Simple Graph Density Inequalities with no Sum of Squares Proofs
Grigoriy Blekherman, Annie Raymond, Mohit Singh, Rekha R. Thomas
aa r X i v : . [ m a t h . C O ] D ec SIMPLE GRAPH DENSITY INEQUALITIESWITH NO SUM OF SQUARES PROOFS
GRIGORIY BLEKHERMAN, ANNIE RAYMOND, MOHIT SINGH, AND REKHA R. THOMAS
Abstract.
Establishing inequalities among graph densities is a central pursuit in extremalcombinatorics. A standard tool to certify the nonnegativity of a graph density expression isto write it as a sum of squares. In this paper, we identify a simple condition under which agraph density expression cannot be a sum of squares. Using this result, we prove that theBlakley-Roy inequality does not have a sum of squares certificate when the path length isodd. We also show that the same Blakley-Roy inequalities cannot be certified by sums ofsquares using a multiplier of the form one plus a sum of squares. These results answer twoquestions raised by Lov´asz. Our main tool is used again to show that the smallest open caseof Sidorenko’s conjectured inequality cannot be certified by a sum of squares. Finally, weshow that our setup is equivalent to existing frameworks by Razborov and Lov´asz-Szegedy,and thus our results hold in these settings too. Introduction
A graph G has vertex set V ( G ) and edge set E ( G ). All graphs are assumed to be simple,without loops or multiple edges. The homomorphism density of a graph H in a graph G ,denoted by t ( H ; G ), is the probability that a random map from V ( H ) to V ( G ) is a graphhomomorphism, i.e., it maps every edge of H to an edge of G . An inequality betweenhomomorphism densities refers to an inequality between t ( H i ; G ), for some finite graphs H i ,that is valid for all graphs G .Many results and problems in extremal graph theory can be restated as inequalities be-tween homomorphism densities [10, 14]. The Cauchy-Schwarz inequality has been one of thepowerful tools used to verify density inequalities for graphs and hypergraphs [5, 6, 10, 15].This proof method is equivalent to the general sum of squares (sos) proof method that hasbeen widely used in optimization [2]. Moreover, sos proofs naturally yield to a computerizedsearch via semidefinite programming. It was shown in [11] that every true inequality betweenhomomorphism densities is a limit of Cauchy-Schwarz inequalities.On the other hand, Hatami and Norine [7] show significant computational limitations onverifying inequalities between homomorphism densities. Firstly, they show that the problemof verifying the validity of an inequality between homomorphism densities is undecidable.Moreover, they also show that there are valid linear inequalities between graph homomor-phism densities that do not have a finite sos proof. Date : December 24, 2018.Grigoriy Blekherman was partially supported by NSF grant DMS-1352073. This material is partially basedupon work supported by the National Science Foundation under Grant No. 1440140, while Annie Raymondand Rekha Thomas were in residence at the Mathematical Sciences Research Institute in Berkeley, California,during the fall of 2017. Mohit Singh was partially supposed by NSF grant CCF-1717947 and Rekha Thomaswas partially supported by NSF grant DMS-1719538. espite the above negative results, the limitations of the sos proof method in proving aparticular graph density inequality of interest has been unclear. The examples arising from[7] do not shed much light on natural graph density inequalities in extremal graph theory.In this paper, we give a simple criterion that rules out sos proofs for the validity of a givengraph density inequality. As a corollary of our method, we obtain that certain classical graphdensity inequalities cannot be proven via the sos method. Moreover, we also show that thesmallest unresolved instance of the celebrated Sidorenko’s conjecture cannot be resolved viathe sos method.To describe our results, we begin with a few definitions about the gluing algebra of graphs.We refer the reader to Lov´asz [10] for a broader exposition. A graph is partially labeled ifa subset of its vertices are labeled with elements of N := { , , , . . . } such that no vertexreceives more than one label. If no vertices of H are labeled then H is unlabeled .Let A denote the vector space of all formal finite R -linear combinations of partially labeledgraphs without isolated vertices, including the empty graph with no vertices which we denoteas 1. We call an element a = P α i H i of A a graph combination , each α i H i a term of a , andeach H i a constituent graph of a . The degree of a term α i H i , α i = 0, is the number of edgesin H i . We say that a is homogeneous of degree d if all its terms have degree d .Let A ∅ denote the subalgebra of A spanned by unlabeled graphs. We view elements a ∈ A ∅ as functions that can be evaluated on unlabeled graphs G via homomorphism densities. Anelement a = P α i H i of A ∅ is called nonnegative if P α i t ( H i ; G ) ≥ G .The vector space A has a product defined as follows. For two labeled graphs H and H , form the new labeled graph H H by gluing together the vertices in the two graphswith the same label, and keeping only one copy of any edge that may have doubled in theprocess. Equipped with this product, A becomes an R -algebra with the empty graph as itsmultiplicative identity.The algebra A admits a simple linear map into A ∅ that removes the labels in a graphcombination to create a graph combination of unlabeled graphs. We call this map unlabeling and denote it by [[ · ]].A sum of squares (sos) in A ∅ is a finite sum of unlabeled squares of graph combinations a i ∈ A , namely, P [[ a i ]]. It can be easily seen that a sos is a nonnegative graph combination. Definition 1.1.
An unlabeled graph F is called a trivial square if whenever F = [[ H ]] forsome labeled graph H , then H is a fully labeled copy of F .In Lemma 2.8, we give a characterization of trivial squares in terms of automorphismsof the underlying graph. Our main result is the following theorem that gives a sufficientcondition for when a graph combination is not a sos. Theorem 1.2.
Let f = P ts =1 λ s F s be a graph combination of unlabeled graphs F s and d min be the minimum degree of any F s . Suppose f satisfies the following conditions:(1) there exists an s such that the degree of F s is equal to d min and λ s < , and(2) for every s such that degree of F s equals d min and λ s > , F s is a trivial square.Then f is a not a sos. As a first application of our theorem, we consider the Blakley-Roy inequality [1]. Let P k denote the path of length k and e denote the graph with a single edge. Then the Blakley-Royinequality asserts that for every k , the combination P k − e k is nonnegative. Indeed variousproofs of this inequality have been obtained, for instance [1, 8]. We show the following result. orollary 1.3. For any odd integer k ≥ , P k is a trivial square. Therefore, for every odd k ≥ and for all λ ∈ R , λP k − e k is not a sos. The above result answers Question 17(b) in Lov´asz [9] which asked whether the Blakley-Roy inequality has a sos proof. In [9] Lov´asz also considered a more general certificate ofnonnegativity: it is easy to see that f is a nonnegative graph combination if there exists asos graph combination g such that f (1 + g ) is sos. Theorem 1.2 shows that for homogeneousgraph combinations, such nonnegativity certificates are no more powerful than usual sos. Inparticular, the Blakley-Roy inequality for odd paths cannot be certified in this way. Corollary 1.4.
For any λ ∈ R and for any k ≥ and odd, there is no sos g such that ( λP k − e k )(1 + g ) is a sos. This resolves question 21 of [9] which asked for an explicit example of a valid homo-morphism density inequality without such multiplicative certificates. The existence of suchinequalities already followed from the undecidability result of [7].As a final corollary, we consider Sidorenko’s conjecture [16] that states that for everybipartite graph H , the graph combination H − e | E ( H ) | is nonnegative. A special case of thisconjecture, which is known to be true, is the Blakely-Roy inequality. While it has beenverified for various graph families, the smallest H for which the conjecture remains open is H = K , \ C where K , is the complete bipartite graph where both parts contain fivevertices, and C is a Hamiltonian cycle with 10 vertices [4]. We show that Theorem 1.2implies that the above inequality cannot be resolved using the sos proof method. Corollary 1.5. If H = K , \ C , then H is a trivial square, and H − e is not a sos.Moreover, there is no sos g such that ( H − e )(1 + g ) is a sos. Our main technical tool is Lemma 2.4, which shows that for any sos f = P λ s F s = P [[ a i ]]with a i = P α ij H ij , there exists a term λ s F s of minimal degree in f such that F s only arisesas a square [[ H ij ]], and consequently, λ s >
0. From this, we derive Theorem 2.7, which showsthat in decomposing homogeneous graph combinations as sums of squares, we are severelyrestricted in the types of graphs that can be used in the underlying squares. In forthcomingwork we will show how these restrictions can be used to classify all homogeneous sums ofsquares of degrees 3 and 4 [3].This paper is organized as follows. In Section 2 we prove our main results on sums ofsquares in the gluing algebra A . In Section 3 we discuss the relation between our gluingalgebra and the Cauchy-Schwarz calculus of Razborov as well as the very closely relatedgluing algebras of Lov´asz-Szegedy. These connections prove that the three results presentedin Corollaries 1.3, 1.4 and 1.5 also hold in any of these settings. Acknowledgments.
We thank Prasad Tetali for bringing to our attention the smallestopen case of Sidorenko’s conjecture. We also want to thank Alexander Razborov for usefuldiscussions about this paper.2.
The Gluing Algebra and its Sums of Squares
Recall the algebra A from the introduction spanned by partially labeled graphs as a R -vector space. We call A a gluing algebra since multiplication in it works by gluing graphsalong vertices with the same labels. For example,
31 2 · =
23 1 . For a fixed nite set of labels L ⊂ N , let A L denote the subalgebra of A spanned by all graphs whoselabel sets are contained in L . Then A ∅ is the subalgebra of A spanned by unlabeled graphs. Lemma 2.1.
Let H and H be two partially labeled graphs such that deg( H H ) ≤ min { deg( H ) , deg( H ) } . Then deg( H H ) = deg( H ) = deg( H ) . Further, deg( H ) = deg( H ) and H and H havethe same set of fully labeled edges.Proof. Suppose H i has degree d i and l i fully labeled edges. Then deg( H i ) = 2 d i − l i . Let c bethe number of fully-labeled edges that are common to both H and H . Then c ≤ min { l , l } and deg( H H ) = d + d − c .We are given that d + d − c ≤ min { d − l , d − l } , which implies that0 ≤ l − c ≤ d − d and 0 ≤ l − c ≤ d − d . The extremes of the two inequalities give that d = d , while adding the two inequalitiesgives that l + l − c = 0. Since c ≤ min { l , l } , it follows that l = l = c . (cid:3) The unlabeling map [[ · ]] : A → A ∅ removes the labels in a graph combination. Notethat for any partially labeled graph H , deg( H ) = deg([[ H ]]). A sum of squares (sos) in A is a finite sum of unlabeled squares of graph combinations a i ∈ A , namely, P [[ a i ]]. Bydefinition, a sos in A lies in A ∅ . Example 2.2.
The Blakley-Roy inequality for a path of length two, − ≥
0, has asum of squares proof as follows ([10], pages 28-29).[[( − ) ]] = [[ − + ]] = − We will now investigate the structure of homogeneous graph combinations that are sos.Their properties and limitations are the key ingredients in the proof of our main results. Webegin with the following lemma whose proof will be postponed to the end of this section.
Lemma 2.3. (1) If F and H are two partially labeled graphs such that [[( F − H ) ]] = 0 ,then F = H .(2) Suppose P ki =1 α i [[( F i − H i ) ]] =0 with α i ≥ and F i = H i for each i , then α i = 0 forall i . Lemma 2.4.
Let f = P λ s F s = P [[ a i ]] be a sos in A with a i = P α ij H ij . Let d be theminimum degree of any cross product H ij H ik within any a i . Then there exists a term λ s F s in f of degree d such that F s only arises via squares [[ H ij ]] , and consequently, λ s > .Proof. Let A d ∅ denote the vector space spanned by all unlabeled graphs of degree d . Let C be the cone in A d ∅ generated by all unlabeled squares of the form [[( F − H ) ]] where F and H are distinct partially labeled graphs such that deg( F ) = deg( H ) = deg( F H ) = d . Since d is fixed, there are only finitely many possibilities for the generators [[( F − H ) ]] of C andhence C is polyhedral, and therefore, closed. Furthermore, C is pointed by Lemma 2.3 (2).Since C is closed and pointed, its dual cone C ∗ is full-dimensional in ( A d ∅ ) ∗ . Therefore, wemay pick a sufficiently generic linear functional L : A d ∅ → R from the interior of C ∗ thatwill not only have the property that L ( a ) > a ∈ C , but also takes distinctvalues on the finitely many unlabeled graphs of degree d in A d ∅ . onsider all distinct graphs [[ H ij H ik ]] of degree d that can be formed by multiplying twoconstituent graphs in any a i and then unlabeling, including the unlabeled squares [[ H ij ]].Let F be the unique largest graph in this list in the total order induced by L , and suppose F = [[ H ij H ik ]] for some i and j = k . By Lemma 2.1 we have that d = deg( F ) = deg( H ij ) =deg( H ik ). Since H ij = H ik , by Lemma 2.3 (1), [[( H ij − H ik ) ]] is a nonzero generator ofthe cone C , and hence, L ([[( H ij − H ik ) ]]) >
0. This implies that L ([[ H ij ]]) + L ([[ H ik ]]) > L ([[ H ij H ik ]]). Therefore, at least one of L ([[ H ij ]]) or L ([[ H ik ]]) is strictly greater than L ( F )which contradicts the choice of F . Thus F only arises via squares of the form [[ H ij ]] forsome H ij . Therefore, it must be a constituent graph of f and setting F s = F proves thelemma. (cid:3) Corollary 2.5.
Let f = P [[ a i ]] be a sos in A with a i = P α ij H ij and let d be the lowestdegree of a term in f . Then the degree d component of f is again a sos, P [[ c i ]] , where allcross products of terms in each c i have degree d .Proof. By Lemma 2.4, we know that d is the lowest degree of any cross product [[ H ij H ik ]] ofgraphs H ij , H ik in an a i . Let f d be the degree d component of f . For each i , let b i denote thegraph combination obtained from a i by deleting all terms α ij H ij for which deg[[ H ij ]] > d . ByLemma 2.1, a deleted term α ij H ij from a i could not have cross multiplied with another term α ik H ik in a i to produce a term of degree d . Therefore, the degree d component of P [[ b i ]] isprecisely f d .Suppose G, H, K are three partially labeled graphs in some b i with deg( GH ) = deg( HK ) = d . Then by Lemma 2.1, deg( G ) = deg( H ) = deg( K ) = d and G, H, K all have the sameset of fully labeled edges. Let c be the number of fully labeled edges in G, H, K . Then d = deg( G ) = 2 deg( G ) − c which implies that c = 2 deg( G ) − d . Similarly, c = 2 deg( K ) − d and hence deg( G ) = deg( K ). Therefore, deg( GK ) = deg( G ) + deg( K ) − c = 2 deg( G ) − G ) + d = d . To conclude, if deg( GH ) = deg( HK ) = d , then also deg( GK ) = d . Thismeans that we may define an equivalence relation by saying G ∼ H if deg( GH ) = d .Now group the terms in each b i so that all constituent graphs in a group are equiva-lent in the above sense. Suppose the graph combinations corresponding to each group are b i , b i , . . . , b it i . By construction, all cross products of terms in any b ij have degree d . Con-sider the new sos expression g := P i P t i j =1 [[ b ij ]]. By construction, deg( g ) = d . For each i ,all terms in P t i j =1 [[ b ij ]] occur among the terms of [[ b i ]]. By our regrouping of terms in a b i ,a term in the expansion of [[ b i ]] is absent from P t i j =1 [[ b ij ]] if and only if its degree is largerthan d . Therefore, g = f d , and we have obtained an sos expression P [[ c i ]] := P i P t i j =1 [[ b ij ]]for f d of the desired form. (cid:3) We illustrate the previous corollary with the following example.
Example 2.6. f = 2 + − − − − ) ]] ere, d = 3 and f = 2 + − { , } and { } . Therefore, we see that f = [[( − ) ]] + [[( ) ]] . The above results prove an important structural property of homogeneous graph combi-nations that are sos which we record in the following theorem. This property will play acrucial role in this paper.
Theorem 2.7.
Every homogeneous graph combination of degree d that is a sos has a sosexpression of the form P [[ a i ]] where all cross products of terms in any a i have degree d . Recall the definition of a trivial square of Definition 1.1. We now prove a characterizationof trivial squares using automorphisms of the graph.
Lemma 2.8.
An unlabeled connected graph G is a non-trivial square, i.e., G = [[ F ]] forsome partially labeled graph F that is not a fully labeled copy of G , if and only if there is anautomorphism ϕ of G such that(1) ϕ is an involution,(2) ϕ fixes a non-empty proper subset of the vertices of G , and(3) the vertices v that are not fixed by ϕ can be partitioned into two groups, each groupconsisting of one vertex from the pair { v, ϕ ( v ) } , such that there are no edges betweenvertices in the two groups.Proof. Suppose G = [[ F ]] is a non-trivial square. Consider the map ϕ : V ( G ) → V ( G )that fixes all vertices of G that were labeled in F , and sends an unlabeled vertex v of F to v ′ where v and v ′ are copies of the same unlabeled vertex in F . Then ϕ is an automorphismof G that is also an involution. Since G is connected, F is not unlabeled, and it is not fullylabeled by assumption. Therefore, ϕ fixes a non-empty proper subset of the vertices of G .The first group of vertices in (3) is made up of the unlabeled vertices v in F and the secondgroup is made up of the duplicates v ′ of unlabeled vertices in F that exist in F .Conversely, suppose G has an automorphism ϕ with properties (1)-(3). Then consider thegraph F obtained by identifying vertices v and ϕ ( v ) and deleting a second copy of any edgethat gets doubled in this process. Observe that Property (3) ensures that no edge is morethan doubled, and no loops are created, and therefore F is a simple graph. Label all verticesof F that identified with themselves with distinct labels to create a partially labeled graph ˜ F .It follows by construction that G = [[ ˜ F ]]. Since ϕ fixes a non-empty proper set of verticesof G , only a proper set of vertices of ˜ F are labeled, hence G is a non-trivial square. (cid:3) Example 2.9 (Paths) . Let P k be an unlabeled path with k edges and k + 1 vertices v , . . . , v k +1 . Using Lemma 2.8 one can argue that a path P k of odd length k is a trivialsquare. Every graph automorphism ϕ of P k has to send v to either itself or to v k +1 . Eachchoice completely determines ϕ since adjacent vertices have to be sent to adjacent vertices.If v is sent to v then ϕ is the trivial involution that fixes all vertices in P k . If v is sent to v k +1 then v is sent to v k , etc until v k +1 is sent to v . This involution doesn’t fix any verticesin P k . Either way, we see from Lemma 2.8 that P k is a trivial square. n the other hand, if k is even, then P = [[ F ]] where F is a path of length k with thefirst vertex labeled 1.The notion of trivial squares together with Theorem 2.7 will provide us with a tool torecognize homogeneous graph combinations that are not sos.We can now prove Theorem 1.2 from the introduction. Proof of Theorem 1.2.
Let f d min be the (homogeneous) lowest degree component of f . ByCorollary 2.5, if f is a sos, then f d min is a sos. By Theorem 2.7, f d min = P [[ a i ]] where allunlabeled cross products of constituent graphs in each a i have degree d min . We also knowfrom Lemma 2.4 that one of the F s in f d min only arises as [[ H ij ]] for some H ij in some a i and then λ s >
0. By assumption, whenever λ s > F s is a trivial square, which means that F s = [[ H ij ]] for some i and j and H ij is fully labeled.Pick a trivial square F s in f d min and suppose F s = [[ H ij ]]. For the same i and j , considerthe cross product [[ H ij H ik ]] for some j = k . Since H ij = H ik as partially labeled graphs,their product has degree larger than d which is a contradiction. So it must be that H ij isthe only constituent graph of a i and a i = α ij H ij . Therefore, we may remove all occurrencesof fully labeled graphs that square and unlabel to F s from the sos decomposition P [[ a i ]] toget an sos expression for f ′ = f d min − λ s F s . Repeating this procedure, we may remove alltrivial squares from f d min to get a graph combination ¯ f with only negative coefficients thatis still an sos. This is a contradiction since we showed in Lemma 2.4 that a sos always hasa term with a positive coefficient. (cid:3) We will now apply Theorem 1.2 to prove our main results. The first application is to showthat the Blakley-Roy inequality P k − e k ≥
0, for k ≥ Proof of Corollary 1.3. If λ ≤ λP k − e k is not sos by Lemma 2.4 which says thatevery homogenous sos has a term with a positive coefficient. If λ >
0, the result follows fromTheorem 1.2 and Example 2.9 which showed that P k is a trivial square. (cid:3) Problem 21 in [9] asks the general question as to whether it is always possible to certifythe nonnegativity of a graph combination f by multiplying it with (1 + g ) where g is sosand having the product be a sos? It was shown in [7] that the answer is no. We provide thefirst explicit example of this by showing that for f = λP k − e k there are no sos g ∈ A ∅ suchthat f (1 + g ) is a sos. Using results of Section 3, it will follow that this answers Lov´asz’squestion negatively. Proof of Corollary 1.4.
Let f = λP k − e k for any λ ∈ R where k is odd which we just showedis not a sos. Suppose there was a sos g ∈ A ∅ such that f (1 + g ) is sos. Then the lowestdegree part of f (1 + g ) is precisely f which is not a sos. This contradicts Corollary 2.5 whichsays that the lowest degree part of a sos is again sos. (cid:3) idorenko’s conjecture is that H − e | E ( H ) | ≥ H is a bipartite graph. Note that P k − e k ≥ H = K , \ C where K , is the complete bipartite graph with two color classes of size five and C is aHamiltonian cycle through the 10 vertices of K , . Our tools show that it is not possible touse sos to establish the nonnegativity of H − e | E ( H ) | when H = K , \ C . edcba j i h g fH = K , \ C labeled as in the proof of Corollary 1.5 Proof of Corollary 1.5.
We use Lemma 2.8 to show that H = K , \ C is a trivial square.The conclusion follows from Theorem 1.2 with the same argument as in Corollary 1.4.We first argue that the automorphism group of H is the dihedral group D which is theautomorphism group of the 10-cycle C . Observe that there is a unique (up to swappingcolors) two-coloring of H . Any involution of H either permutes the vertices within eachcolor class, or swaps the two color classes. The complement ¯ H of H is a 10-cycle with twocomplete graphs K on the even and odd vertices respectively. Any automorphism ϕ of H is also an automorphism of ¯ H . The above argument shows that ϕ sends edges of the unionof the two K ’s to themselves. Therefore ϕ sends the edges of the 10-cycle to itself, andthus the automorphism group of H is a subgroup of D . However, it is easy to see thatautomorphisms of the 10-cycle are also automorphisms of H , and the automorphism groupof H is D .There are 11 involutions in D , ten of which are reflections and one is rotation by 180degrees. We want to argue that each of these involutions violate at least one of the properties(1)-(3) of Lemma 2.8. Five of the reflections and the rotation do not fix any vertices whichviolates property (2). The remaining five reflections about the diagonals are involutions thatfix two vertices of H . We will argue that these reflections violate property (3). It suffices toargue this for one of them. Consider the reflection of H about its horizontal diagonal. Thisinvolution fixes vertices a and f , but sends b j , c i , d h and e g . Now we checkwhether the vertices that are not fixed by the involution can be partitioned as in (3). Wesee that vertices b, c, d, e have to be in the same group since there are edges connecting oneto the next. But then g, h, i, j also belong to this group because of the diagonals. Thus it isnot possible to divide the vertices that are not fixed by the involution into two groups as in(3), and H is a trivial square. (cid:3) We note that the proof of Lemma 2.4 says something special about the sos decompositionof homogeneous graph combinations of the form F − F . Recall the cone C from theproof of the lemma that was generated by unlabeled squares of the form [[( F − H ) ]] where eg( F ) = deg( H ) = deg( F H ) = d . Since C is polyhedral, it has a finite inequalitydescription which allows one to test for membership in C . Proposition 2.10.
A graph combination f = F − F where F , F are two unlabeled graphs ofthe same degree is a sos if and only if it has a sos decomposition of the form P λ ij [[( H i − H j ) ]] with λ ij ≥ .Proof. Suppose f is a sos but f C . Since C is pointed, there is a linear functional L such that L ( f ) < L ( c ) > c ∈ C . Since L ( F ) < L ( F ), the proof ofLemma 2.4 says that F is a square and its coefficient in f is positive which is a contradiction.Therefore, f lies in C which means that f = P λ ij [[( H i − H j ) ]] for some λ ij ≥ (cid:3) The rest of this section is devoted to the proof of Lemma 2.3. This needs the notion ofnonnegativity of a graph combination which we saw briefly in the introduction. For this, wewill need to view elements a ∈ A as functions that can be evaluated on unlabeled graphs G ,including those with isolated vertices. We closely follow the exposition in [7].Recall that a graph homomorphism between two unlabeled graphs H and G is an adja-cency preserving map h : V ( H ) → V ( G ) such that h ( i ) h ( j ) ∈ E ( G ) if ij ∈ E ( H ). Thehomomorphism density of H in G , denoted as t ( H ; G ) is the probability that a random mapfrom V ( H ) → V ( G ) is a homomorphism. Define t (1; G ) := 1 for all G . Now suppose H isa partially labeled graph and L H is its set of labels. Given a map ϕ : L H → V ( G ), definethe homomorphism density t ( H ; G, ϕ ) as the probability that a random map from V ( H )to V ( G ) is a homomorphism conditioned on the event that the labeled vertices in H aremapped to V ( G ) according to ϕ . Then, by the rules of conditional probability, t ([[ H ]]; G ) isthe (positively) weighted average of the conditional probabilities t ( H ; G, ϕ ) over all maps ϕ .For a combination of partially labeled graphs a = P ti =1 α i H i , let L a = ∪ ti =1 L H i be the unionof all label sets of all constituent graphs of a . Then for a fixed map ϕ : L a → V ( G ), define t ( a ; G, ϕ ) := P α i t ( H i ; G, ϕ | L Hi ).Suppose we now fix a label set L and a map ϕ : L → V ( G ). Then if H and H are twopartially labeled graphs whose label sets L H and L H are contained in L , t ( H H ; G, ϕ ) = t ( H ; G, ϕ | L H ) t ( H ; G, ϕ | L H ). Recall that A L was the subalgebra of A consisting of allpartially labeled graphs whose label sets are contained in L . Then we have that t ( − ; G, ϕ )is a homomorphism from A L to R .We say that a ∈ A is nonnegative if t ( a ; G, ϕ ) ≥ G and maps ϕ : L a → V ( G ). Note that any partially labeled graph H is nonnegative since t ( H ; G, ϕ ) is aprobability. By the same reason, H and [[ H ]] are also nonnegative, but graph combinations a ∈ A are not necessarily nonnegative since they have arbitrary coefficients. However, if a ∈ A , then a is nonnegative since t is a homomorphism. In particular, any sos P [[ a i ]] ∈ A is nonnegative.To prove Lemma 2.3, we will need the notion of weighted graph homomorphisms as in[10, § node-weighted graph G is one with node weights ω u ( G ) on the nodes u ∈ V ( G ).To a map h : V ( H ) → V ( G ), define ω h ( H, G ) := Q u ∈ V ( H ) ω h ( u ) ( G ). The number of weightedhomomorphisms (resp. weighted injective homomorphisms) from H to G is then P ω h ( H, G )where the sum varies over all homomorphisms h : nonzeroV ( H ) → V ( G ) (resp. all injectivehomomorphisms h : V ( H ) → V ( G )). roof of Lemma 2.3. (1) We need to show that if [[( F − H ) ]] = 0 then F = H . Suppose0 = [[( F − H ) ]] = [[ F ]] + [[ H ]] − F H ]]. Then [[ F ]] = [[ H ]] = [[ F H ]] sincedensity functions of unlabeled graphs are linearly independent [10, Corollary 5.45].This implies that F and H must have the same number of vertices, and the same setof labels L . Indeed, you see this by counting the number of vertices in each of thethree graphs [[ F ]] , [[ H ]] and [[ F H ]], differentiated by how many are labeled in F and H , and how many labels are shared between F and H . If for a graph G and amap ϕ : L → V ( G ), t ( F ; G, ϕ ) = t ( H ; G, ϕ ), then t (( F − H ) ; G, ϕ ) > t ([[( F − H ) ]] , G ) >
0, contradicting that [[( F − H ) ]] = 0. Our proof strategywill be to show that if F = H then there is a graph G and a map ϕ : L → V ( G )such that t ( F ; G, ϕ ) = t ( H ; G, ϕ ).In order to construct the graph G , we follow the proof of [10, Proposition 5.44].Weigh the unlabeled vertices in both F and H with distinct variables x i and call theseweighted graphs ˜ F and ˜ H . Then consider the 2 × M with rows indexedby F and H , with columns indexed by ˜ F and ˜ H and entries equal to the number ofweighted homomorphisms from F, H to ˜
F , ˜ H , where the (commonly) labeled verticesof F (resp. H ) and ˜ F (resp. ˜ H ) map to each other. The matrix M is filled withpolynomials and hence, det( M ) is a polynomial. We observe that rank( M ) = 2, i.e.,det( M ) is not identically zero. Indeed, since the variables x i are all distinct, themultilinear component of det( M ) is det( M ′ ), where M ′ is the matrix with entriesequal to the number of weighted injective homomorphisms from F (resp. H ) to ˜ F (resp. ˜ H ), which is nonzero, since M ′ is upper/lower triangular, and its diagonalentries are nonzero polynomials. Hence, det( M ) is not the zero polynomial, whichshows that for an algebraically independent substitution of the x i ’s, det( M ) will notvanish.This means that there is some choice of positive integer values for the x i ’s thatwill keep the matrix M non-singular. Substitute each x i with such a positive integerto get a new matrix M and replace a vertex v of weight m in ˜ F or ˜ H with m copiesof itself (each with the same neighborhood that v had) to obtain graphs G and G .The matrix with entries equal to the number of homomorphisms from H and F , to G and G , such that the commonly labeled vertices are mapped to each other isstill M . Here ϕ is again the map that sends L to V ( G ) and V ( G ). Convertingthese entries to homomorphism densities involves dividing each column in the matrixby a constant which keeps the resulting matrix again non-singular. Therefore, itsrows are not scalar multiples of each other, and either t ( F ; G , ϕ ) = t ( H ; G , ϕ ), or t ( F ; G , ϕ ) = t ( H ; G , ϕ ).(2) For λ i ≥ P λ i [[( F i − H i ) ]] = 0 if and only if for each i , either λ i = 0 or [[( F i − H i ) ]] = 0. We have that [[( F − H ) ]] = 0 if and only if F = H . Since F i = H i foreach i , it must be that λ i = 0 for all i . (cid:3) Translation of our gluing algebra to other settings
The goal of this section is to show that the existence of a sum of squares certificate is equiv-alent in Razborov’s flag algebra and Lov´asz-Szegedy’s gluing algebra and Hatami-Norine’s luing algebra, and the gluing algebra we presented in Section 2. Therefore Corollaries 1.3,1.4 and 1.5 hold in these settings as well. Relation to Lov´asz-Szegedy.
A family of gluing algebras very similar to ours was in-troduced in the work of Lov´asz and Szegedy [11]. The only difference is that they allowedcombinations of graphs with isolated vertices, and the graphs in each algebra were requiredto have the same labels. However, the resulting unlabeled sos expressions are the same upto removing isolated vertices, which does not affect homomorphism densities.Let f be a graph combination in A ∅ . Suppose that we have a sos expression f = P [[ a i ]]where each a i is in A L where L = [ k ] = { , . . . , k } and where a i = P tj =1 α ij H ij . For eachconstituent partially labeled graph H ij with label set L H ij , define ¯ H ij to be the graph H ij with k − | L H ij | labeled isolated vertices attached, where new vertices are labeled with thelabels from [ k ] \ L H ij . We thus obtain a sos P ¯ a i in the Lov´asz-Szegedy algebra of k -labeledquantum graphs. Observe that after unlabeling the expressions, P [[ a i ]] and P [[¯ a i ]] areequivalent up to adding or removing isolated vertices.Similarly, start with a sos expression in the algebra of k -labeled quantum graphs P ¯ a i .For a constituent partially k -labeled graph ¯ G define G to be the partially labeled graphobtained from ¯ G by removing the isolated vertices. This gives a sum of squares expressionin A , which agrees with the Lov´asz-Szegedy sum of squares up to removing isolated vertices. Relation to Hatami-Norine.
A variant of Lov´asz-Szegedy gluing algebra was defined byHatami and Norine in [7]. Our gluing algebra A is isomorphic to the quotient algebra intheir paper. The partially labeled graph H with no isolated vertices is just an explicit cosetrepresentative of the quotient by the ideal K generated by all differences of the form emptygraph minus 1-vertex graph with a label (or unlabeled). We refer to [7] for more details. Relation to Razborov’s Flag Algebras.
A different algebra was used in the work ofRazborov in [14]. There, partially labeled graphs are called flags . The main differenceis that flag algebras are concerned with induced subgraph density , while homomorphismdensity is known to be asymptotically equal to non-induced subgraph density . A well-knownM¨obius transformation relates induced and non-induced subgraph densities via a changeof basis. Multiplication in the flag algebras looks syntactically different from the gluingalgebra, however after passing through the M¨obius transformation and its inverse, the twomultiplications are the same. Therefore Cauchy-Schwarz proofs in the flag algebras areequivalent to sos proofs in the gluing algebra. We refer to [12] and [13] for more details.
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