A fractional approach to minimum rank and zero forcing. Kevin Francis Palmowski. A dissertation submitted to the graduate faculty

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1 A factional appoach to minimum ank and zeo focing by Kevin Fancis Palmowski A dissetation submitted to the gaduate faculty in patial fulfillment of the equiements fo the degee of DOCTOR OF PHILOSOPHY Majo: Applied Mathematics Pogam of Study Committee: Leslie Hogben, Majo Pofesso Fitz Keinet Ryan Matin Jennife Newman Namata Vaswani Iowa State Univesity Ames, Iowa 2015 Copyight c Kevin Fancis Palmowski, All ights eseved.

2 ii DEDICATION Hey Mom, Dad, Kimbely, Eica, Jason, and Megan, I did it! Thanks fo all of the love and suppot that you ve given me ove the yeas, especially while I ve been away at gad school. I ve leaned a lot at Iowa State, but pehaps the most impotant lesson of all is that ou family is amazing and we e all daned lucky to have each othe. No one deseves this dedication as much as you guys. Love, Kevin

3 iii TABLE OF CONTENTS LIST OF FIGURES v ACKNOWLEDGEMENTS ABSTRACT vi vii CHAPTER 1. INTRODUCTION Oveview Oganization of the thesis CHAPTER 2. ORTHOGONAL REPRESENTATIONS, PROJECTIVE RANK, AND FRACTIONAL MINIMUM POSITIVE SEMIDEFI- NITE RANK: CONNECTIONS AND NEW DIRECTIONS Intoduction Applications A factional appoach Backgound, definitions, and notation Othogonal subspace epesentations and pojective ank Othogonal subspace epesentations and -fold othogonal ank Pojective ank as factional othogonal ank Factional minimum positive semidefinite ank Faithful othogonal subspace epesentations and factional minimum positive semidefinite ank Faithful d/-pojective epesentations

4 iv Relation to positive semidefinite matices Popeties of m + [] (G) and m+ f (G) Factional minimum positive semidefinite ank and pojective ank 34 CHAPTER 3. FRACTIONAL ZERO FORCING VIA THREE-COLOR FORCING GAMES Intoduction Zeo focing games Motivation and method Definitions and notation Contibution and oganization of the pape Factional positive semidefinite focing The -fold positive semidefinite focing game and factional positive semidefinite focing numbe Global intepetation of -fold positive semidefinite focing Thee-colo intepetation of factional positive semidefinite focing Results fo factional positive semidefinite focing numbe Factional positive semidefinite focing numbes fo gaph families Thee-colo intepetation of skew zeo focing The thee-colo skew zeo focing game Geneal esults fo skew zeo focing Skew zeo focing as factional zeo focing Leaf-stipping and skew zeo focing numbe CHAPTER 4. CONCLUSIONS Geneal conclusions Recommendations fo futue eseach

5 v LIST OF FIGURES Figue 3.1 Standad zeo focing game example Figue 3.2 Positive semidefinite zeo focing game example (fist steps) Figue 3.3 Skew zeo focing game example Figue 3.4 AON 2-fold PSD focing sets fo K Figue 3.5 Optimal AON -fold PSD focing sets Figue 3.6 Gaph G with unique pefect matching and Z (G) >

6 vi ACKNOWLEDGEMENTS Fist and foemost, I owe an enomous amount of cedit to my adviso, Leslie Hogben, who has watched ove me fom the stat of my gaduate caee and has always had my best inteest at heat. Leslie, you ae the best. The membes of my committee have been vey patient and flexible duing thei yeas of sevice, and fo that I am quite gateful. Many thanks to my collaboatos Simone, David, and Michael. It was geat woking with each of you, and I wish you evey success as you go fowad in you caees. The Mathematics faculty at Iowa State Univesity ae all geat and it has been a blast leaning fom and getting to know them, but I must specially acknowledge Steve Butle and Kis Docta Lee fo going the exta mile to become my vey close fiends. It would be a gievous eo to fail to acknowledge Melanie Eickson and Ellen Olson, each of whom has done a temendous amount fo me duing my time at ISU. Not a week went by when they did not solve a last-minute cisis o egale me with an amusing anecdote. Ladies, take a bow you have eaned it. Rod Ragne deseves cedit fo ensuing that I have always had access to top-notch computing hadwae in my office. Witing papes and making slides fo talks was infinitely easie thanks to my speedy compute and dual-monito setup, and I am gateful to Rod fo all that he has done ove the yeas fo the gaduate students in the depatment. Lastly, I would like to thank all of my fellow ISU Math gads especially Aaon, AnnaVictoia, Bian, Gabi, John, Kistin, Maia, and Steve fo poviding me with entetainment and suppot though the yeas. Gad school would not have been the same without you, and I m thankful fo the memoies.

7 vii ABSTRACT This thesis applies techniques fom factional gaph theoy to develop factional vesions of gaph paametes elated to minimum ank and zeo focing. Pojective ank, a gaph paamete with applications to quantum infomation, is fomally elated to - fold genealizations of othogonal epesentations fo gaphs. Using simila techniques, factional minimum positive semidefinite ank is defined via -fold genealizations of faithful othogonal epesentations and -fold minimum positive semidefinite ank, and it is shown that the factional minimum positive semidefinite ank of any gaph equals the pojective ank of the complement of the gaph. An altenate chaacteization of -fold minimum positive semidefinite ank that consides the anks of cetain Hemitian matices is also pesented. Motivated by the connections between zeo focing games and minimum ank poblems, an -fold analogue of the positive semidefinite zeo focing pocess is intoduced and used to define the factional positive semidefinite focing numbe of a gaph. An analysis of the -fold positive semidefinite focing game leads to a thee-colo focing game that allows computation of factional positive semidefinite focing numbe without appealing to the -fold game. The thee-colo appoach is applied to the standad zeo focing game and it is shown that the skew zeo focing numbe of a gaph is exactly the paamete obtained by applying the factionalization technique to the standad zeo focing game. Gaphs whose skew zeo focing numbe equals zeo ae chaacteized via the thee-colo appoach and an algoithm.

8 1 CHAPTER 1. INTRODUCTION This thesis applies techniques fom factional gaph theoy [4] to develop factional vesions of gaph paametes elated to minimum ank and zeo focing. While the oveall theme of intoducing and analyzing factional vesions of peviously-studied gaph paametes is gaph theoetic in natue, linea algeba also plays a key ole in the development of these new paametes. The undelying theoy elated to this wok has connections to quantum infomation, contol of quantum systems, and modeling the spead of disease in a netwok, as descibed in Chaptes 2 and 3. Biefly, the factionalization pocess that we implement begins by defining an fold vesion of a gaph paamete. The -fold paamete should be an extension of the egula paamete to some highe dimension; fo example, if the egula paamete is elated to single objects, then the -fold vesion may conside sets of objects. Ceating a easonable definition fo the -fold paamete is an impotant and challenging aspect of this pocess; thee may be many ways to define an -fold paamete, so a definition that impats popeties simila to those of the egula gaph paamete is desiable. The factional gaph paamete is then defined to be the infimum (o, if appopiate, supemum) ove the natual numbes of the atio of the -fold paamete to. Depending on the popeties of the -fold paamete, thee may be equivalent ways to expess o define the factional paamete. As descibed in Section 1.1, defining factional minimum positive semidefinite ank and exploing its connection with pojective ank wee ou initial motivations. Since zeo focing pocesses ae elated to minimum ank poblems, a natual application of some

9 2 of the theoy developed to deive factional minimum positive semidefinite ank was to define -fold and factional zeo focing pocesses. As a final note, we emphasize that the tem factional gaph paamete is a nod to the method with which ou new paametes ae developed, and not a claim that any paticula paamete is ational-valued. Indeed, we will see that poving ationality of one paamete is an open poblem in quantum infomation, and the othe paametes consideed tun out to be intege-valued. 1.1 Oveview In Chapte 2, a factional analogue of minimum positive semidefinite ank is developed. This investigation was motivated by a desie to develop moe theoy elated to pojective ank, a paamete that was intoduced in 2012 and has connections to poblems aising in quantum infomation (see, fo example, [3]). Thee ae numeous open questions elated to pojective ank, most notably whethe thee exists a gaph whose pojective ank is iational; if so, then the infamous Tsielson s poblem can be answeed in the negative, thus solving an impotant open poblem in the ealm of quantum infomation (see Chapte 2 fo moe infomation and efeences). Given a gaph G = (V, E) with V (G) = {1, 2,..., n}, a symmetic matix A = [a ij ] C n n is said to fit G if a ij = 0 if and only if ij / E(G). The minimum ank of a symmetic matix that fits G is the minimum ank of the gaph G, denoted hee by m(g). By futhe esticting to positive semidefinite matices that fit G, we can similaly define the minimum positive semidefinite ank of G, denoted hee by m + (G). It is also possible to define minimum ank poblems using matices ove fields othe than C (specifically, R is moe often consideed in the liteatue); moe infomation on minimum ank poblems and an extensive bibliogaphy can be found in [1]. Ou use of

10 3 C is necessitated by connections to existing paametes aising fom poblems based in quantum physics. An equivalent definition of minimum positive semidefinite ank is based on faithful othogonal epesentations fo a gaph. A faithful othogonal epesentation fo a gaph G is a set of vectos {x u } u V (G) C d (fo some d) such that x ux v = 0 if and only if uv / E(G). It can be shown that the minimum d such that G has a faithful othogonal epesentation in C d is equal to m + (G). Though thei names ae simila, faithful othogonal epesentations ae complementay to a diffeent type of vecto epesentation fo a gaph known as an othogonal epesentation. An othogonal epesentation fo a gaph G is a set of vectos {x u } u V (G) C d (fo some d) such that if uv E(G), then x ux v = 0. Note that, in contast to the definition of a faithful othogonal epesentation, the vectos in an othogonal epesentation ae othogonal when they coespond to edges in the gaph, and the othogonality condition is not an if and only if. The othogonal ank of G, denoted ξ(g), is the minimum d such that G has an othogonal epesentation in C d. Because a faithful othogonal epesentation fo a gaph G is an othogonal epesentation fo the gaph s complement G, it is clea that ξ(g) m + (G) fo any gaph G. The pojective ank of a gaph is a paamete of inteest to those woking in quantum infomation (see Section fo a mathematical definition of pojective ank). Peviously, it was infomally conjectued and assumed to be tue that pojective ank can be consideed though the lens of factional gaph theoy as factional othogonal ank. In Sections and 2.2.2, we develop the necessay machiney to establish this claim (Theoem ). Due to the elationship between othogonal ank and minimum positive semidefinite ank, it was also conjectued that pojective ank would be elated to factional minimum positive semidefinite ank, should definition of such a paamete be feasible. In Section 2.3, we define factional minimum positive semidefinite ank and use this new

11 4 paamete to investigate the conjectue. A main and somewhat unexpected esult in that section, Theoem , is that the pojective ank of a gaph equals the factional minimum positive semidefinite ank of its complement. This connection, as well as the theoy developed in Chapte 2, should povide an avenue though which esults petaining to minimum ank can be extended and adapted to infom new developments petaining to pojective ank. The focus of Chapte 3 is developing factional analogues of zeo focing paametes. The zeo focing numbe (positive semidefinite zeo focing numbe) of a gaph G, denoted hee by Z(G) (Z + (G)), can be obtained by playing a vetex coloing game on the gaph. By coloing some vetices of G blue and the est white and epeatedly applying a focing ule (by which white vetices can be tuned blue), the playe seeks to colo the entie gaph blue; if this is possible, the initial set of blue vetices is a focing set. Zeo focing numbes ae of inteest because of thei applications to contol of quantum systems and maximum nullity poblems (see Chapte 3 fo moe infomation and efeences). As noted in Section 3.1.2, the mateial in Chapte 3 was initially motivated by analysis of matices that -fit a gaph, intoduced in Section Note that any gaph G is the gaph associated with a matix that fits G, and zeo focing pocesses take place on G. Ou -fold zeo focing pocess consides application of zeo focing ules to the gaph of a matix that -fits G; while thee can be many such matices, Poposition allows us to focus on matices that ae associated with the (independent) -blowup of G. With an -fold paamete established, we ae able to define a factional vesion of the positive semidefinite zeo focing numbe. An examination of the focing game played on the gaph blowup eveals that -fold focing sets with cetain stuctue must always exist (Theoem 3.2.6). This esult leads to the altenate definition of factional positive semidefinite focing numbe pesented in Theoem Section builds on these esults by intoducing a new focing game that uses thee colos dak blue, light blue, and white. It is then shown that this new

12 5 game allows diect computation of the factional positive semidefinite focing numbe of a gaph without appealing to the -fold game, a main esult of Chapte 3. Section 3.3 applies the thee-colo appoach consideed fo factional positive semidefinite focing to (standad) zeo focing, poviding an altenate chaacteization of the skew zeo focing game, a specific type of zeo focing game that was peviously consideed in [2]. This intepetation is used thoughout the chapte to gain new insight into the skew zeo focing game and to pove new esults about skew zeo focing. In Section 3.3.3, we apply the factionalization pocess to the standad zeo focing game, and Theoem shows that the factional focing numbe of a gaph is actually the skew zeo focing numbe of the gaph. The thee-colo appoach also shows its use in Section 3.3.4, whee an algoithm is used to completely chaacteize gaphs whose skew zeo focing numbe equals zeo (Theoem ); this is anothe main esult of Chapte Oganization of the thesis This thesis is a collection of eseach papes submitted to jounals. Chapte 1 povides an oveview of the topics discussed in the emaining chaptes and seves to elucidate the common themes of this wok. Note that all papes follow the mathematical convention of alphabetizing the authos names. Chapte 2 contains the pape Othogonal Repesentations, Pojective Rank, and Factional Minimum Positive Semidefinite Rank: Connections and New Diections, which is joint wok of Kevin F. Palmowski with Leslie Hogben, David E. Robeson, and Simone Seveini. Kevin Palmowski was esponsible fo most of the eseach and almost all of the witing fo this pape. A vesion of this pape was submitted to Electonic Jounal of Linea Algeba; the diffeences between the submitted vesion and that pesented in this thesis ae mino, non-mathematical editoial changes.

13 6 Chapte 3 contains the pape Factional Zeo Focing via Thee-colo Focing Games, which is joint wok of Kevin F. Palmowski with Leslie Hogben, David E. Robeson, and Michael Young. The peliminay eseach fo this pape was conducted jointly by all authos ove the couse of one week duing a visit of David E. Robeson to Iowa State Univesity. Results wee subsequently efined and poof details wee filled in by Kevin Palmowski, who was esponsible fo almost all of the witing of this pape. A vesion of this pape was submitted to Discete Applied Mathematics; the diffeences between the submitted vesion and that pesented in this thesis ae mino, non-mathematical editoial changes. Concluding emaks and a discussion of futue avenues fo eseach ae pesented in Chapte 4.

14 7 Bibliogaphy [1] S. Fallat and L. Hogben, Minimum Rank, Maximum Nullity, and Zeo Focing Numbe of Gaphs, in Handbook of Linea Algeba, 2nd ed., L. Hogben, ed., CRC Pess, Boca Raton, FL, [2] IMA-ISU eseach goup on minimum ank (M. Allison, E. Bodine, L. M. DeAlba, J. Debnath, L. DeLoss, C. Ganett, J. Gout, L. Hogben, B. Im, H. Kim, R. Nai, O. Pypoova, K. Savage, B. Shade, A. Wangsness Wehe), Minimum ank of skewsymmetic matices descibed by a gaph, Linea Algeba Appl., 432: , [3] D. E. Robeson and L. Mančinska, Gaph Homomophisms fo Quantum Playes, to appea in J. Combin. Theoy Se. B (2014), axiv: [quant-ph], [4] E. Scheineman and D. Ullman, Factional Gaph Theoy, Dove, Mineola, NY, 2011; also available online fom

15 8 CHAPTER 2. ORTHOGONAL REPRESENTATIONS, PROJECTIVE RANK, AND FRACTIONAL MINIMUM POSITIVE SEMIDEFINITE RANK: CONNECTIONS AND NEW DIRECTIONS Modified fom a pape submitted to Electonic Jounal of Linea Algeba Leslie Hogben, Kevin F. Palmowski, David E. Robeson, and Simone Seveini 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 fo 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. Ameican Institute of Mathematics, 600 E. Bokaw Rd., San Jose, CA 95112, USA (hogben@aimath.og). Depatment of Mathematics, Iowa State Univesity, Ames, IA 50011, USA ({LHogben,kpalmow}@iastate.edu). Division of Mathematical Sciences, Nanyang Technological Univesity, SPMS-MAS-03-01, 21 Nanyang Link, Singapoe (dobeson@ntu.edu.sg). Depatment of Compute Science, Univesity College London, Gowe Steet, London WC1E 6BT, United Kingdom (simoseve@gmail.com).

16 9 2.1 Intoduction This pape deals with factional vesions of gaph paametes defined by othogonal epesentations, including minimum positive semidefinite ank. In Section 2.2, 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 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 (e.g., in [14, 15]) but not igoously pesented (fomal definitions of pojective ank and othe paametes ae given in Section 2.1.3). In Section 2.3, 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 2.3.5, 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 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 2.1.1), give a bief intoduction to the factional appoach

17 10 to chomatic numbe to motivate ou definitions (Section 2.1.2), and povide necessay notation and teminology (Section 2.1.3) 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, 11], 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 [12]. 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

18 11 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 [15] 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 [15] that pojective ank is monotone with espect to quantum homomophisms, i.e., 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 [14]. 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 find a sepaation between quantum chomatic numbe and a ecently defined semidefinite elaxation of this paamete, answeing a question posed in [12]. 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.

19 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, we can define 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.

20 13 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 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. 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.

21 14 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, i.e., 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, i.e., 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 1X 2 = 0, whee X 1 and X 2 ae basis matices fo S 1 and S 2, espectively.

22 15 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, ank P 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 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 [14] and [15] 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 m + (G) = min { d : G has a faithful othogonal epesentation in C d}. (2.1)

23 16 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 ank A 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, e.g., [7]) of m + (G) is m + (G) = min{ank A : 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 2.3), which facilitates handling cases with isolated vetices sepaately.

24 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 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).

25 18 Lemma ξ [] is a subadditive function of, i.e., 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 {Su} u V (G), and a (d s ; s) othogonal subspace epesentation containing s-dimensional subspaces of C ds, say {Su} s 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 u = X 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 ) = ank X u = ank X u + ank X s u = + s. Suppose u, v V (G) and let Xu, Xv, Xu, s Xv, s X u, and X v be as above; then X ux v = (X u) (Xv) 0 0 (Xu) s (Xv) s. Suppose uv E(G). Since {S u} is an othogonal subspace epesentation, we have (Xu) (Xv) = 0; similaly, (Xu) s (Xv) s = 0, so XuX 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 Fo evey gaph G and all N, ξ [](G) ξ(g). Poof. Since ξ [1] (G) = ξ(g), we have ξ [] (G) ξ [ 1] (G) + ξ(g)... ξ(g).

26 19 Obsevation Fo evey gaph G and all N, ξ [] (G) ω(g). Poposition 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 Suppose N and G = ti=1 G i fo some gaphs {G i } t i=1. Then ξ [] (G) = max i { ξ[] (G i ) }. Poof. Since each G i is an induced subgaph of G, we have ξ [] (G i ) ξ [] (G) fo each i, so max i { ξ[] (G i ) } ξ [] (G). Fo each i [1 : t], let d i = ξ [] (G i ) and let d = max i {d i }. Let {S i u} u V (Gi ) be a (d i ; ) othogonal subspace epesentation fo G i and fo each vetex u V (G i ) let X i u C d i be a basis matix fo Su. i Fo each u V (G), we have u V (G i ) fo some i; define S u = ange X i u 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 S k u S k v, 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 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 )}.

27 20 While the maximum popety obseved in Poposition 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 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 d i 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. If u V (G 1 ) \ V (G 2 ), let X u = X1 u 0 d2 ; if u V (G 2 ) \ V (G 1 ), let and if u V (G 1 ) V (G 2 ), let X u = X u = 0 d 1 X 2 u X1 u X 2 u ;. Fo each u V (G), let S u = ange(x u ). Each S u is an -dimensional subspace of C d 1+d 2. We conside multiple cases to show that if uv E(G), then XuX 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 XuX v = (Xu) 1 Xv 1. Since Su 1 Sv, 1 this quantity equals the zeo matix, so S u S v. The case whee u V (G 2 ) \ V (G 1 ) is simila.

28 21 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 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 span { e (i 1)+1,..., e (i 1)+ 1, e i }. Poof. By Obsevation 2.2.3, d ω(g) t. 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 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 2.2.8, fo i = 1, 2, each G i has a (d 1 ; ) othogonal subspace epesentation, {Su} i u V (G), in which vetex v t is epesented by Sv i = span { e (v 1)+1,..., } e (v 1)+ 1, e v. Thus fo v [1 : t], S 1 v = Sv; 2 denote this common subspace by S v. Fo vetices u V (G i ) \ [1 : t], define S u = Su i (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.

29 22 Poposition 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, e.g., [14]). Theefoe, ω(g) ξ [] (G) ξ(g) χ(g) = ω(g) and thus equality holds thoughout. We note that pefect gaphs and chodal gaphs ae among those that satisfy ω(g) = χ(g), and so Poposition applies to these classes. Remak Since ξ [1] (G) = ξ(g) fo evey gaph G, the pevious popeties of -fold othogonal ank also apply to othogonal ank, whee appopiate Pojective ank as factional othogonal ank It is easy to see that (d; ) othogonal subspace epesentations ae closely elated to d/-epesentations; in fact, they ae in one-to-one coespondence. Poposition A gaph G has a (d; ) othogonal subspace epesentation if and only if G has a d/-epesentation. Poof. Suppose that G has a (d; ) othogonal subspace epesentation {S u } u V (G), so each S u is an -dimensional subspace of C d. Fo each u V (G), define P u = X u Xu, whee X u C d is a basis matix fo S u. It is then easy to veify that P u C d d, ank P u = ank X u =, Pu = P u, and Pu 2 = 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.

30 23 Fo the convese, 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 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 1 Recall that χ f (G) is ational fo any gaph G.

31 24 othogonal subspace epesentation poblem 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 ([16], Lemma A.4.1). 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 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 Fo evey gaph G: i) [14, 15] ξ f (G) ω(g). ii) If H is a subgaph of G, then ξ f (H) ξ f (G). iii) If G = ti=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).

32 25 Poof. Conside the second claim. By Poposition 2.2.4, fo any N, ξ [] (H) ξ [] (G), so ξ [](H) ξ [](G). Taking the limit as appoaches and applying Coollay , we have ξ f (H) ξ f (G). The emaining claims follow by applying simila aguments to the coesponding -fold esults. 2.3 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) 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 S u S v if and only if uv / E(G). A faithful othogonal epesentation (as defined in Section 2.1.3) 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.

33 26 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 (2.1) of m + ; we caution the eade that this coincides with the definitions of faithful othogonal epesentation and minimum positive semidefinite ank in the liteatue (e.g. [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 and is omitted, as ae the poofs fo othe esults in this section that paallel those fo the non-faithful case (i.e., the ξ-family of paametes). Lemma m + [] is a subadditive function of, i.e., 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 Fo evey gaph G and all N, as m + [] (G) m + (G). Fo any gaph G, we define the factional minimum positive semidefinite ank of G m + f (G) = inf { m + [] (G) }. Notice that if G has a (d; ) faithful othogonal subspace epesentation, then m + [] (G) d, so m + f (G) d. We can uppe bound factional minimum positive semidefinite ank by the nonfactional vesion by using Coollay Again, ecall that this coincides with the liteatue if and only if the gaph G has no isolated vetices.

34 27 Coollay Fo evey gaph G, m + f (G) m+ (G). Since m + [](G) is subadditive, we have the following coollay, which follows fom Lemma ([16], Lemma A.4.1). Coollay 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 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).

35 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, ank P 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 evese 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 ; as befoe, we will omit such paallel poofs. Poposition 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 Fo evey gaph G, m + f { d (G) = inf d, } : G has a faithful d/-epesentation. Coollay Fo any gaph G with complement G, ξ f (G) m + f (G) m+ (G). 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