A unified construction of semiring-homomorphic graph invariants
aa r X i v : . [ m a t h . C O ] J u l A UNIFIED CONSTRUCTION OF SEMIRING-HOMOMORPHICGRAPH INVARIANTS
TOBIAS FRITZ
Abstract.
It has recently been observed by Zuiddam that finite graphs form a pre-ordered commutative semiring under the graph homomorphism preorder together withjoin and disjunctive product as addition and multiplication, respectively. This led to anew characterization of the Shannon capacity Θ via Strassen’s Positivstellensatz: Θ( ¯ G ) =inf f f ( G ), where f : Graph → R + ranges over all monotone semiring homomorphisms.Constructing and classifying graph invariants Graph → R + which are monotone undergraph homomorphisms, additive under join, and multiplicative under disjunctive prod-uct is therefore of major interest. We call such invariants semiring-homomorphic . Theonly known such invariants are all of a fractional nature: the fractional chromatic num-ber, the projective rank, the fractional Haemers bounds, as well as the Lov´asz number(with the latter two evaluated on the complementary graph). Here, we provide a unifiedconstruction of these invariants based on linear-like semiring families of graphs. Alongthe way, we also investigate the additional algebraic structure on the semiring of graphscorresponding to fractionalization.Linear-like semiring families of graphs are a new notion of combinatorial geometrydifferent from matroids which may be of independent interest. Contents
1. Introduction 22. The preordered commutative semiring of graphs 43. The algebraic structure of blowup and fractionalization 84. Semiring families of graphs and the linear-like condition 105. Recovering all known semiring-homomorphic graph invariants 166. Some open problems 21References 23
Mathematics Subject Classification.
Primary: 05C72, 05C69; Secondary: 06F25.
Acknowledgements.
Most of this work was conducted while the author was affiliated with the MaxPlanck Institute for Mathematics in the Sciences. We thank Chris Cox, Chris Godsil, Matilde Marcolli,David Roberson, and Jeroen Zuiddam for various combinations of useful discussions and detailed feedbackon a draft. Introduction
The Shannon capacity of graphs is a notorious graph invariant with high relevance toinformation theory [17]. For technical convenience we state the definition of the Shannoncapacity of the complementary graph,Θ( ¯ G ) := sup n ∈ N n p ω ( G ∗ n ) . Here, ω is the clique number, and G ∗ n is the n -fold disjunctive product of G with itself.Although the very definition of Θ( ¯ G ) provides an algorithm for computing a sequence oflower bounds converging to Θ( ¯ G ), already finding an algorithm computing a convergentsequence of upper bounds seems to be an open problem. In other words, the computabilityof Θ( ¯ G ) itself is open. The simplest graph for which the Shannon capacity is not known isthe 7-cycle [22].As an application of Strassen’s Positivstellensatz [27, Corollary 2.6], it was recentlyshown by Zuiddam how to obtain a tight family of upper bounds on the Shannon capacity,improving substantially on our earlier characterization [10, Example 8.25]. Theorem 1.1 (Zuiddam) . The Shannon capacity satisfies
Θ( ¯ G ) = inf η η ( G ) , (1.1) where η ranges over all graph invariants η : Graph → R + which satisfy the normalization η ( K ) = 1 as well as the following conditions: ◦ η is monotone under graph homomorphisms: G → H = ⇒ η ( G ) ≤ η ( H ); ◦ η is additive under graph joins: η ( G + H ) = η ( G ) + η ( H ); ◦ η is multiplicative under disjunctive products: η ( G ∗ H ) = η ( G ) η ( H ) . Here,
Graph is the set of all (isomorphism classes of) finite graphs. So if a complete classi-fication of such graph invariants was available, our understanding of the Shannon capacitywould improve dramatically. For example, if these invariants could even be enumeratedalgorithmically, then we would have an algorithm for computing Θ.Graph invariants satisfying these three conditions are therefore of major interest, andwe dedicate the present paper to their study. It is convenient to introduce a new term forthem; we call them semiring-homomorphic graph invariants, since the preservation ofaddition and multiplication is what characterizes semiring homomorphisms. The most basicexamples of semiring-homomorphic graph invariants are the fractional chromatic number χ f and the complementary Lov´asz number G ϑ ( ¯ G ) [19]. Zuiddam observed that theonly known additional examples are the projective rank [20] and the fractional Haemersbounds of [6], both also applied to the complement. UNIFIED CONSTRUCTION OF SEMIRING-HOMOMORPHIC GRAPH INVARIANTS 3
What we achieve in this paper is a general construction of semiring-homomorphic graphinvariants which recovers all the known examples. Curiously, all of these are of a fractional nature, and in particular are monotone not only under graph homomorphisms, but evenunder the more abundant fractional graph homomorphisms (in our sense, Lemma 6.7).Correspondingly, our considerations involve fractional graph theory [25] in an essentialway.We now give a brief outline of our construction. Many graph invariants η are constructedby labelling the vertices of a graph by certain other objects, in such a way that the labelsassociated to adjacent vertices satisfy a certain relation. Essentially by definition, thistype of labelling is secretly a graph homomorphism, namely to the graph with allowedlabels as vertices and the specified relation as adjacency. It has been well-recognized thatfractionalizing a graph invariant η amounts to considering vertex labels given by sets oflabels of cardinality d , dividing the resulting invariant by d , and taking the limit d → ∞ ;see e.g. [13, Section 1.2]. Here we formalize the general pattern behind this construction:if F = ( F n ) n ∈ N is a sequence of graphs, then the F -number η F is the graph invariant givenby η F ( G ) := min { n ∈ N | G → F n } , where G → F n means that there exists a graph homomorphism from G to F n , i.e. a labellingof the vertices of G by the vertices of F n such that adjacent vertices have labels that areadjacent in F n . The corresponding fractional invariant is η frac F ( G ) := inf n nd (cid:12)(cid:12)(cid:12) G → F n /d o , where F n /d for d, n ∈ N > is the graph of d -cliques in F n (Definition 3.4). This formalizesthe intuitive idea that a d -fractional labelling assigns to each vertex a set of labels ofcardinality d . It specializes to the definition of the fractional chromatic number in the casewhere F n = K n is the complete graph (Example 5.1).We find that this definition makes the resulting fractional invariant η frac F into a semiring-homomorphic invariant as soon as the family of graphs ( F n ) satisfies a number of conditionsthat make it into a linear-like semiring family (Definitions 4.1 and 4.6). The role playedby these graph families is akin to that of the family of Euclidean spaces ( R n \ { } ), withorthogonality as adjacency, and has a strong geometrical flavour. We therefore think oflinear-like semiring families of graphs as a new concept of combinatorial geometry, distinctfrom matroid theory, which could be of independent interest. Summary.
In Section 2, we introduce our setting, emphasizing that our graphs can also beinfinite. We recall the best known version of Strassen’s Positivstellensatz as Theorem 2.5and apply it to
Graph , the preordered semiring of graphs, in order to state Zuiddam’sTheorem 1.1 as Theorem 2.6 and the more general Theorem 2.7 on rates. In Section 3,we consider operations of blowup and fractionalization of graphs as additional algebraicstructure on
Graph . In Section 4, we introduce semiring families of graphs F = ( F n )and prove Theorem 4.15 as our main result, stating in particular that η frac F is semiring-homomorphic. The proof mirrors the more specific arguments of Bukh and Cox [6] in TOBIAS FRITZ our more general situation. In Section 5, we show how this recovers all known examplesof semiring-homomorphic graph invariants as particular instances, thereby also giving thefirst (and so far only) examples of linear-like semiring families. In Section 6, we end witha discussion of some open problems.2.
The preordered commutative semiring of graphs
Definition 2.1.
Throughout this paper, a graph G is an undirected simple graph G =( V ( G ) , E ( G )) , where both the set of vertices V ( G ) and the set of edges E ( G ) may be infinite.As usual, we denote adjacency ( v, w ) ∈ E ( G ) for vertices v, w ∈ V ( G ) by v ∼ w . So in contrast to [29], we also include infinite graphs in some of our considerations.We do this in order to obtain e.g. the Lov´asz number (Example 5.4) from our construc-tion, although it is open whether infinite graphs are really necessary in order to do so(Problem 6.4).We now get to the definition of
Graph (resp.
FinGraph ), the preordered commutativesemiring of (finite) graphs. The essential pieces of structure on
Graph and
FinGraph are thefollowing standard definitions [25, 11]:
Definition 2.2.
Let G and H be graphs. (a) A graph homomorphism f : G → H is a map f : V ( G ) → V ( H ) such that if v ∼ w in G , then also f ( v ) ∼ f ( w ) in H . (b) The graph join G + H is the graph with vertex set the disjoint union V ( G + H ) := V ( G ) ⊔ V ( H ) , the induced edges on each summand, and v ∼ w for every v ∈ V ( G ) and w ∈ V ( H ) . (c) The disjunctive product G ∗ H is the graph with vertex set the cartesian product V ( G ∗ H ) := V ( G ) × V ( H ) , and ( v, w ) ∼ ( v ′ , w ′ ) if and only if v ∼ v ′ or w ∼ w ′ . (d) The lexicographic product G ⋉ H is the graph with vertex set the cartesian product V ( G ⋉ H ) := V ( G ) × V ( H ) , and ( v, w ) ∼ ( v ′ , w ′ ) if and only if v ∼ v ′ , or v = v ′ and w ∼ w ′ . We write G → H to denote the statement that there is a graph homomorphism from G to H . For example, we always have G ⋉ H → G ∗ H . We say that two graphs G and H are homomorphically equivalent if G → H and H → G , and denote this by G ≃ H ; weoccasionally also write G ∼ = H to denote graph isomorphism. We will frequently use the clique number ω ( G ) := sup { n ∈ N | K n → G } and the chromatic number χ ( G ) := inf { n ∈ N | G → K n } . Both of these invariants are additive in the sense that ω ( G + H ) = ω ( G ) + ω ( H ) and χ ( G + H ) = χ ( G ) + χ ( H ). For infinite graphs, either or both of these may be infinite.Besides working with potentially infinite graphs, we additionally deviate in our con-ventions slightly from [29] in that our definitions correspond to those of Zuiddam undertaking graph complements. Our reasons are that this allows us not to take complements UNIFIED CONSTRUCTION OF SEMIRING-HOMOMORPHIC GRAPH INVARIANTS 5 when talking about graph homomorphisms, and that this results in a more straightforwardinformation-theoretic interpretation in terms of zero-error communication [28].We write
Graph for the collection of all isomorphism classes of graphs.
Graph becomesa preordered set if we put G ≤ H if and only if there is a graph homomorphism G → H .Even upon considering finite graphs only, the resulting preordered set is known to be highlycomplex [24, Section IV.3]. In order to keep our notation intuitive, we will continue to write G → H in order to indicate the existence of a graph homomorphism from G to H andavoid the notation G ≤ H . We refer to [29, 9] for the definition of preordered commutativesemiring. The following is again due to Zuiddam [29] in the case of finite graphs. Proposition 2.3.
With graph join as addition and disjunctive product as multiplication,
Graph is a preordered commutative semiring.Proof.
Straightforward to check; e.g. the distributivity of multiplication over addition isgiven by the canonical isomorphism G ∗ H + G ∗ H ∼ = −→ ( G + G ) ∗ H, which is given by the identity on the level of vertices for all graphs G , G and H . (cid:3) It is noteworthy that our version of
Graph is technically a proper class rather than a set,since it contains graphs of arbitrarily large cardinality . This type of situation is closelyfamiliar from category theory, and here we also adopt the standard workaround usingGrothendieck universes [26, Section 3.3], which lets us ignore this issue from now on.We also write FinGraph ⊆ Graph for the preordered subsemiring of finite graphs, withrespect to the restricted preorder relation and algebraic operations.The objects of study of this paper are the following:
Definition 2.4. A semiring-homomorphic graph invariant is an order-preserving semiringhomomorphism η : FinGraph → R + . Concretely, a semiring-homomorphic graph invariant η therefore assigns a number η ( G )to every graph G ∈ FinGraph such that η ( K ) = 1, and such that the following hold forany two finite graphs G, H ∈ FinGraph : ◦ If G → H , then η ( G ) ≤ η ( H ); ◦ η ( G + H ) = η ( G ) + η ( H ); ◦ η ( G ∗ H ) = η ( G ) η ( H ). Recall that a preorder “ ≤ ” is a reflexive and transitive relation, not necessarily antisymmetric. This is because if we only consider graphs of cardinality at most κ for some infinite cardinal κ , thenthere are at most 2 κ × κ × κ = 2 κ isomorphism classes of graphs. Here, the first factor 2 κ arises fromchoosing a set of vertices, and the second factor 2 κ × κ from choosing a set of edges. TOBIAS FRITZ
The set of semiring-homomorphic graph invariants corresponds to the space of points de-fined over R + of the real spectrum of FinGraph . We study the known examples of semiring-homomorphic graph invariants as Examples 5.1 to 5.4.The purpose of this paper is to contribute to the problem of classifying all semiring-homomorphic graph invariants, a project which has been initiated by Zuiddam [29]. It isa curious fact that all known semiring-homomorphic graph invariants are of a fractionalnature; the fact that fractional graph theory plays an important role in this paper ismerely due to this heuristic observation, and we do not know whether fractional graphtheory is intrinsically important for the desired classification. In particular, it is openwhether all semiring-homomorphic graph invariants are also monotone under fractionalgraph homomorphisms (Problem 6.8).The rest of this section is mainly motivational, and dedicated to illustrating the rele-vance of trying to achieve a classification of semiring-homomorphic graph invariants. Webegin by recalling a variant of Strassen’s Positivstellensatz recently obtained by us in [9,Corollary 2.14]. This theorem adds two further equivalent characterization to Zuiddam’simproved version [29, Theorem 2.2] over Strassen’s original result.
Theorem 2.5 ([9]) . Let S be a preordered commutative semiring such that N ⊆ S via theunique semiring homomorphism N → S , and such that for every nonzero x ∈ S there is n ∈ N with x ≤ n and ≤ nx . Then for nonzero x, y ∈ S , the following are equivalent: (a) f ( x ) ≥ f ( y ) for every monotone semiring homomorphism f : S → R + . (b) For every ε > , there are m, n ∈ N > and nonzero z ∈ S such that m ≤ εn and n z x + mz ≥ n z y. (c) For every ε > , there are m, n ∈ N > and nonzero z ∈ S such that mn ≤ ε and mz x ≥ n z y. (d) For every ε > there are k, n ∈ N > such that k ≤ εn and k x n ≥ y n . (2.1) Moreover, if f ( x ) > f ( y ) for every monotone semiring homomorphism f : S → R + , thenthere are n ∈ N and nonzero z, w ∈ S with zx + w ≥ zy + w . In the case of
FinGraph , the canonical homomorphism N → FinGraph takes n ∈ N to thecomplete graph K n , so that the “scalars” in FinGraph are precisely the complete graphs.The product n G is equal to the n -fold join of G with itself. The boundedness conditions x ≤ n and 1 ≤ nx therefore mean that for every G ∈ FinGraph with at least one vertex,there is n ∈ N such that G → K n and K → n G . While the latter condition simply statesthat n G has at least one vertex, the former corresponds to having finite chromatic number, Strassen [27] has coined the term asymptotic spectrum for this kind of set in light of condition (2.1).Due to the equivalence with the other conditions in Theorem 2.5, which are arguably less of an asymptoticnature, and also due to the close relation with the real spectrum as used in real algebraic geometry [21,Section 2.4], we no longer find Strassen’s terminology to be optimally descriptive.
UNIFIED CONSTRUCTION OF SEMIRING-HOMOMORPHIC GRAPH INVARIANTS 7 which is trivial for a finite graph. Therefore
FinGraph is a preordered semiring satisfyingthe hypotheses of Theorem 2.5.We now focus on the equivalence of conditions (a) and (d) in the case of
FinGraph ;the implications of the other statements of Theorem 2.5 for
FinGraph have not yet beenexplored.
Theorem 2.6 (Zuiddam) . The Shannon capacity satisfies
Θ( ¯ G ) = inf η η ( G ) , (2.2) where η ranges over all semiring-homomorphic graph invariants. Before the proof, we note that it is also of interest to compute rates for finite graphs
G, H ∈ FinGraph with at least one vertex, R ( G → H ) := sup (cid:26) mn (cid:12)(cid:12)(cid:12)(cid:12) H ∗ m → G ∗ n (cid:27) , as originally defined in [10, Examples 8.1 and 8.18], and subsequently rediscovered as the graph information ratio in [28], where an operational interpretation in terms of zero-errorinformation theory has been given. As a special case, we have R ( G → K ) = log Θ( ¯ G ) [10,Example 8.1]. These rates can also be computed in terms of an optimization over semiring-homomorphic graph invariants: Theorem 2.7. If G has at least one edge and H at least one vertex, then R ( G → H ) = inf η log η ( G )log η ( H ) , where η ranges over all semiring-homomorphic graph invariants. This is in the spirit of the rate formulas for resource efficiency in the sense of [10,Theorem 8.24]. Theorem 2.6 follows from this upon using R ( G → K ) = log Θ( ¯ G ) asnoted above together with η ( K ) = 2 for all η . Proof.
This holds for regularized rates as per [9, Corollary 2.17], which coincide with the R ( G → H ) as per [10, Example 8.18]. (cid:3) By Theorem 2.6, if a classification of semiring-homomorphic graph invariants was avail-able in such a way that one could enumerate them algorithmically, this would resolvethe long-standing open question of computability of Θ in the positive. So far, only fewsemiring-homomorphic graph invariants are known at all; we will discuss them in Section 5,explaining how they are instances of the general construction given in Section 4, for whichwe prepare in the next section.
Remark . Both Theorems 2.6 and 2.7 are still correct, with the same proofs remainingvalid, if we include in the definition of
FinGraph not only the finite graphs, but all graphswith finite chromatic number. The degenerate cases where G has no edge or even no vertex must be treated separately, but it isobvious that Theorem 2.6 holds for them. TOBIAS FRITZ The algebraic structure of blowup and fractionalization
The following considerations make explicit the algebraic structure of fractionalization,in the general case of possibly infinite graphs. They apply to the finite case in particular inthe sense that all the following constructions preserve finiteness. In this context, some ofthe following observations also appear in the proofs in Zuiddam’s thesis [30, Section 3.3.2],in particular (3.1) and Lemma 3.9.
Definition 3.1.
For a graph G and d ∈ N > , the d -fold blowup is given by the lexicographicproduct G ⋉ K d , and we denote it by G ⋉ d . In other words, G ⋉ d arises from G by replacing each vertex v ∈ V ( G ) by d adjacentcopies v , . . . , v d ∈ V ( G ⋉ d ). It is easy to see that the map G G ⋉ d is functorial inthe sense that it takes graph homomorphisms to graph homomorphisms, and therefore alsopreserves homomorphic equivalence.It is important to distinguish G ⋉ d from the d -fold scalar multiple of G in Graph , whichis dG = K d ∗ G . The blowup approximates dG in that G ⋉ d → K d ∗ G , while the conversedoes generally not hold: Example . Let C be the 5-cycle. Then 2 C C ⋉ χ (2 C ) = χ ( K ∗ C ) = 6,while χ ( C ⋉
2) = 5. (This is essentially the well-known observation that the fractionalchromatic number of C is strictly smaller than its chromatic number, since χ alwayscommutes with scalar multiplication, χ ( dG ) = dχ ( G ). This is easy to see directly or fromProposition 4.10 and Example 5.1.) Definition 3.3.
Given a graph G , its power graph 2 G has as vertices the subsets of V ( G ) ,with adjacency S ∼ T for two subsets S, T ⊆ V ( G ) if and only if S ∩ T = ∅ and s ∼ t forevery s ∈ S and t ∈ T . Definition 3.4.
For a graph G and d ∈ N , the d -fractionalization G/d is the inducedsubgraph of G on the set of d -cliques in G . For example, the d -fractionalization of a complete graph is the corresponding Knesergraph, K n /d = KG n,d . Again the operation G G/d is obviously functorial in G undergraph homomorphisms, and therefore preserves homomorphic equivalence.The notation G/d is motivated by the following observation:
Lemma 3.5.
The d -fold blowup and the d -fractionalization are Galois adjoints: G ⋉ d → H ⇐⇒ G → H/d.
Proof.
It is straightforward to see that either type of homomorphism can be reinterpretedas the other. (cid:3)
The adjunction implies that G → ( G ⋉ d ) /d and ( H/d ) ⋉ d → H , although the converserelations generally do not hold: Example . Starting with K , we have ( K ⋉ / ∼ = K / ∼ = KG , , and χ ( KG , ) = 4by the Lov´asz–Kneser theorem [2, Chapter 38]. Therefore ( K ⋉ / K . For the otherstatement, since C / C ( C / ⋉ UNIFIED CONSTRUCTION OF SEMIRING-HOMOMORPHIC GRAPH INVARIANTS 9
Lemma 3.7.
For all G and d, d ′ ∈ N > , ( G ⋉ d ) ⋉ d ′ ∼ = G ⋉ dd ′ , G/d ′ /d ≃ G/ ( dd ′ ) . Proof.
The first isomorphism is easy to check directly. The second homomorphic equiv-alence then follows from Lemma 3.5 by the uniqueness of adjoints up to equivalence, to-gether with composition of adjoints: ( − ⋉ d ) ⋉ d ′ ⊣ ( − ) /d ′ /d and ( − ⋉ d ) ⋉ d ′ ∼ = ( − ) ⋉ dd ′ ⊣ ( − ) / ( dd ′ ). (cid:3) Despite the homomorphic equivalence, the isomorphism
G/d ′ /d ∼ = G/ ( dd ′ ) does generallynot hold, since there are generally many ways of partitioning a ( dd ′ )-clique into a d -cliqueof d ′ -cliques. Lemma 3.8.
For any graphs G and H and d ∈ N > , G ⋉ d + H ⋉ d ∼ = ( G + H ) ⋉ d, (3.1) G/d + H/d → ( G + H ) /d, (3.2) but in general ( G + H ) /d G/d + H/d .Proof.
Again the first isomorphism is straightforward to check . The second homomor-phism G/d + H/d → ( G + H ) /d is clear since the images of the two canonical homomor-phisms G/d → ( G + H ) /d and H/d → ( G + H ) /d are pairwise adjacent.In the other direction, taking G = H = K and d = 2 gives a (rather degenerate)counterexample, since K / ∼ = ∅ , while K / ∼ = K . (cid:3) Let us emphasize again that the operation ( − ) ⋉ d does not coincide with scalar multi-plication in Graph . More specifically, although G ⋉ ( d + d ′ ) → G ⋉ d + G ⋉ d ′ holds, the converse does generally not, as e.g. the example G = C and d = d ′ = 1 shows(Example 3.2).Finally, there is a compatibility inequality between the blow-up and the disjunctiveproduct: Lemma 3.9.
For any graphs G and H and d ∈ N , ( G ∗ H ) ⋉ d → ( G ⋉ d ) ∗ H. but the converse does not hold in general.Proof. The adjacency is given by ( i, g, h ) ∼ ( i ′ , g ′ , h ′ ) on the left-hand side if and only if h ∼ h ′ ∨ g ∼ g ′ ∨ ( g = g ′ ∧ h = h ′ ∧ i = i ′ )and on the right-hand side if and only if h ∼ h ′ ∨ g ∼ g ′ ∨ ( g = g ′ ∧ i = i ′ ) . It is clear that the first implies the second. It is also an instance of the general categorical fact that left adjoints preserve colimits.
As a counterexample to the other direction, we have ( K ⋉ ∗ C ( K ∗ C ) ⋉
2. Oneway to see this is to note that χ (( K ⋉ ∗ C ) = χ (4 C ) = 4 ·
3, while χ (( K ∗ C ) ⋉ ⋉
2) = 10,where the latter is per explicit computation in
Sage . (cid:3) We finish our investigations of the algebraic structure of fractionalization with a resulton the compatibility of blow-up with fractionalization:
Lemma 3.10.
For a graph G and d, d ′ ∈ N > , we have ( G/d ′ ) ⋉ d → ( G ⋉ d ) /d ′ .Proof. We already know (
G/d ′ ) ⋉ d ′ → G , which implies ( G/d ′ ) ⋉ dd ′ → G ⋉ d by functo-riality of blow-up and Lemma 3.7. Another application of Lemma 3.7 and the adjunctionof Lemma 3.5 gives indeed ( G/d ′ ) ⋉ d → ( G ⋉ d ) /d ′ . (cid:3) Semiring families of graphs and the linear-like condition
The essential ingredients of our upcoming construction of semiring-homomorphic graphinvariants will be semiring families of graphs:
Definition 4.1. A semiring family of graphs is given by a sequence of graphs ( F n ) n ∈ N suchthat F = ∅ and F = ∅ and for all n, m ∈ N , F n + F m → F n + m , F n ∗ F m → F nm . (4.1)These two operations are analogues of the addition and multiplication on a single semir-ing; we could say that they are equivalent to specifying a lax semiring homomorphism N → Graph by analogy with lax monoidal functors [1, Chapter 3]. The condition F = ∅ then corresponds to unitality in the form K → F . We intentionally do not consider theparticular homomorphisms implementing (4.1) as part of the structure of a semiring fam-ily. In particular, we do not require the existence of homomorphisms which would satisfycompatibility equations like associativity or distributivity, since not all of these actuallyhold e.g. in the upcoming Example 5.4. Finally, one can also imagine indexing a semiringfamily by semirings other than N , but we will not consider such more general families here.Since F = ∅ , we have ω ( F ) ≥
1. We can then use induction and F n + F → F n +1 toshow that ω ( F n ) ≥ n for all n ∈ N . Next, we show that it’s possible to take “commondenominators”: Lemma 4.2.
Let ( F n ) be a semiring family. Then for every n, d, m ∈ N > , F n /d → F mn /md. Proof.
By Lemma 3.7 and functoriality of − /d , it is enough to consider the case d = 1. Bythe adjunction of Lemma 3.5, we only need to prove F n ⋉ m → F mn . This arises as thecomposite F n ⋉ m → K m ∗ F n → F m ∗ F n → F mn . (cid:3) We will prove in Theorem 4.15 that every semiring family which satisfies an additional linear-like condition gives rise to a semiring-homomorphic graph invariant by fractionaliza-tion. In general, there are three basic invariants that can be constructed from a semiringfamily:
UNIFIED CONSTRUCTION OF SEMIRING-HOMOMORPHIC GRAPH INVARIANTS 11
Definition 4.3.
Let F = ( F n ) be a semiring family and G ∈ FinGraph . (a) The F -number of G is η F ( G ) := min { n ∈ N | G → F n } . (4.2)(b) The fractional F -number of G is η frac F ( G ) := inf n nd (cid:12)(cid:12)(cid:12) G → F n /d o . (4.3)(c) The asymptotic F -number of G is η ∞ F ( G ) := inf n n p η F ( G ∗ n ) (4.4)Here, the relation of η F and η frac F can also be considered an instance of general fraction-alization of graph parameters [30, (3.2)].If we think of a semiring family ( F n ) as a combinatorial generalization of finite-dimensionalinner product spaces over a field, with orthogonality as adjacency, then a homomorphism G → F n is a combinatorial generalization of an orthogonal representation (or vector colour-ing), so that η F ( G ) has the flavour of a vector chromatic number. In the case F n = K n (Example 5.1), we recover usual colourings and the usual chromatic number χ ( G ).All three invariants are finite since G is finite, and therefore has finite chromatic number,which implies that there is n ∈ N with G → K n → F n . Also all three invariants aremonotone under graph homomorphisms by construction. Moreover: Proposition 4.4.
All three invariants η ∗ F ∈ { η F , η frac F , η ∞ F } are subadditive under joins, η ∗ F ( G + H ) ≤ η ∗ F ( G ) + η ∗ F ( H ) and submultiplicative under lexicographic and disjunctive products, η ∗ F ( G ⋉ H ) ≤ η ∗ F ( G ∗ H ) ≤ η ∗ F ( G ) η ∗ F ( H ) . Proof.
The inequality η ∗ F ( G ⋉ H ) ≤ η ∗ F ( G ∗ H ) is by monotonicity of η ∗ F and G ⋉ H → G ∗ H ,so that it only remains to prove subadditivity and submultiplicativity with respect to thedisjunctive product.This is clear for the F -number η F , as it follows directly from the definition of a semiringfamily.We prove the subadditivity for the fractional F -number η frac F , which follows upon choosingcommon denominators as follows. For ε >
0, choose n, d, n ′ , d ′ ∈ N with d, d ′ > ε -approximations to the defining infima of η frac F , namely η frac F ( G ) ≥ nd − ε, η frac F ( H ) ≥ n ′ d ′ − ε, arising from G → F n /d and H → F n ′ /d ′ . By Lemma 4.2, we can assume d = d ′ withoutloss of generality. But then we have G ⋉ d → F n and H ⋉ d → F n ′ by the Galois adjunctionof Lemma 3.5. Then G ⋉ d + H ⋉ d → F n + F n ′ → F n + n ′ , and hence ( G + H ) ⋉ d → F n + n ′ by the distributivity of Lemma 3.8. Now G + H → F n + n ′ /d again by the adjunction, whichgives η frac F ( G + H ) ≤ n + n ′ d ≤ η frac F ( G ) + η frac F ( H ) + 2 ε . The claim follows in the limit ε → The submultiplicativity of η frac F works similarly to the subadditivity, using the same data.We have G ⋉ d → F n and H ⋉ d ′ → F n ′ , and therefore( G ∗ H ) ⋉ dd ′ → ( G ⋉ d ) ∗ ( H ⋉ d ′ ) → F n ∗ F n ′ → F nn ′ , where the first homomorphism is by Lemmas 3.7 and 3.9. Therefore η frac F ( G ∗ H ) ≤ nn ′ dd ′ ≤ (cid:16) η frac F ( G ) + ε (cid:17)(cid:16) η frac F ( H ) + ε (cid:17) , so that submultiplicativity follows in the limit ε → η ∞ F , we first note that the infimum inf n n p η F ( G ∗ n ) is a limit by Fekete’s lemmaand submultiplicativity of η F . Using this, submultiplicativity of η ∞ F is then clear again bysubmultiplicativity of η F , n p η F (( G ∗ H ) ∗ n ) = n p η F ( G ∗ n ∗ H ∗ n ) ≤ n p η F ( G ∗ n ) n p η F ( H ∗ n ) , since then the inequality also holds in the limit. Finally we treat subadditivity of η ∞ F . Forgiven ε >
0, we fix k ∈ N such that for all m ≥ k , η ∞ F ( G ) ≥ m p η F ( G ∗ m ) − ε, η ∞ F ( H ) ≥ m p η F ( H ∗ m ) − ε. Now we have that G ∗ j → G ∗ k for j ≤ k , and therefore for sufficiently large n , n p η F (( G + H ) ∗ n ) = n vuuut η F n X j =0 (cid:18) nj (cid:19) G ∗ j ∗ H ∗ ( n − j ) ≤ n vuut n X j =0 (cid:18) nj (cid:19) η F ( G ∗ j ) η F ( H ∗ ( n − j ) ) ≤ n vuut k (cid:18) nk (cid:19) ( η F ( G ∗ k ) η F ( H ∗ n ) + η F ( G ∗ n ) η F ( H ∗ k )) + n − k X j = k (cid:18) nj (cid:19) η F ( G ∗ j ) η F ( H ∗ ( n − j ) ) ≤ n q O (poly( n ))(( η ∞ F ( H ) + ε ) n + ( η ∞ F ( G ) + ε ) n ) + (cid:0) η ∞ F ( G ) + η ∞ F ( H ) + 2 ε (cid:1) n Since lim n →∞ n √ α n + β n + γ n = max( α, β, γ ), the expression is dominated by the thirdexponential, and we conclude η ∞ F ( G + H ) ≤ η ∞ F ( G ) + η ∞ F ( H ) + 2 ε by taking the limit n → ∞ . The claim now follows as ε → (cid:3) Next, we will introduce our linear-like condition on a semiring family and prove that thismakes η frac F additive under joins and multiplicative under disjunctive products, and there-fore semiring-homomorphic. Introducing this condition requires a bit more preparation.For S ⊆ V ( G ), we write S ⊥ for the set of all vertices that are adjacent to all vertices in S . The map S S ⊥ implements a contravariant Galois adjunction, meaning that S ⊆ T ⊥ if and only if T ⊆ S ⊥ if and only if S and T are pairwise adjacent. This is equivalent to UNIFIED CONSTRUCTION OF SEMIRING-HOMOMORPHIC GRAPH INVARIANTS 13 adjacency S ∼ T in the power graph 2 G . It follows that the map S S ⊥⊥ is a closureoperation. We call those sets S which satisfy S ⊥⊥ = S flats , by analogy with flats inmatroid theory. Since S ⊆ T ⊥ implies T ⊥⊥ ⊆ S ⊥ , we have that S ∼ T implies S ∼ T ⊥⊥ ,and by the same token applied again, also S ⊥⊥ ∼ T ⊥⊥ . Definition 4.5.
The rank of a subset S ⊆ V ( G ) is rk( S ) := ω ( S ⊥⊥ ) . Thus S has rank ≥ r if and only if there is an r -clique C in G such that for every T ∼ S in 2 G , we also have T ∼ C . Note that C does not need to be contained in S . An interestingspecial case is when S is itself a clique, where it may happen that rk( S ) > | S | , for exampleif G consists of the clique S and a disjoint clique C larger than S together with a separatecut vertex adjacent to all vertices in both S and C , which gives S ⊥⊥ = S ∪ C .For a subset S ⊆ V ( G ), we write G | S for the induced subgraph on S . Definition 4.6.
A sequence of graphs ( F n ) n ∈ N is linear-like if for every flat S ⊆ V ( F n ) ,we have a homomorphic equivalence F n | S ≃ F rk( S ) . In some cases, the homomorphic equivalence F n | S ≃ F rk( S ) can be strengthened toisomorphism F n | S ∼ = F rk( S ) , but this is not always the case (Example 5.3).The definition of linear-like sequence is strongly reminiscent of Brunet’s orthomatroids [5],but weaker in a sense. In fact, the closure operation S S ⊥⊥ may not even make F n intoa matroid, even if we have isomorphisms F n | S ∼ = F rk( S ) . For example in Example 5.4, thetwo vectors (1 , ±√ , , . . . ) ∈ V ( F ) have empty orthogonal complement, so that the onlyflat which contains both is F itself, which already has rank 3. Hence F is not a matroid.However, the linear-like condition implies that if φ : G → F n is any homomorphism,then it factors as G → F rk(im( φ )) → F n , which we will make frequent use of. Lemma 4.7.
For a linear-like semiring family ( F n ) , we have ω ( F n ) = n .Proof. ω ( F n ) ≥ n follows from F + . . . + F → F n , as already noted above. For the otherinequality, applying the linear-like condition to the whole graph implies F n ≃ F ω ( F n ) . Butsince F n + F ω ( F n ) − n → F ω ( F n ) , we have ω ( F n ) = ω ( F ω ( F n ) ) ≥ ω ( F n ) + ω ( F ω ( F n ) − n ) ≥ ω ( F n ) + ( ω ( F n ) − n ) . Therefore ω ( F n ) ≤ n . (cid:3) We now continue assuming that ( F n ) is a linear-like semiring family of graphs. Lemma 4.8. If F k + F ℓ → F n , then k + ℓ ≤ n .Proof. By monotonicity of ω together with ω ( F k + F ℓ ) = k + ℓ and ω ( F n ) = n . (cid:3) Lemma 4.9. If G + H → F n , then there is a decomposition n = k + ℓ with G → F k and H → F ℓ . Proof.
Let φ : G + H → F n be a homomorphism. Then im( φ | G ) ∼ im( φ | H ), and thereforealso im( φ | G ) ⊥⊥ ∼ im( φ | H ) ⊥⊥ . The claim now follows from the linear-like property andLemma 4.8. (cid:3) Proposition 4.10. η F and η frac F are additive under joins.Proof. We have already shown subadditivity under joins for both invariants in Proposi-tion 4.4, so that it remains to prove superadditivity. For η F , this is by Lemma 4.9.For η frac F , the superadditivity is now similar to the previous arguments: choose n and d > η frac F ( G + H ) ≥ nd − ε, as witnessed by some homomorphism G + H → F n /d . Then again G ⋉ d + H ⋉ d ≃ ( G + H ) ⋉ d → F n , which decomposes as G ⋉ d → F k and H ⋉ d → F ℓ for suitable k and ℓ .Hence η frac F ( G ) + η frac F ( H ) ≤ kd + ℓd ≤ nd ≤ η frac F ( G + H ) + ε. Since ε was arbitrary, we must have η frac F ( G ) + η frac F ( H ) ≤ η frac F ( G + H ). (cid:3) Next, we derive an alternative characterization of η frac F , closely following the argumentsof Bukh and Cox in the case of the fractional Haemers bound [6, Proposition 7]. Definition 4.11. A rank- r -representation of a graph G with values in a linear-like semiringfamily ( F n ) is a homomorphism φ : G → F m for some m ∈ N such that rk( φ ( v )) ≥ r in F m for every v ∈ V ( G ) . Proposition 4.12. η frac F ( G ) = inf φ,r ( rk( S v ∈ V ( G ) φ ( v )) r : φ is a rank- r -representation of G ) Proof.
We will show that every nd in the defining infimum (4.3) is dominated by some rk( S v ∈ V ( G ) φ ( v )) r and vice versa.Given G → F n /d , we get φ as the composite G → F n /d → F n , where F n /d → F n isthe canonical inclusion homomorphism. This lands in rank d by definition, and triviallysatisfies rk( S v ∈ V ( G ) φ ( v )) d ≤ nd .Conversely, let φ : G → F m be a rank- r -representation. We can assume without loss ofgenerality that each φ ( v ) is a flat, since extending from φ ( v ) to φ ( v ) ⊥⊥ does not decreasethe rank and preserves the adjacency relations as well. Then we can restrict each φ ( v ) toan r -clique, which is guaranteed to exist by the definition of rank. Doing so results in ahomomorphism G → F m /r , or equivalently G ⋉ r → F m . Since the rank of the image ofthis homomorphism is upper bounded by n := rk( S v ∈ V ( G ) φ ( v )) by construction, it factorsthrough F n by the linearity-like condition. (cid:3) Proposition 4.13. η frac F is multiplicative on lexicographic and on disjunctive products. UNIFIED CONSTRUCTION OF SEMIRING-HOMOMORPHIC GRAPH INVARIANTS 15
Here, the statement about the lexicographic product is due to Chris Cox, who observedthat the following proof still goes through; the argument is again an adaptation of the oneof Bukh and Cox [6].
Proof.
Thanks to Proposition 4.4, we only need to prove supermuliplicativity with respectto lexicographic product. So let η frac F ( G ⋉ H ) ≥ nd − ε , corresponding to some homomorphism φ : G ⋉ H → F n /d . For fixed v ∈ V ( G ), consider the associated homomorphism φ ( v, − ) : H ⋉ d → F n , and put r := min v rk(im( φ ( v, − ))) . Then η frac F ( H ) ≤ rd by the linear-like property. Finally, consider the homomorphism G → F n , v im( φ ( v, − )) . It is a rank- r -representation by definition of r . Proposition 4.12 therefore gives η frac F ( G ) ≤ nr .In total, we have η frac F ( G ) η frac F ( H ) ≤ nr · dd = nd ≤ η frac F ( G ⋉ H ) + ε, which is enough. (cid:3) Comparing the three invariants, the inequality η ∞ F ( G ) ≤ η F ( G ) is trivial, and likewise η frac F ( G ) ≤ η F ( G ). The latter inequality can now be strenghtened: Corollary 4.14.
For every finite graph G , we have η frac F ( G ) ≤ η ∞ F ( G ) .Proof. Apply the trivial inequality η frac F ( G ) ≤ η F ( G ) to the powers G ∗ n , use multiplicativityof η frac F , and take the limit n → ∞ . (cid:3) We now summarize the results obtained so far into our main theorem, of which (c) isthe main part.
Theorem 4.15.
For any linear-like semiring family F = ( F n ) , the three graph invariants η F , η frac F and η ∞ F have the following properties: (a) All are monotone under graph homomorphisms, subadditive under joins, and sub-multiplicative under both lexicographic and disjunctive products. (b)
For η F , subadditivity holds with equality. (c) For η frac F , subadditivity and both forms of submultiplicativity hold with equality. Inparticular, η frac F is semiring-homomorphic . (d) For any finite graph G , η frac F ( G ) ≤ η ∞ F ( G ) ≤ η F ( G ) . (e) On K n , all three invariants take the value n . We do not know whether the inequality η frac F ≤ η ∞ F is strict, or whether it is necessarilyan equality, perhaps under suitable additional conditions on the semiring family ( F n ).Before getting to our examples, we state a comparison criterion for the semiring-homomorphicgraph invariants induced by two linear-like semiring families: Proposition 4.16.
Let F = ( F n ) and F ′ = ( F ′ n ) be linear-like semiring families such thatthere is C ∈ N together with a homomorphism F n → F ′ Cn for every n ∈ N which takes setsof rank r to sets of rank at least Cr for every r ∈ N . Then we have η frac F ′ ( G ) ≤ η frac F ( G ) for every finite graph G .Proof. We show that if G → F n /d , then also G → F ′ Cn /Cd , from which the claim fol-lows. But this is because of F n /d → F ′ Cn /Cd , which we show like this: the assumedhomomorphisms φ n : F n → F ′ Cn take d -cliques C , C ⊆ V ( F n ) with C ∼ C to d -cliques φ n ( C ) ∼ φ n ( C ), which implies φ n ( C ) ⊥⊥ ∼ φ n ( C ) ⊥⊥ . Since these flats are of rank atleast Cd , each of them must itself contain a Cd -clique. Choosing such Cd -cliques arbitrarilyresults in the desired graph homomorphism F n /d → F ′ Cn /Cd . (cid:3) Recovering all known semiring-homomorphic graph invariants
We now illustrate how Theorem 4.15(c) lets us reconstruct all known examples ofsemiring-homomorphic graph invariants. The corresponding examples of linear-like semir-ing families underline their geometric character.
Example . The very simplest semiring family ( F n ) is F n = K n , and it is easy to see thatit is indeed linear-like. In this case, we get the fractional chromatic number as η frac F , andit is well-known that η frac F = η ∞ F . The properties now follow from Theorem 4.15 are well-known in this case; for example, the multiplicativity of the fractional chromatic numberunder lexicographic products is [25, Corollary 3.4.5]. Example . Let F be a field with involution α ¯ α such that the subfield of fixpoints ofthe involution is a Euclidean field. The paradigmatic examples are F = R with the identityinvolution, or F = C with complex conjugation.These assumptions guarantee that every finite-dimensional Hilbert space over F hasan orthonormal basis, where a Hilbert space is defined as a vector space equipped witha positive definite hermitian sesquilinear form. The existence of an orthonormal basisfollows e.g. using Gram–Schmidt orthogonalization, resulting in the well-known fact thata hermitian form over any field with involution can be diagonalized [14, p. 543], and thennormalizing by the relevant square roots. We use F n with its standard hermitian innerproduct as a standard n -dimensional Hilbert space. We then define a graph F n withvertex set given by the projective space V ( F n ) := P n − ( F ), and we declare two verticesrepresented by vectors α, β ∈ F n \ { } to be adjacent if and only if h α, β i = 0. Up tohomomorphic equivalence, we can also work with the set of unit vectors with respect toorthogonality as adjacency, or even with all of F n \ { } . UNIFIED CONSTRUCTION OF SEMIRING-HOMOMORPHIC GRAPH INVARIANTS 17
Next, we show that ( F n ) is a semiring family. Using the canonical inclusions F n → F n ⊕ F m and F m → F n ⊕ F m together with the canonical isomorphism F n ⊕ F m ∼ = F n + m ,we obtain the desired homomorphism F n + F m → F n + m in the form of maps P n − ( F ) + P m − ( F ) → P n + m − ( F ) which preserve orthogonality. Similarly, the tensor product map F n × F m −→ F n ⊗ F m ∼ = F nm , ( α, β ) α ⊗ β also has the property that if α ∼ α ′ or β ∼ β ′ , then also α ⊗ β ∼ α ′ ⊗ β ′ . This proves that F n ∗ F m → F nm as well. We thus have a semiring family of graphs.Now for a set of vertices S , the flat S ⊥⊥ coincides with the linear span of S : it mustcontain the linear span since every vector orthogonal to S is also orthogonal to every linearcombination; and conversely, Gram–Schmidt orthogonalization shows that S ⊥ is a subspacewhose dimension is n − rk( S ), and is therefore complementary to the span lin F ( S ). So inthis case, the linearity-like condition of Definition 4.6 is straightforward to see.Applying Theorem 4.15 therefore produces a semiring-homomorphic graph invariant forevery field F satisfying the present assumptions. For F = C , this specializes to the projec-tive rank introduced by Manˇcinska and Roberson [20, Section 6], essentially by definitionof the latter. For F = R , the resulting semiring-homomorphic invariant turns out to be thesame [20, Section 6], since Proposition 4.16 implies bidirectional inequality: complexifica-tion induces graph homomorphisms P n − ( R ) → P n − ( C ) satisfying the hypotheses with C = 1; conversely, regarding C n as a real vector space of twice the dimension (and takingthe real part of the inner product) results in a graph homomorphism P n − ( C ) → P n − ( R )satisfying the hypotheses with C = 2. Thus using F = R recovers projective rank as well.Theorem 4.15 in this case also recovers known properties, such as the multiplicativity [8,Theorem 27].More generally, if F is a real closed field or the imaginary quadratic extension of a realclosed field, then the resulting invariants coincide with those arising from F = R or F = C ,respectively, since the existence of a graph homomorphism G → P n − ( F ) is a sentencein the first-order logic of ordered fields, and all real closed field have the same first-ordertheory [7, Section 5.4].The previous example involves inner product spaces for symmetric (or hermitian) innerproducts. It therefore seems natural to ask whether there could also be symplectic versionsof these semiring families which give semiring-homomorphic graph invariants. So far wehave not been able to make this idea work; the main problem is that the tensor product ofantisymmetric matrices is symmetric rather than antisymmetric.Nevertheless, also the Haemers bounds [12] and fractional Haemers bounds [6] can beinterpreted in terms of bilinear forms [23]: Example . Let F be any field, and suppose that U and W are vector spaces over F together with a bilinear form β : U × W → F . Choosing bases for U and W represents β by a matrix M with entries M ij := β ( u i , w j ). Following Peeters [23, p. 423], we associateto this data a graph O β with vertex set V ( O β ) := { ( x, y ) ∈ U × W | β ( x, y ) = 1 } , and adjacency ( x, y ) ∼ ( x ′ , y ′ ) if and only if β ( x, y ′ ) = β ( x ′ , y ) = 0. The left kernel K ⊆ U is the subspace K := { x ∈ U | β ( x, y ) = 0 ∀ y ∈ W } , and similarly the right kernel of β is the subspace of W given by L := { y ∈ W | β ( x, y ) = 0 ∀ x ∈ U } . Choosing complementary subspaces U ′ and W ′ results in direct sum decompositions U = K ⊕ U ′ and W = L ⊕ W ′ . We write β ′ : U ′ × W ′ → F for the restriction of β to thesecomplementary subspaces, and claim that there is a homomorphic equivalence O β ≃ O β ′ .As a homomorphism O β ′ → O β , we can simply choose the inclusion map. For O β → O β ′ ,we use the projection maps U → U ′ and W → W ′ , which commute with the bilinear formby the complementarity to kernels assumption, and therefore indeed O β ≃ O β ′ . Moreover,since β ′ is nondegenerate by construction, we also must have dim U ′ = dim W ′ ; by choosingsuitable bases, we can assume U ′ = W ′ = F n , and that β ′ is the standard inner product h− , −i n : F n × F n → F without loss of generality. In this way, we see that every O β is homomorphically equivalentto O h− , −i n for n = rk( β ).So for n ∈ N , we consider the graph F n := O h− , −i n . We thus have V ( F n ) := { ( x, y ) ∈ F n × F n | h x, y i = 1 } , with ( x, y ) ∼ ( x ′ , y ′ ) if and only if h x, y ′ i = h x ′ , y i = 0. Then a k -clique consists of x , . . . , x k ∈ F n and y , . . . , y k ∈ F n such that h x i , y j i = δ ij , and it follows that the ( x i )and ( y i ) are each linearly independent. This implies ω ( F n ) ≤ n , and ω ( F n ) = n followssince the standard basis achieves the bound. In the situation of the graph O β above, weconclude ω ( O β ) = rk( β ).Using the obvious isomorphism F n ⊕ F m ∼ = F n + m gives a homomorphism F n → F n + m upon sending ( x, y ) ∈ F n to ( x ⊕ , y ⊕ F m → F n + m using the secondcomponent. These two homomorphisms assemble to F n + F m → F n + m .Concerning compatibility with the disjunctive product, we similarly fix an isomorphism F nm ∼ = F n ⊗ F m induced by a bijection between the components, and consider the map( x , y ) × ( x , y ) ( x ⊗ x , y ⊗ y ) . The resulting compatibility between tensor product and canonical pairing shows that thisindeed respects the edges of our graphs, resulting in a homomorphism F n ∗ F m → F nm .Concerning the linear-like property, we use the fact that every flat is the orthogonalcomplement of some collection of vertices { ( x i , y i ) } i ∈ I with some potentially infinite indexset I . Then the orthogonal complement consists of all vertices ( x ′ , y ′ ) with x ′ ∈ U and y ′ ∈ W , where U ⊆ F n is the subspace U = { x ′ ∈ F n | h x ′ , y i i = 0 ∀ i } , and similarly, W = { y ′ ∈ F n | h x i , y ′ i = 0 ∀ i } . UNIFIED CONSTRUCTION OF SEMIRING-HOMOMORPHIC GRAPH INVARIANTS 19
As we saw above, the resulting graph is indeed homomorphically equivalent to F k , where k is the rank of the bilinear pairing between U and W . This shows that the linear-likecondition indeed holds, where the rank is the correct one due to the discussion of cliquenumbers above.We now claim that the associated invariants η F ( G ) and η frac F ( G ) coincide with theHaemers bound [12] and the fractional Haemers bound [4, 6] , respectively, of the com-plementary graph ¯ G (see also [23]). Using the notation of [6], the Haemers bound of ¯ G isthe smallest rank of a matrix M ∈ F V ( G ) × V ( G ) with M aa = 1 for all a ∈ V ( G ) and M ab = 0for all adjacencies a ∼ b in G . Let M be such a matrix achieving the smallest rank, forwhich we write k . Then M defines a bilinear pairing β : F V ( G ) × F V ( G ) → F given by( x, y ) x t M y . We have O β ≃ F k per the above. Since G → O β by the assumptions on M , we also conclude G → F k .Conversely, suppose that we have a homomorphism G → F k , encoded in vertex labellings x : G → F k and y : G → F k . Then the matrix M vw := h x v , y w i has the required propertiesby construction. Hence our η F ( G ) coincides with the Haemers bound of ¯ G . Using [6,Proposition 6], it now also follows that our η frac F ( G ) coincides with the fractional Haemersbound of ¯ G .Intuitively, we can now also understand the (fractional) chromatic number of Example 5.1in a new light: it plays the role of the (fractional) Haemers bound over the ‘field with oneelement’ [18]. Example . We now explain how the Lov´asz number [19] arises via our construction.Let ℓ be the real Hilbert space of square integrable sequences ( x i ) i ∈ N , and consider F n := { x ∈ ℓ | x = 1 , k x k = n } , considered as a graph with orthogonality as adjacency. Equivalently, we can define F n byusing the unit vectors x in any separable Hilbert space which satisfy in addition h c, x i = n − / for some fixed unit vector c .We get F n + F m → F n + m via, using the shorthand notation x = ( x , x + ), F n → R ⊕ ℓ ⊕ ℓ , x (cid:18) , r mn , r mn · x + , (cid:19) F m → R ⊕ ℓ ⊕ ℓ , y (cid:18) , − r nm , , r nm · y + (cid:19) (5.1)and using some orthogonal isomorphism R ⊕ ℓ ⊕ ℓ ∼ = ℓ which takes the first componentto the first component. In the above expressions, the components have been constructedprecisely in such a way that the conditions for membership in F n + m are satisfied, such thatthe images of the two maps are elementwise orthogonal, and such that the orthogonalityrelations within F n and F m are preserved.Showing F n ∗ F m → F nm is even easier: we similarly choose some identification ℓ ⊗ ℓ ∼ = ℓ with e ⊗ e e , and simply use the tensor product map ( x, y ) → x ⊗ y . The fractional Haemers bound was first introduced as fractional minrank in [4].
Concerning the linear-like condition, we first show ω ( F n ) = n . For ω ( F n ) ≥ n , it isenough to consider the standard basis vectors e , . . . , e n ∈ ℓ and c = n − / P ni =1 e i inthe alternative description mentioned above. We now show ω ( F n ) ≤ n . Suppose thatthere were vectors x (1) , . . . , x ( n +1) ∈ F n which are pairwise orthogonal. Then putting y := n e − P n +1 i =1 x ( i ) results in a vector whose norm squared is given by h y, y i = n h e , e i + n +1 X i =1 h x ( i ) , x ( i ) i − n n +1 X i =1 h e , x ( i ) i = n + ( n + 1) · n − · ( n + 1) · n = − n, which is absurd. Overall, we have proven ω ( F n ) = n .It remains to be proven that every flat in F n is homomorphically equivalent to some F k for k ≤ n , which is enough because of ω ( F k ) = k . As in Example 5.3, we consider afamily of vertices { x ( i ) } i ∈ I and show that its orthogonal complement is homomorphicallyequivalent to some F k . That complement consists of all y ∈ F n with h y, x ( i ) i = 0 for all i ∈ I .We can assume without loss of generality that the x ( i ) are linearly independent. Then byinduction , it is enough to show that the orthogonal complement is isomorphic to F n − inthe case | I | = 1, in which case we write x := x (1) . Applying an orthogonal transformationwhich preserves the first component, we can assume x = (1 , −√ n − , , . . . ) without lossof generality. Then the orthogonal complement coincides exactly with the image of theembedding F n − → F n , (1 , y + ) , r n − , r nn − · y + ! which is an instance of (5.1) above for F n − + F → F n .We now claim that already the resulting graph invariant η F ( G ) is closely related to theLov´asz number, in that η F ( G ) = ⌈ ϑ ( ¯ G ) ⌉ . (5.2)This is because F n is isomorphic to the graph of unit vectors in ℓ , where adjacency x ∼ y means that h x, y i = − n − . One way to see this isomorphism is to map x (1 , √ n − · x ). If we use this definitionof F n together with n ∈ R + being an arbitrary nonnegative number, then it is a knownfact that ϑ ( ¯ G ) is the infimum over all n ∈ R + with G → F n [15, Theorem 8.1]. Thisproves (5.2).Finally, we claim that both η frac F ( G ) and η ∞ F ( G ) coincide with ϑ ( ¯ G ). To this end, weprove the inequalities ϑ ( ¯ G ) ≤ η frac F ( G ) ≤ η ∞ F ( G ) ≤ ϑ ( ¯ G ) . Our argument shows in particular that if there are infinitely many linearly independent x ( i ) , then thecomplement is empty. UNIFIED CONSTRUCTION OF SEMIRING-HOMOMORPHIC GRAPH INVARIANTS 21
The first inequality follows from ϑ ( ¯ G ) ≤ η frac F ( G ) + 1 by (5.2), using as well the fact that ϑ ( G ⋉ d ) = dϑ ( ¯ G ) (as an instance of [16, Corollary, § ϑ ( ¯ G ) = d − ϑ ( G ⋉ d ) ≤ η frac F ( G ⋉ d ) + 1 d = η frac F ( G ) + d − , which implies the claim in the limit d → ∞ . The third inequality η ∞ F ( G ) ≤ ϑ ( ¯ G ) followsdirectly from the fact that both sides preserve disjunctive powers G G ∗ n . Now The-orem 4.15 recovers some standard properties of the Lov´asz number of the complement,including some of its multiplicativity properties [16, § η frac F ( G ) = inf d η F ( G ⋉ d ) d is attained, making η frac F ( G ) into a rational number.Since the Lov´asz number can be irrational, this statement does not generalize to arbitrarylinear-like semiring families. 6. Some open problems
Clearly the most interesting open question is:
Problem 6.1.
Does every semiring-homomorphic graph invariant
FinGraph → R + arisefrom a linear-like semiring family? If not, are the ones which arise in this way at leastdense in the topology of pointwise convergence in the set of functions FinGraph → R + ? Another problem which may also be of interest from the perspective of combinatorialgeometries is:
Problem 6.2.
Find more examples of linear-like semiring families of graphs.
The fact that the cases F = C and F = R in Example 5.2 both define projective rankraises another nontrivial question: Problem 6.3.
When to two given linear-like semiring families result in the same semiring-homomorphic graph invariants?
Applying Proposition 4.16 in both directions provides a sufficient criterion; the sepa-ration result for the fractional Haemers bounds of Bukh and Cox [6, Theorem 19] mayprovide some hints on how to find conditions which guarantee that two given linear-likesemiring families induce different semiring-homomorphic graph invariants.A closely related question is:
Problem 6.4.
Can the graph invariants reconstructed in Examples 5.2 to 5.4 also beobtained from linear-like semiring families consisting only of finite graphs?
This is of interest since it would make these semiring families live themselves in thepreordered semiring
FinGraph . In light of Remark 2.8, we could also replace
FinGraph bythe preordered semiring of all graphs with finite chromatic number, and then we would wantto know whether the above invariants can be obtained from linear-like semiring familiesconsisting of graphs with finite chromatic number. We do not know whether the semiring families from Example 5.3 have finite chromatic number (except in special cases, e.g. forfinite fields F ). But we do know what happens in the case of Example 5.2 and Example 5.4: Remark . The semiring families ( F n ) of Example 5.2 have finite chromatic number, aswe show now.We first prove that if x, y, z ∈ F n are unit vectors with |h x, y i| > / √ |h y, z i| > / √
2, then also |h x, z i| >
0. Rescaling by suitable phases implies that we can assume h x, y i > / √ h y, z i > / √
2. Then the inner product of w := x + z − h x + z, y i y withitself evaluates to 0 ≤ h w, w i = 2 − ( h x, y i + h y, z i ) + 2Re( h x, z i ) , which indeed implies |h x, z i| ≥ Re( h x, z i ) >
0. Hence to show that F n has finite chromaticnumber, it is enough to establish the existence of a finite set of unit vectors b , . . . , b m ∈ F n \{ } such that for every other unit vector x ∈ F n \{ } , there is i with |h x, b i i| > / √
2. Inthe Euclidean case ( α = ¯ α for all α ∈ F ), we note that the components of every unit vectormust lie in the subring F fin ⊆ F of finite elements α ∈ F characterized by − n ≤ α ≤ n forsome n ∈ N . Since F fin is totally ordered and contains Q , every α ∈ F fin has a standard part st( α ) ∈ R , where st : F fin → R is a ring homomorphism; st( α ) is defined to be the uniquereal contained in all rational intervals which contain α . Since st is a ring homomorphism,it commutes with the standard inner product, st( h x, y i ) = h st × n ( x ) , st × n ( y ) i . Hence it isenough to prove that there are finitely many unit vectors b , . . . , b m ∈ Q n satisfying theabove condition with respect to x ∈ R n ; but this follows from compactness of the unitsphere and density of rational points on the unit sphere. In the non-Euclidean case, F is adegree two extension over its Euclidean subfield , the claim follows in the same way using b , . . . , b m ∈ Q [ i ] n satisfying the above condition with respect to all unit vectors x ∈ C n .The following observation and its proof were communicated to us by David Roberson. Remark . In Example 5.4, already the graph F has infinite chromaticnumber, and therefore so do all F n with n ≥
3. The reason is that if F had finitechromatic number C ∈ N , then by (5.2), every finite graph G with ϑ ( ¯ G ) ≤ G → F , and therefore χ ( G ) ≤ C . But this is absurd, because the Knesergraphs KG r − ,r satisfy ϑ ( KG r − ,r ) = r − r < ϑ ( KG n,k ) = | V ( KG n,k ) | ϑ ( KG n,k ) − = (cid:0) nk (cid:1)(cid:0) n − k − (cid:1) − = nk [19, Theorems 8 and 13]. Taking n = 3 k − ϑ ( KG k − ,k ) <
3, but χ ( KG k − ,k ) = (3 k − − k + 2 = k + 1 by theLov´asz–Kneser theorem [2, Chapter 38], which is unbounded.Nevertheless, we do not know whether the Lov´asz number can be reconstructed from adifferent linear-like semiring family involving only graphs which are finite or at least havefinite chromatic number, so that Problem 6.4 is still open. There is α = ¯ α . Replacing α by α − ¯ α shows that we can assume α to be purely imaginary, ¯ α = − α ,which means that α is in the Euclidean subfield; using the existence of square roots implies that we canassume α = ±
1. Since the only square roots of 1 are ±
1, we must have α = −
1. It is now easy to seethat every element of F is a linear combination of 1 and α over the Euclidean subfield. UNIFIED CONSTRUCTION OF SEMIRING-HOMOMORPHIC GRAPH INVARIANTS 23
Last but not least, it is very curious that all known semiring-homomorphic graph invari-ants are of a fractional nature. Before phrasing our concrete question on this observation,we need to briefly discuss fractional graph homomorphisms. We write G H if there is a fractional graph homomorphism from G to H , by which we mean that there is d ∈ N suchthat G → ( H ⋉ d ) /d , or equivalently G ⋉ d → H ⋉ d . This does not coincide with theexisting fractional homomorphism notion characterized by ω ( G ) ≤ ω f ( H ) [3, Theorem 7],where ω f is the fractional clique number, as we will see in Example 6.9. For an example ofa pair of graphs with a fractional homomorphism in our sense but no homomorphism, wehave e.g. KG , K but KG , K by Example 3.6. We now have: Lemma 6.7.
Semiring-homomorphic graph invariants η frac F constructed via Theorem 4.15are monotone under fractional graph homomorphisms in our sense.Proof. If G H , then G ⋉ d → H ⋉ d for some d , so that the claim follows from Proposi-tion 4.13. (cid:3) However, we do not know whether this stronger kind of monotonicity applies to allsemiring-homomorphic graph invariants:
Problem 6.8.
Does G H imply η ( G ) ≤ η ( H ) for every semiring-homomorphic graphinvariant η : FinGraph → R + ? A negative answer would clearly imply a negative answer to both versions of Problem 6.1.
Example . To see that our notion of fractional graph homomorphisms differs from the oneof [3], it is enough to show that the semiring-homomorphic invariant χ f is not monotonewith respect to the latter notion of fractional homomorphisms. For example although ω ( C ) = 2 = ω f ( K ), we clearly have χ f ( C ) = χ f ( K ).Our last open problem was communicated to us by David Roberson: Problem 6.10 (Roberson) . Is every semiring-homomorphic graph invariant multiplicativewith respect to lexicographic product?
As a special case, this would mean η ( G ⋉ d ) = dη ( G ) for every semiring-homomorphic η , and therefore imply a positive answer to Problem 6.8, since G ⋉ d → H ⋉ d then yields dη ( G ) ≤ dη ( H ) and therefore η ( G ) ≤ η ( H ). A negative answer would imply a negativeanswer to both versions of Problem 6.1 thanks to the multiplicativity with respect tolexicographic product of Theorem 4.15, together with the fact that the multiplicativityequation is a closed condition in the topology of pointwise convergence. References [1] Marcelo Aguiar and Swapneel Mahajan.
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