentations and pojective ank In this section, we intoduce and discuss (d; ) othogonal subspace epesentations fo a gaph G, which ae extensions of othogonal epesentations in the spiit of factional gaph theoy [16] The -fold othogonal ank of a gaph, ξ [] (G), is defined and some popeties of this quantity ae examined We then elate these epesentations to d/pojective epesentations and tie pojective ank into the new theoy, fomalizing the existing undestanding that pojective ank and factional othogonal ank ae one and the same Unless othewise specified, all matices and vectos in this section ae assumed to be complex-valued

7 Electonic Jounal of Linea Algeba, ISSN A publication of the Intenational Linea Algeba Society Volume 32, pp , Apil Factional Appoach to Othogonal Repesentations 21 Othogonal subspace epesentations and -fold othogonal ank Let G be a gaph and let d, N with d A (d; ) othogonal subspace epesentation, o (d; )-OSR, fo G is a set of subspaces {S u } u V (G) such that fo each u V (G), S u is an -dimensional subspace of C d ; and if uv E(G), then S u S v The -fold othogonal ank of a gaph G is defined by ξ [] (G) = min {d : G has a (d; ) othogonal subspace epesentation} An othogonal epesentation in C d natually geneates a (d; 1) othogonal subspace epesentation, and vice vesa, so ξ(g) = ξ [1] (G) We now exploe some popeties of ξ [] (G) Lemma 21 ξ [] is a subadditive function of, ie, fo evey gaph G and all, s N, ξ [+s] (G) ξ [] (G) + ξ [s] (G) Poof Let d = ξ [] (G) and d s = ξ [s] (G) Then G has a (d ; ) othogonal subspace epesentation containing -dimensional subspaces of C d, say {S u} u V (G), and a (d s ; s) othogonal subspace epesentation containing s-dimensional subspaces of C ds, say {S s u} u V (G) We show by constuction that thee exists an othogonal subspace epesentation fo G containing ( + s)-dimensional subspaces of C d+ds Fo each u V (G), let Xu C d and Xu s C ds s be basis matices fo Su and Su, s espectively Define [ ] X X u = u 0 d s 0 ds Xu s C (d+ds) (+s), and let S u = ange(x u ) We immediately see that S u is a subspace of C d+ds, X u is a basis matix fo S u, and dim(s u ) = ankx u = ankx u + ankx s u = + s Suppose u, v V (G) and let X u, X v, X s u, X s v, X u, and X v be as above; then X ux v = [ (X u ) (X v) 0 0 (X s u) (X s v) Suppose uv E(G) Since {S u} is an othogonal subspace epesentation, we have (X u) (X v) = 0; similaly, (X s u) (X s v) = 0, so X ux v = 0 Since X u and X v ae basis matices fo S u and S v, espectively, we conclude that if uv E(G), then S u S v Thus, {S u } u V (G) is a (d + d s ; + s) othogonal subspace epesentation fo G, so ξ [+s] (G) d + d s = ξ [] (G) + ξ [s] (G) Coollay 22 Fo evey gaph G and all N, ξ [](G) Poof Since ξ [1] (G) = ξ(g), we have ξ(g) ξ [] (G) ξ [ 1] (G) + ξ(g) ξ(g) Obsevation 23 Fo evey gaph G and all N, ξ [] (G) ω(g) ]

8 Electonic Jounal of Linea Algeba, ISSN A publication of the Intenational Linea Algeba Society Volume 32, pp , Apil L Hogben, KF Palmowski, DE Robeson, and S Seveini 104 Poposition 24 Let N and let H be a subgaph of G Then ξ [] (H) ξ [] (G) Poof Since evey edge of H is an edge of G, any (d; ) othogonal subspace epesentation fo G povides a (d; ) othogonal subspace epesentation fo H, and the esult is immediate Poposition 25 Suppose N and G = t i=1 G i fo some gaphs {G i } t i=1 Then ξ [](G) = max i { ξ[] (G i ) } Poof Since each G { i max i ξ[] (G i ) } ξ [] (G) is an induced subgaph of G, we have ξ [] (G i ) ξ [] (G) fo each i, so Fo each i [1 : t], let d i = ξ [] (G i ) and let d = max i {d i } Let {Su} i u V (Gi) be a (d i ; ) othogonal subspace epesentation fo G i and fo each vetex u V (G i ) let Xu i C di be a basis matix fo Su i Fo each u V (G), we have u V (G i ) fo some i; define [ ] Xu i S u = ange 0 (d di) Each S u is an -dimensional subspace of C d, and if uv E(G), then uv E(G k ) fo some k, so Su k Sv k, which implies that S u S v (by constuction) Theefoe, {S u } u V (G) is a (d; )-OSR fo G, so ξ [] (G) d = max i {ξ [] (G i )} and equality follows This esult does not hold fo abitay gaph unions, as the following example fo the = 1 case shows Example 26 Let G = C 5 with V (G) = {1, 2, 3, 4, 5} and E(G) = {12, 23, 34, 45, 51} Define G 1 = P 4 with V (G 1 ) = {1, 2, 3, 4} and E(G 1 ) = {12, 23, 34} and define G 2 = P 3 with V (G 2 ) = {4, 5, 1} and E(G 2 ) = {45, 51} We see that G = G 1 G 2, but since ξ(p 3 ) = ξ(p 4 ) = 2 and ξ(c 5 ) = 3, it is not tue that ξ(g) = max{ξ(g 1 ), ξ(g 2 )} While the maximum popety obseved in Poposition 25 may not cay ove to the case when G is a nondisjoint union of gaphs, we ae still able to obtain a weake esult, which follows Poposition 27 Suppose N and G = t i=1 G i, whee G i is an induced subgaph of G fo each i Then ξ [] (G) t i=1 ξ [](G i ) Poof We pove the esult fo the case whee t = 2 and note that ecusive application of this case will pove the moe geneal one Fo each i {1, 2}, let d i = ξ [] (G i ) and {S i u} u V (Gi) be a (d i ; )-OSR fo G i, and fo each u V (G i ), let X i u C di be a basis matix fo S i u We patition V (G) = V (G 1 ) V (G 2 ) into thee disjoint sets and conside vetices in each set u V (G 1 ) \ V (G 2 ), let [ ] X 1 X u = u ; if u V (G 2 ) \ V (G 1 ), let and if u V (G 1 ) V (G 2 ), let X u = 0 d2 [ 0d1 X 2 u [ X 1 X u = u X 2 u ] ; ] If

9 Electonic Jounal of Linea Algeba, ISSN A publication of the Intenational Linea Algeba Society Volume 32, pp , Apil Factional Appoach to Othogonal Repesentations Fo each u V (G), let S u = ange(x u ) Each S u is an -dimensional subspace of C d1+d2 We conside multiple cases to show that if uv E(G), then X ux v = 0, so S u S v Thoughout, we assume that uv E(G) Fist, suppose that u V (G 1 ) \ V (G 2 ); then eithe v V (G 1 ) \ V (G 2 ) o v V (G 1 ) V (G 2 ) In eithe case, uv E(G 1 ) (since G 1 is an induced subgaph), and block multiplication yields X ux v = (X 1 u) X 1 v Since S 1 u S 1 v, this quantity equals the zeo matix, so S u S v The case whee u V (G 2 ) \ V (G 1 ) is simila If u, v V (G 1 ) V (G 2 ), then uv E(G 1 ) E(G 2 ) since G 1 and G 2 ae induced subgaphs Then XuX v = (Xu) 1 Xv 1 + (Xu) 2 Xv 2 Since Su 1 Sv 1 and Su 2 Sv, 2 this quantity is again the zeo matix, so S u S v Theefoe, {S u } u V (G) is a (d 1 + d 2 ; )-OSR fo G, so ξ [] (G) d 1 + d 2 = ξ [] (G 1 ) + ξ [] (G 2 ) Lemma 28 Suppose that the complete gaph K t is a subgaph of G with V (K t ) = [1 : t] and G has a (d; ) othogonal subspace epesentation Then d t and G has a (d; ) othogonal subspace epesentation in which the vetex i V (K t ) is epesented by Poof By Obsevation 23, d ω(g) t span { e (i 1)+1,, e (i 1)+ 1, e i } If M C d l fo some l d and the columns of M ae othonomal, then by a change of othonomal basis thee exists a unitay matix U C d d such that UM = [e 1,, e l ] Let {S u } u V (G) be a (d; ) othogonal subspace epesentation fo G and fo each u V (G) let X u be a basis matix fo S u Define M = [X 1,, X t ] and choose U so that UM = [e 1,, e t ] Define S u = ange(ux u ) Then {S u} u V (G) is a (d; ) othogonal subspace epesentation fo G with the desied popety Theoem 29 If G = G 1 K t G 2 and N, then ξ [] (G) = max { ξ [] (G 1 ), ξ [] (G 2 ) } Poof Without loss of geneality, let d 1 = ξ [] (G 1 ) d 2 = ξ [] (G 2 ) and V (K t ) = [1 : t] Then by Lemma 28, fo i = 1, 2, each G i has a (d 1 ; ) othogonal subspace epesentation, {S i u} u V (G), in which vetex v t is epesented by S i v = span { e (v 1)+1,, e (v 1)+ 1, e v } Thus, fo v [1 : t], S 1 v = S 2 v; denote this common subspace by S v Fo vetices u V (G i ) \ [1 : t], define S u = S i u (obseve that u > t is in only one of V (G 1 ) o V (G 2 )) Then {S u } u V (G) is a (d 1 ; ) othogonal subspace epesentation fo G Poposition 210 If G is a gaph with ω(g) = χ(g), then ξ [] (G) = ω(g) fo evey N Poof It is well-known that ξ(g) χ(g) (see, eg, [15]) Theefoe, and thus, equality holds thoughout ω(g) ξ [] (G) ξ(g) χ(g) = ω(g), We note that pefect gaphs and chodal gaphs ae among those that satisfy ω(g) = χ(g), and so Poposition 210 applies to these classes Remak 211 Since ξ [1] (G) = ξ(g) fo evey gaph G, the pevious popeties of -fold othogonal ank also apply to othogonal ank, whee appopiate

10 Electonic Jounal of Linea Algeba, ISSN A publication of the Intenational Linea Algeba Society Volume 32, pp , Apil L Hogben, KF Palmowski, DE Robeson, and S Seveini Pojective ank as factional othogonal ank It is easy to see that (d; ) othogonal subspace epesentations ae in one-to-one coespondence with d/-epesentations Poposition 212 A gaph G has a (d; ) othogonal subspace epesentation if and only if G has a d/-epesentation Poof Let {S u } u V (G) be a (d; ) othogonal subspace epesentation fo a gaph G, so each S u is an -dimensional subspace of C d Fo each u V (G), define P u = X u X u, whee X u C d is a basis matix fo S u It is then easy to veify that P u C d d, ankp u = ankx u =, P u = P u, and P 2 u = P u Let uv E(G), so S u S v We see that S u S v X ux v = 0 X u X ux v X v = 0 P u P v = 0 Thus, if uv E(G), then P u P v = 0 We conclude that {P u } u V (G) is a d/-epesentation fo G Convesely, suppose that {P u } u V (G) is a d/-epesentation fo G Fo each u V (G), let P u = X u I Xu be a educed singula value decomposition of the pojecto P u (whee X u C d ) and define S u = ange(p u ) = ange(x u ) Clealy S u is an -dimensional subspace of C d If uv E(G), then P u P v = 0, so by the above chain of equivalences S u S v Theefoe, {S u } u V (G) is a (d; ) othogonal subspace epesentation fo G With this in mind, we obtain the following factional definition of pojective ank Theoem 213 Fo evey gaph G, ξ f (G) = inf { ξ[] (G) } Poof inf { } ξ[] (G) = inf = inf = inf d, = inf d, { } min{d : G has a (d; )-OSR} { { }} d min : G has a (d; )-OSR d { } d : G has a (d; )-OSR { } d : G has a d/-epesentation = ξ f (G) Given that this expession of ξ f (G) is simila to that of χ f (G) given in [16], it is not uneasonable to hope that this could shed some light on the question of the ationality of ξ f (G) fo all gaphs 1 Unfotunately, finding a b-fold coloing with c colos fo G is ultimately a fa diffeent poblem fom finding a (d; ) othogonal subspace epesentation fo G In the b-fold coloing poblem, we have a estiction on the numbe of available colos, which adds a cetain finiteness to the poblem: each vetex is assigned a subset of the available c < colos In contast, esticting the subspaces to lie in C d in the othogonal subspace epesentation poblem 1 Recall that χ f (G) is ational fo any gaph G

11 Electonic Jounal of Linea Algeba, ISSN A publication of the Intenational Linea Algeba Society Volume 32, pp , Apil Factional Appoach to Othogonal Repesentations does not impose this same type of finiteness: each vetex is assigned a finite dimensional subspace of C d, and d <, but thee ae infinitely many subspaces that can be assigned to each vetex We povide one additional equivalent definition of pojective ank, fo which we need the following utility esult fom [16], also commonly known as Fekete s Lemma Lemma 214 ([16], Lemma A41) Suppose g : N R is subadditive and g(n) 0 fo all n Then the limit g(n) lim n n exists and is equal to the infimum of g(n)/n (n N) Since ξ [] is subadditive, this yields the following coollay to the pevious theoem Coollay 215 Fo evey gaph G, and this limit exists ξ f (G) = inf { } ξ[] (G) ξ [] (G) = lim, With this esult, we see that many of the popeties of ξ [] (G) also apply to ξ f (G) Theoem 216 Fo evey gaph G: i) [15, 11] ξ f (G) ω(g) ii) If H is a subgaph of G, then ξ f (H) ξ f (G) iii) If G = t i=1 G i fo some gaphs {G i } t i=1, then ξ f (G) = max i {ξ f (G i )} iv) If G = t i=1 G i fo some induced subgaphs {G i } t i=1, then ξ f (G) t i=1 ξ f (G i ) v) If G = G 1 K t G 2, then ξ f (G) = max {ξ f (G 1 ), ξ f (G 2 )} vi) If G satisfies ω(g) = ξ(g), then ξ f (G) = ω(g) Poof Conside the second claim By Poposition 24, fo any N, ξ [] (H) ξ [] (G), so ξ [](H) ξ [] (G) Taking the limit as appoaches and applying Coollay 215, we have ξ f (H) ξ f (G) The emaining claims follow by applying simila aguments to the coesponding -fold esults 3 Factional minimum positive semidefinite ank In this section, we intoduce (d; ) faithful othogonal subspace epesentations, -fold minimum positive semidefinite ank, and factional minimum positive semidefinite ank, extending the definitions of faithful othogonal epesentations and minimum positive semidefinite ank We then intoduce faithful d/-pojective epesentations and connect eveything to pojective ank A connection to positive semidefinite matices is exploed, and popeties of ou new quantities ae poven Unless othewise specified, all matices and vectos in this section ae assumed to be complex-valued (the liteatue on minimum positive semidefinite ank is mixed, with both eal and complex cases studied) 31 Faithful othogonal subspace epesentations and factional minimum positive semidefinite ank Given a gaph G and d, N with d, a (d; ) faithful othogonal subspace epesentation, o (d; )-FOSR, fo G is a set of subspaces {S u } u V (G), whee fo each u V (G), S u is an -dimensional subspace of C d ; and

12 Electonic Jounal of Linea Algeba, ISSN A publication of the Intenational Linea Algeba Society Volume 32, pp , Apil L Hogben, KF Palmowski, DE Robeson, and S Seveini 108 S u S v if and only if uv / E(G) A faithful othogonal epesentation (as defined in Section 13) geneates a (d; 1) faithful othogonal subspace epesentation, and vice vesa Futhe, a (d; )-FOSR fo a gaph G is a (d; )-OSR fo its complement G, but the evese statement is not tue in geneal Now that we have defined an -fold analogue of a faithful othogonal epesentation, it is natual to conside a coesponding vesion of m + (G) The -fold minimum positive semidefinite ank of G is m + [](G) = min{d : G has a (d; ) faithful othogonal subspace epesentation} In paticula, we have m + [1] (G) = m+ (G), using definition (11) of m + ; we caution the eade that this coincides with the definitions of faithful othogonal epesentation and minimum positive semidefinite ank in the liteatue (eg [1, 7]) if and only if G has no isolated vetices We note that m + [](G) is subadditive The poof is analogous to the poof of Lemma 21 and is omitted, as ae the poofs fo othe esults in this section that paallel those fo the non-faithful case (ie, the ξ-family of paametes) Lemma 31 m + [] is a subadditive function of, ie, fo evey gaph G and all, s N, m + [+s] (G) m+ [] (G) + m+ [s] (G) As in the non-faithful case, an immediate coollay elates m + [] to m+ Coollay 32 Fo evey gaph G and all N, m + [] (G) m + (G) 2 Fo any gaph G, we define the factional minimum positive semidefinite ank of G as { m + m + [] f (G) = inf (G) } Notice that m + [] (G) d if G has a (d; ) faithful othogonal subspace epesentation, so m+ f (G) d Coollay 32 implies that the non-factional minimum positive semidefinite ank is an uppe bound on the factional vesion Again, ecall that this coincides with the liteatue if and only if the gaph G has no isolated vetices Coollay 33 Fo evey gaph G, m + f (G) m+ (G) 3 Since m + [](G) is subadditive, we have the following coollay, which follows fom Lemma 214 ([16], Lemma A41) 2 To apply this esult using the definition of m + (G) in the minimum ank liteatue equies that G have no isolated vetices 3 To apply this esult using the definition of m + (G) in the minimum ank liteatue equies G have no isolated vetices

13 Electonic Jounal of Linea Algeba, ISSN A publication of the Intenational Linea Algeba Society Volume 32, pp , Apil Factional Appoach to Othogonal Repesentations Coollay 34 Fo evey gaph G, and this limit exists m + f m + [] (G) = lim (G), We conclude this section with an example that gives futhe insight into these new paametes Example 35 Let N and conside the gaph G = P 4 with V (P 4 ) = {1, 2, 3, 4} and E(P 4 ) = {12, 23, 34} With e i as the i th standad basis vecto in C 2+1, we can veify that the following is a valid (2+ 1; )-FOSR fo P 4 : S 1 = ange([e 1, e 2,, e ]), S 2 = ange([e 2, e 3,, e +1 ]), S 3 = ange([e +1, e +2,, e 2 ]), S 4 = ange([e +2, e +3,, e 2+1 ]) Theefoe, m + [] (P 4) Suppose that {Q u } u V (P4) is a (2; )-FOSR fo P 4 ; we show that such a epesentation cannot exist Since 13, 14 / E(P 4 ), Q 1 Q 3 and Q 1 Q 4 The undelying space is C 2 and each subspace Q i is -dimensional, so we must theefoe have Q 3 = Q 4 = Q 1 Now, 23 E(P 4 ), so Q 2 Q 3, but 24 / E(P 4 ), so it also follows that Q 2 Q 4 Since Q 3 = Q 4, this is a contadiction; thus, thee is no (2; )-FOSR fo P 4, and so m + [] (P 4) = Using the limit chaacteization of m + f, it follows that m+ f (P 4) = lim 2+1 = 2 This example demonstates that the infimum in the definition of the factional minimum positive semidefinite ank cannot be eplaced with a minimum, even when m + f is a ational numbe Additionally, since m + (P 4 ) = 3, the gaph G = P 4 satisfies m + f (G) < m+ (G) 32 Faithful d/-pojective epesentations Let G be a gaph and d, N with d A faithful d/-pojective epesentation, o faithful d/-epesentation fo shot, is an assignment of matices {P u } u V (G) to the vetices of G such that fo each u V (G), P u C d d, ankp u =, P u = P u, and P 2 u = P u ; and P u P v = 0 if and only if uv / E(G) A faithful d/-epesentation fo G is a d/-epesentation fo G, but the convese is not necessaily tue It is convenient to note that a (d; ) faithful othogonal subspace epesentation fo G is equivalent to a faithful d/-epesentation The poof is analogous to that of Poposition 212; as befoe, we will omit such paallel poofs Poposition 36 A gaph G has a (d; ) faithful othogonal subspace epesentation if and only if G has a faithful d/-epesentation An immediate coollay gives an altenate definition fo m + f (G) Coollay 37 Fo evey gaph G, m + f (G) = inf d, { d : G has a faithful d/-epesentation Coollay 38 Fo any gaph G with complement G, ξ f (G) m + f (G) m+ (G) 4 Poof This follows fom the fact that any faithful d/-epesentation fo G is also a d/-epesentation fo G, as well as fom Coollay 33 4 To apply this esult using the definition of m + (G) in the minimum ank liteatue equies G have no isolated vetices }

14 Electonic Jounal of Linea Algeba, ISSN A publication of the Intenational Linea Algeba Society Volume 32, pp , Apil L Hogben, KF Palmowski, DE Robeson, and S Seveini Relation to positive semidefinite matices In this section, we connect (d; ) faithful othogonal subspace epesentations to positive semidefinite matices, thus genealizing the known esults fo the = 1 case (when the gaph in question has no isolated vetices) and connecting m + [](G) to the ank of a positive semidefinite matix We begin with some definitions Let G be a gaph on n vetices and suppose that V (G) = [1 : n] Fo some N, let A C n n be patitioned into an n n block matix [A ij ], whee A ij is the submatix in (block) ow i and (block) column j of A We say that the matix A -fits G if A ii = I fo each i V (G) and A ij = 0 if and only if ij / E(G), and define the set H + [] (G) = { A C n n : A 0 and A -fits G } Example 39 We povide a simple example fo the = 2 case Let G = P 3, the path on 3 vetices, with V (G) = {1, 2, 3} and E(G) = {12, 23} Choosing X = [e 1 e 2 e 1 e 4 e 3 e 4 ], whee e j is the j th standad basis vecto in C 4, we can veify that A = X X = H + [2] (P 3) This constuctive example gives an intuitive feel fo one diection of the poof of the main esult of this section Theoem 310 Fo evey gaph G on n vetices and any N, m + [] (G) = min { anka : A H + [] (G) } Poof Let d = m + [] (G), and let l = min { anka : A H + [] (G) } Fist, assume that {S i } is a (d; ) faithful othogonal subspace epesentation fo G and fo each i V (G) let X i C d be a basis matix fo S i Define X = [X 1 X 2 X n ] C d n and let B = X X C n n We see immediately that B 0 and ankb = ankx d Patitioning B into an n n block matix with blocks [B ij ] of size, we have B ij = Xi X j Since S i S j if and only if Xi X j = 0, we have B ij = 0 if and only if S i S j, which occus if and only if ij / E(G) Additionally, { since X i has } othonomal columns, we have B ii = I fo each i Theefoe, B H + [] (G), so min anka : A H + [] (G) ankb d = m + [] (G) Fo the evese inequality, suppose that B H + [](G) and ankb = l Then thee exists a matix X C l n such that B = X X Patition B into blocks [B ij ] and patition X into l blocks as X = [X 1 X 2 X n ] Fo each vetex i V (G), let S i = ange(x i ) C l Since Xi X i = I, we have ankx i =, so each S i is an -dimensional subspace of C l Additionally, Xi X j = B ij = 0 if and only if ij / E(G), so S i S j if and only if ij / E(G) { Theefoe, {S i } is an (l; ) faithful othogonal subspace epesentation fo G, so m + [] (G) l = min anka : A H + [] }, (G) and thus, equality holds This matix-based epesentation is a poweful theoetical tool that allows us to simplify the poofs of some popeties of -fold minimum positive semidefinite ank, as well as to moe clealy daw paallels to

15 Electonic Jounal of Linea Algeba, ISSN A publication of the Intenational Linea Algeba Society Volume 32, pp , Apil Factional Appoach to Othogonal Repesentations the existing and well-established = 1 case (although again, the connection to the liteatue equies that the gaph in question has no isolated vetices) The condition that A ii = I if A -fits a gaph G is a stong one, so we conclude this section with a weake condition that will be used to futhe simplify poofs without sacificing utility We say that A weakly -fits G if A ii is a diagonal matix with stictly positive diagonal enties fo each i V (G) and A ij = 0 if and only if ij / E(G) Clealy, any matix that -fits G also weakly -fits G Remak 311 Suppose that A weakly -fits a gaph G and let D = D 1 D n, whee each D i is the invese of the positive squae oot of A ii, ie, D i = A 1 2 ii Then the matix B = DAD -fits G, since D is a diagonal matix with stictly positive diagonal enties, so multiplication by D does not change the zeo patten of A Futhe, ankb = anka, since D has full ank This emak yields an immediate coollay to the pevious theoem Coollay 312 Fo evey gaph G on n vetices and any N, m + [] (G) = min { anka : A C n n, A 0 and A weakly -fits G } 34 Popeties of m + [] (G) and m+ f (G) In this section, we pove numeous esults egading popeties of -fold and factional minimum positive semidefinite ank, many of which extend known popeties of m + to the new paametes Obsevation 313 Fo evey gaph G and all N, m + [](G) α(g) Poposition 314 Let N and let H be an induced subgaph of G Then m + [] (H) m+ [] (G) Poof Fo any u, v V (H), uv E(H) if and only if uv E(G), since H is induced Theefoe, any (d; ) faithful othogonal subspace epesentation fo G povides a (d; ) faithful othogonal subspace epesentation fo H, and the esult follows immediately Poposition 315 If G = t i=1 G i fo some gaphs {G i } t i=1, then m+ [] (G) = t i=1 m+ [] (G i) fo each N Poof [ Suppose that V (G) = [1 : n] and that V (G i ) = n i fo i = 1, 2,, t Futhe assume that V (G i ) = 1 + i 1 j=1 n j : ] i j=1 n j, so that if A H + [] (G), then A = A 1 A 2 A t, whee A i H + [] (G i) fo each i Note that anka = t i=1 anka i We theefoe have { } m + [] (G) = min anka : A H + [] (G) { t } = min anka i : A i H + [] (G i) fo each i = = i=1 i=1 t { } min anka i : A i H + [] (G i) t m + [] (G i) i=1 Theoem 316 If G = t i=1 G i fo some gaphs {G i } t i=1, then m+ [] (G) t i=1 m+ [] (G i) fo each N

16 Electonic Jounal of Linea Algeba, ISSN A publication of the Intenational Linea Algeba Society Volume 32, pp , Apil L Hogben, KF Palmowski, DE Robeson, and S Seveini 112 Poof We pove the esult fo the case whee t = 2 and note that ecusive application of this case will pove the moe geneal one Let V (G) = [1 : n] whee n > 0 and assume that V (G 1 ) \ V (G 2 ) = [1 : n 1 ], V (G 1 ) V (G 2 ) = [n : n 1 + c], and V (G 2 ) \ V (G 1 ) = [n 1 + c + 1 : n 1 + c + n 2 ], whee n 1, n 2, c 0 (it is not assumed that each of these is stictly nonzeo) Note that n = n 1 + c + n 2, and this odeing assets that the fist n 1 vetices (enumeating in the natual ode) lie exclusively in G 1, the next c ae common to both gaphs, and the last n 2 lie exclusively in G 2 Fo i = 1, 2, let m + [] (G i) = d i and let A i H + [] (G i) be chosen so that anka i = d i Notice that A 1 C (n1+c) (n1+c) has its ows and columns indexed by V (G 1 ) = [1 : n 1 + c] and A 2 C (n2+c) (n2+c) has its ows and columns indexed by V (G 2 ) = [n : n] Let  1 = [ A ] [ ] 0 0 C n n,  2 = C n n, 0 A 2 and define A = Â1 + βâ2 C n n, whee β > 0 is chosen so that if A,  1, and Â2 ae patitioned into n n block matices with block size, then A ij = 0 if and only if (Â1) ij = 0 and (Â2) ij = 0 (ie, no cancellation of an entie block occus) Since A is a positive linea combination of positive semidefinite matices, A 0, and by ou choice of β we see that A weakly -fits G Theefoe, m + [] (G) anka ankâ1 + ankâ2 = d 1 + d 2 = m + [] (G 1) + m + [] (G 2) All of the esults we have poven fo -fold minimum positive semidefinite ank can be extended to esults fo factional minimum positive semidefinite ank The poof is analogous to that of Theoem 216 and is omitted Theoem 317 Fo evey gaph G: i) m + f (G) α(g) ii) If H is an induced subgaph of G, then m + f (H) m+ f (G) iii) If G = t i=1 G i fo some gaphs {G i } t i=1, then m+ f (G) = t i=1 m+ f (G i) iv) If G = t i=1 G i fo some gaphs {G i } t i=1, then m+ f (G) t i=1 m+ f (G i) Let G be a connected gaph of ode at least two A standad technique fo computing the minimum positive semidefinite ank of G is cut-vetex eduction [1, 7, 18]: Suppose that v V (G) is a cut-vetex and (G v) has connected components {H i } t i=1 Fo each i, let G i be the subgaph of G induced by the union of the vetices of H i with v, that is, G i = G[V (H i ) {v}] Then m + (G) = t i=1 m+ (G i ) Unfotunately, this technique does not cay ove to the -fold case when > 1, as the following example shows Example 318 Conside the gaph G = P 4, the path on 4 vetices, with V (G) = {x, y, v, z} in path ode; ecall fom Example 35 that m + [](G) = fo any N Taking v as a cut-vetex, we have G 1 = P 3 with V (G 1 ) = {x, y, v} and G 2 = P 2 with V (G 2 ) = {v, z} Fix > 1 Since α(g 1 ) = 2, any valid (d; )-FOSR fo G 1 must have d 2 Futhe, it is easy to see that m + (G 1 ) = 2, so 4 m + [] (G 1) 2 m + (G 1 ) = 2 Hence, equality holds and m + [] (G 1) = 2 Next, since m + (G 2 ) = 1 and d fo any valid (d; )-FOSR, we have m + [] (G 2) m + (G 2 ) =, so m + [] (G 2) = Hence, if > 1, then m + [] (G) = < 2 + = m+ [] (G 1) + m + [] (G 2), so cut-vetex eduction does not apply

17 Electonic Jounal of Linea Algeba, ISSN A publication of the Intenational Linea Algeba Society Volume 32, pp , Apil Factional Appoach to Othogonal Repesentations 35 Factional minimum positive semidefinite ank and pojective ank Recall that any (d; )-FOSR fo G is a (d; )-OSR fo G, but the convese statement does not apply in geneal It thus follows that ξ [] (G) m + [](G) fo any gaph G and N, and the next example demonstates that this inequality can be stict Example 319 Conside the gaph G = P 4 with V (P 4 ) = {1, 2, 3, 4} and E(P 4 ) = {12, 23, 34} and fix N Since ω(p 4 ) = 2, we have ξ [] (P 4 ) 2 With e i as the i th standad basis vecto fo C 2, it is easy to veify that the following is a (2; )-OSR fo P 4 : S 1 = S 3 = ange([e 1, e 2,, e ]), S 2 = S 4 = ange([e +1, e +2,, e 2 ]) Theefoe, ξ [] (P 4 ) = 2 Since P 4 = P 4 and m + [] (P 4) = (Example 35), we have 2 = ξ [] (P 4 ) < m + [] (P 4) = Recall fom Coollay 38 that ξ f (G) m + f (G) fo any gaph G While stict inequality may hold in the -fold case fo an abitay gaph G, we now demonstate that equality always holds in the factional case fo any gaph G Fo this esult, we equie the following lemma Lemma 320 Let G be a gaph with complement G Let {P u } u V (G) be a d/-epesentation fo G and let {R u } u V (G) be a faithful b/1-epesentation fo G Then fo any k N, G has a faithful (kd+b)/(k+1)- epesentation {Q u } u V (G) Futhe, given any ε > 0, k can be chosen such that d kd+b k+1 < ε, ie, the value of the faithful epesentation {Q u } fo G is within ε of the value of the (non-faithful) epesentation {P u } fo G Poof Since {P u } is a d/-epesentation fo G, we have P u C d d with ankp u = fo each u V (G) = V (G), and P u P v = 0 if uv E(G), so P u P v = 0 if uv / E(G) ) Let ε > 0 be abitay and choose k > ( d b 2 ε 1 Fo each vetex u V (G), let Q u C (kd+b) (kd+b) be the block diagonal matix constucted fom k copies of P u and one copy of R u, ie, ( k ) Q u = P u R u i=1 We see immediately that ankq u = k + 1, and since P u and R u ae pojectos, so is Q u Since P u P v = 0 if uv / E(G) and R u R v = 0 if and only if uv / E(G), we conclude that Q u Q v = 0 if and only if uv / E(G) Theefoe, {Q u } u V (G) is a faithful (kd + b)/(k + 1)-epesentation fo G, which veifies the fist claim By choice of k, we have k + 1 > d b ε Conside d kd + b k + 1 = d(k + 1) (kd + b) (k + 1) d b 1 = k + 1 d b ε < d b = ε, which veifies the second claim It was peviously noted that any faithful d/-epesentation fo G is also d/-epesentation fo G Lemma 320 is a patial convese in the sense that, given any d/-epesentation fo G, we can constuct a

18 Electonic Jounal of Linea Algeba, ISSN A publication of the Intenational Linea Algeba Society Volume 32, pp , Apil L Hogben, KF Palmowski, DE Robeson, and S Seveini 114 faithful d 1 / 1 -epesentation fo G such that the two epesentations have essentially the same value This yields the next esult Theoem 321 Fo evey gaph G with complement G, Poof Let ξ f (G) = m + f (G) { } d R = : G has a d/-epesentation, { } d F = : G has a faithful d/-epesentation Fo any d d1 R and ε > 0, Lemma 320 assets that thee exists some 1 F such that follows that inf R = inf F, ie, ξ f (G) = m + f (G) d d1 1 < ε It Acknowledgment Some of this wok was done while Leslie Hogben was a geneal membe of the Institute fo Mathematics and its Applications (IMA) and duing a week-long visit of Kevin Palmowski to IMA; they thank IMA both fo financial suppot (fom NSF funds) and fo poviding a wondeful collaboative eseach envionment REFERENCES [1] M Booth, P Hackney, B Hais, CR Johnson, M Lay, TD Lenke, LH Mitchell, SK Naayan, A Pascoe, and BD Sutton On the minimum semidefinite ank of a simple gaph Linea Multilinea Algeba, 59: , 2011 [2] A Cabello, S Seveini, and A Winte Gaph-theoetic appoach to quantum coelations Phys Rev Lett, 112:040401, 2014 [3] PJ Cameon, A Montanao, MW Newman, S Seveini, and A Winte On the quantum chomatic numbe of a gaph Electon J Combin, 14:Reseach Pape #R81, 2007 [4] V Capao and M Lupini Intoduction to Sofic and Hypelinea Goups and Connes Embedding Conjectue Lectue Notes in Mathematics, Vol 2136, Spinge, 2015 [5] T Cubitt, L Mančinska, DE Robeson, S Seveini, D Stahlke, and A Winte Bounds on entanglement-assisted souce-channel coding via the Lovász theta numbe and its vaiants IEEE Tans Infom Theoy, 60: , 2014 [6] R Duan, S Seveini, and A Winte Zeo-eo communication via quantum channels, non-commutative gaphs and a quantum Lovász function IEEE Tans Infom Theoy, 59: , 2013 [7] S Fallat and L Hogben Minimum ank, maximum nullity, and zeo focing numbe of gaphs In Handbook of Linea Algeba, second edition, L Hogben (edito), CRC Pess, Boca Raton, 2013 [8] S Fallat and L Hogben The minimum ank of symmetic matices descibed by a gaph: A suvey Linea Algeba Appl, 426: , 2007 [9] M Howad, J Wallman, V Veitch, and J Emeson Contextuality supplies the magic fo quantum computation Natue, 510: , 2014 [10] D Leung, L Mančinska, W Matthews, M Ozols, and A Roy Entanglement can incease asymptotic ates of zeo-eo classical communication ove classical channels Comm Math Phys, 311:97 111, 2012 [11] L Mančinska and DE Robeson Gaph homomophisms J Combin Theoy, Se B, 118: , 2016 [12] N Ozawa About the Connes embedding conjectue: Algebaic appoaches Jpn J Math, 8: , 2013 [13] VI Paulsen, S Seveini, D Stahlke, IG Todoov, and A Winte Estimating quantum chomatic numbes J Funct Anal 270: , 2016 [14] VI Paulsen and IG Todoov Quantum chomatic numbes via opeato systems Q J Math, 66: , 2015 [15] DE Robeson Vaiations on a Theme: Gaph Homomophisms PhD Thesis, Univesity of Wateloo, 2013 [16] E Scheineman and D Ullman Factional Gaph Theoy Dove, Mineola, NY, 2011; also available online at http: //wwwamsjhuedu/ es/fgt/