Spectrahedrality of hyperbolicity cones of multivariate matching polynomials

The generalized Lax conjecture asserts that each hyperbolicity cone is a linear slice of the cone of positive semidefinite matrices. We prove the conjecture for a multivariate generalization of the matching polynomial. This is further extended (albeit in a weaker sense) to a multivariate version of the independence polynomial for simplicial graphs. As an application, we give a new proof of the conjecture for elementary symmetric polynomials (originally due to Brändén). Finally, we consider a hyperbolic convolution of determinant polynomials generalizing an identity of Godsil and Gutman.


Introduction
A homogeneous polynomial h(x) ∈ R[x 1 , . . . , x n ] is hyperbolic with respect to a vector e ∈ R n if h(e) = 0, and if for all x ∈ R n the univariate polynomial t → h(te−x) has only real zeros. Note that if h is a hyperbolic polynomial of degree d, then we may write B Nima Amini namini@kth.se 1 Department of Mathematics, Royal Institute of Technology, 100 44 Stockholm, Sweden Remark 1. 5 The exact analogue of Theorem 1.4 fails for n > 3 variables. This may be seen by comparing dimensions. The set of polynomials on R n of the form det(x 1 A 1 + · · · x n A n ) with A i a d × d symmetric matrix for 1 ≤ i ≤ n, has dimension at most n d+1 2 (as an algebraic image (A 1 , . . . , A n ) → det(x 1 A 1 + · · · x n A n ) of a vector space of the same dimension), whereas the set of hyperbolic polynomials of degree d on R n has non-empty interior in the space of homogeneous polynomials of degree d in n variables (see [29]) and therefore has the same dimension n+d−1 d . A convex cone in R n is spectrahedral if it is of the form where A i , i = 1, . . . , n are symmetric matrices such that there exists a vector (y 1 , . . . , y n ) ∈ R n with n i=1 y i A i positive definite. It is easy to see that spectrahedral cones are hyperbolicity cones. A major open question asks if the converse is true. Conjecture 1.6 (Generalized Lax conjecture [18,32]) All hyperbolicity cones are spectrahedral. Remark 1. 7 An important consequence of Conjecture 1.6 in the field of optimization is that hyperbolic programming [30] is the same as semidefinite programming.
We may reformulate Conjecture 1.6 as follows, see [18,32]. The hyperbolicity cone of h(x) with respect to e = (e 1 , . . . , e n ) is spectrahedral if there is a homogeneous polynomial q(x) and real symmetric matrices A 1 , . . . , A n of the same size such that where Λ ++ (h, e) ⊆ Λ ++ (q, e) and n i=1 e i A i is positive definite. If we can choose q(x) ≡ 1, then we say that h(x) admits a definite determinantal representation.
The paper is organized as follows. In Sect. 2, we prove Conjecture 1.6 for a multivariate generalization of the matching polynomial (Theorem 2.16). We also show that this implies Conjecture 1.6 for elementary symmetric polynomials (Theorem 2. 19). Our result may therefore be viewed as a generalization of [5]. In Sect. 3, we generalize further to a multivariate version of the independence polynomial using a recent divisibility relation of Leake and Ryder [21] (Theorem 3.9). The variables of the homogenized independence polynomial do not fully correspond combinatorially (under the line graph operation) to the more refined homogeneous matching polynomial. The restriction of Theorem 3.9 to line graphs is therefore weaker than Theorem 2.16. Finally, in Sect. 4 we consider a hyperbolic convolution of determinant polynomials generalizing an identity of Godsil and Gutman [12] which asserts that the expected characteristic polynomial of a random signing of the adjacency matrix of a graph is equal to its matching polynomial.
Unless stated otherwise, G = (V (G), E(G)) denotes a simple undirected graph. We shall adopt the following notational conventions.
• Sym(S) denotes the symmetric group on the set S. Write S n = Sym([n]).
• G H denotes the disjoint union of the graphs G and H .

Hyperbolicity cones of multivariate matching polynomials
Note that this is indeed a polynomial since m(G, k) = 0 for k > |V (G)| 2 . Heilmann and Lieb [17] studied the following multivariate version of the matching polynomial with variables x = (x i ) i∈V and non-negative weights λ = (λ e ) e∈E , [17]) If λ = (λ e ) e∈E is a sequence of non-negative edge weights, then μ λ (G, x) is stable.
We will consider the following homogeneous multivariate version of the matching polynomial.

Example 2.5
The homogeneous multivariate matching polynomial of the graph G in Fig. 1 is given by
In this section, we prove Conjecture 1.6 in the affirmative for the polynomials μ(G, x⊕ w). We first assert that μ(G, x ⊕ w) is indeed a hyperbolic polynomial.
Proof Clearly, μ(G, 1 ⊕ 0) = 1 = 0. Let x ⊕ w ∈ R G and λ e = w 2 e for all e ∈ E(G). Then, Since μ λ (G, x) is real stable by Heilmann-Lieb theorem it follows that the right hand side is real-rooted. Hence, μ(G, x ⊕ w) is hyperbolic with respect to e = 1 ⊕ 0.
Analogues of the standard recursions for the univariate matching polynomial (see [11,Thm 1.1]) also hold for μ(G, x ⊕ w). In particular, the following recursion is used frequently so we give details.

Lemma 2.8 Let u ∈ V (G). Then, the homogeneous multivariate matching polynomial satisfies the recursion
Proof The identity follows by partitioning the matchings M ∈ M(G) into two parts depending on whether Let G be a graph and u ∈ V (G). The path tree T (G, u) is the tree with vertices labelled by simple paths in G (i.e. paths with no repeated vertices) starting at u and where two vertices are joined by an edge if one vertex is labelled by a maximal subpath of the other.

Example 2.9
Definition 2.10 Let G be a graph and u ∈ V (G). Let φ : R T (G,u) → R G denote the linear change of variables defined by where x = (x p ) p∈V (T ) and w = (w e ) e∈E(T ) .

Remark 2.11
Note that η(T , x ⊕ w) is a polynomial in variables x = (x v ) v∈V (G) and w = (w e ) e∈E(G) .
For the univariate matching polynomial, we have the following rather unexpected divisibility relation due to Godsil [10], Below we prove a multivariate analogue of this fact. A similar multivariate analogue was also noted independently by Leake and Ryder [21]. In fact, they were able to find a further generalization to independence polynomials of simplicial graphs. We will revisit their results in Sect. 3. The arguments all closely resemble Godsil's proof for the univariate matching polynomial. For the convenience of the reader, we provide the details in our setting.

Lemma 2.12
Let u ∈ V (G). Then, so the lemma holds. In particular, the lemma holds for all graphs with at most two vertices. We now argue by induction on the number of vertices of G. We first claim that Let v ∈ N (u). By examining the path tree T (G, u), we note the following isomorphisms following from the fact that T (G\u, n) is isomorphic to the connected component of T (G, u)\u which contains the path un in G. By the definition of φ and the general multiplicative identity the above isomorphisms translate to the following identities from which the claim follows. By Lemma 2.8, induction, above claim and the definition of φ, we finally get which is the reciprocal of the desired identity.

Proof
The argument is by induction on the number of vertices of G. Deleting the root u of T (G, u), we get a forest with |N (u)| disjoint components isomorphic to T (G\u, v), respectively, for v ∈ N (u). This gives (2) In [12], Godsil and Gutman proved the following relationship between the univariate matching polynomial μ(G, t) of a graph G and the characteristic polynomial where the sum ranges over all subgraphs C (including C = ∅) in which each component is a cycle of degree 2 and comp(C) is the number of connected components of C. In particular, if T is a tree, then the only such subgraph is C = ∅, and therefore Next, we will derive a multivariate analogue of this relationship for trees. Lemma 2.14 Let T = (V , E) be a tree. Then, μ(T , x⊕w) has a definite determinantal representation.
Hence by acyclicity of trees, we have that

Remark 2.15
The proof of Lemma 2.14 is not dependent on T being connected so the statement remains valid for arbitrary undirected acyclic graphs (i.e. forests).
We now have all the ingredients to prove our main theorem.

Proof
The proof is by induction on the number of vertices of G. For the base case we have μ(G, x ⊕ w) = x v , so Λ + = {x ∈ R : x ≥ 0} which is clearly spectrahedral. Assume G contains more than one vertex. If G = G 1 G 2 for some non-empty graphs G 1 , G 2 , then Λ ++ (μ(G i , x ⊕ w)) is spectrahedral by induction for i = 1, 2. Therefore, showing that Λ ++ (μ(G, x ⊕ w)) is spectrahedral. We may therefore assume G is connected. Let u ∈ V (G). Since G is connected and has size greater than one, N (u) = ∅. By Lemma 2.13, we may define the polynomial for each graph G and u ∈ V (G). We want to show that By Lemma 2.12, we have that Fixing v ∈ N (u) it follows using (2) that Therefore by Lemma 1.2, for all w ∈ N (u) where the last inclusion follows by inductive hypothesis. Hence, Finally by Lemma 2.14, η(T (G, u), x⊕w) has a definite determinantal representation. Hence, the theorem follows by induction.

Remark 2.17
To show that a hyperbolic polynomial h has a spectrahedral hyperbolicity cone it is by Theorem 2.16 sufficient to show that h can be realized as a factor of a matching polynomial μ(G, , e (possibly after a linear change of variables).

Fig. 2
The star graph S n labelled by vertex and edge variables x n+1 The polynomials e d (x) are hyperbolic (in fact stable) as a consequence of e.g Grace-Walsh-Szegő theorem (see [26,Thm 15.4]).

Example 2.18
The star graph, denoted S n , is given by the complete bipartite graph K 1,n with n + 1 vertices. As an application of Theorem 2.16, we show that several wellknown instances of hyperbolic polynomials have spectrahedral hyperbolicity cones by realizing them as factors of the multivariate matching polynomial of S n under some linear change of variables. With notation as in Fig. 2, using the recursion in Lemma 2.8, the multivariate matching polynomial of S n is given by n , e). Hence, the Lorentz cone is spectrahedral. Of course this (and the preceding example) also follow from the fact that all quadratic hyperbolic polynomials have spectrahedral hyperbolicity cone [28].
Hyperbolicity cones of elementary symmetric polynomials have been studied by Zinchenko [34], Sanyal [31] and Brändén [5]. Brändén proved that all hyperbolicity cones of elementary symmetric polynomials are spectrahedral. As an application of Theorem 2.16, we give a new proof of this fact using matching polynomials. We show that e k (x) = e k ([n]) divides the multivariate matching polynomial of the length k-truncated path tree T n,k of the complete graph K n rooted at a vertex v after a linear change of variables. Let (C k ) k≥0 denote the real sequence defined by Consider the family of multivariate matching polynomials where i ∈ S, k ∈ N and φ S,k,i is the linear change of variables defined recursively (see Fig. 3) via and x v is the variable corresponding to the root of T n,k . 3. w e j → x j for j ∈ S\i where w e j are the variables corresponding to the edges e j incident to the root of T n,k . 4. For each j ∈ S\i make recursively the linear substitutions φ S\i,k−1, j , respectively, to the variables corresponding to the j-indexed copies of the subtrees of T n,k isomorphic to T n−1,k−1 .
We claim

Hyperbolicity cones of multivariate independence polynomials
A subset I ⊆ V (G) is independent if no two vertices of I are adjacent in G. Let I(G) denote the set of all independent sets in G, and let i(G, k) denote the number of independent sets in G of size k. By convention, i(G, 0) = 1. The (univariate) independence polynomial is defined by The line graph L(G) of G is the graph having vertex set E(G) and where two vertices in L(G) are adjacent if and only if the corresponding edges in G are incident. It follows that μ(G, t) = t |V (G)| I (L(G), −t −2 ). Therefore, the independence polynomial can be viewed as a generalization of the matching polynomial. In contrast to the matching polynomial, the independence polynomial of a graph is not real-rooted in general. However, Chudnovsky and Seymour [8] proved that I (G, t) is real-rooted if G is claw free, that is, if G has no induced subgraph isomorphic to the complete bipartite graph K 1,3 . The theorem was later generalized by Engström to graphs with weighted vertices.
Theorem 3.1 (Engström [9]) Let G be a claw-free graph and λ = (λ v ) v∈V (G) a sequence of non-negative vertex weights. Then, the polynomial

A full characterization of the graphs for which I (G, t) is real-rooted remains an open problem.
A natural multivariate analogue of the independence polynomial is given by Leake and Ryder [21] define a strictly weaker notion of stability which they call samephase stability. A polynomial p(z) ∈ R[z 1 , . . . , z n ] is (real) same-phase stable if for every x ∈ R n + , the univariate polynomial p(tx) is real-rooted. The authors prove that I (G, x) is same-phase stable if and only if G is claw free. In fact, the same-phase stability of I (G, x) is an immediate consequence of Theorem 3.1.
The added variables in a homogeneous multivariate independence polynomial should preferably have labels carrying combinatorial meaning in the graph. For line graphs, it is additionally desirable to maintain a natural correspondence with the homogeneous multivariate matching polynomial μ (G, x ⊕ w). Unfortunately, we have not found a hyperbolic definition that satisfies both of the above properties. We have thus settled for the following definition.  e = (0, . . . , 0, 1)

Lemma 3.3 If G is a claw-free graph, then I (G, x ⊕ t) is a hyperbolic polynomial with respect to
By Theorem 3.1 the polynomial I λ (G, s) is real-rooted. Clearly, all roots are negative which implies I λ (G, −s −2 ) is real-rooted. Hence, the univariate polynomial s → I (G, se − x ⊕ t) is real-rooted which shows that I (G, x ⊕ t) is hyperbolic with respect to e.
An induced clique K in G is called a simplicial clique if for all u ∈ K the induced subgraph N [u] ∩ (G\K ) of G\K is a clique. In other words, the neighbourhood of each u ∈ K is a disjoint union of two induced cliques in G. Furthermore, a graph G is said to be simplicial if G is claw free and contains a simplicial clique.
In this section, we prove Conjecture 1.6 for the polynomial I (G, x ⊕ t) when G is simplicial. The proof unfolds in a parallel manner to Theorem 2.16 by considering a different kind of path tree. Before the results can be stated, we must outline the necessary definitions from [21].
A connected graph G is a block graph if each 2-connected component is a clique. Given a simplicial graph G with a simplicial clique K , we recursively define a block graph T (G, K ) called the clique tree associated to G and rooted at K (see Fig. 4).
We begin by adding K to T (G, K ). Let K u = N [u]\K for each u ∈ K . Attach the disjoint union u∈K K u of cliques to T (G, K ) by connecting u ∈ K to every v ∈ K u . Finally, recursively attach T (G\K , K u ) to the clique K u in T (G, K ) for every u ∈ K . Note that the recursion is made well-defined by the following lemma.

Lemma 3.4 (Chudnovsky-Seymour [8]) Let G be a claw-free graph, and let K be a simplicial clique in G. Then, N [u]\K is a simplicial clique in G\K for all u ∈ K .
It is well known that a graph is the line graph of a tree if and only if it is a clawfree block graph [16,Thm 8.5]. In [21], it was demonstrated that the block graph T (G, K ) is the line graph of a certain induced path tree T ∠ (G, K ). Its precise definition is not important to us, but we remark that it is a subtree of the usual path tree defined in Sect. 2 that avoids traversed neighbours. This enables us to find a definite determinantal representation of I (T (G, K ), x ⊕ t) via Lemma 2.14. The second important fact is that I (G, x) divides I (T (G, K ), x) where T (G, K ) is relabelled according to the natural graph homomorphism φ K : T (G, K ) → G. Hence using the recursion provided by the simplicial structure of G, we have almost all the ingredients to finish the proof of Conjecture 1.6 for I (G, x ⊕ t). structure of simplicial graphs and removing cliques instead of vertices. For the proof to go through in the homogeneous setting, we must replace the usual recursion by Theorem 3.6 (Leake-Ryder [21]) Let K be a simplicial clique of the simplicial graph G. Then, where T (G, K ) is relabelled according to the natural graph homomorphism φ K : The following lemma ensures the hyperbolicity cones behave well under vertex deletion.

Lemma 3.7
Let v ∈ V (G). Then, Λ ++ (I (G, x ⊕ t)) ⊆ Λ ++ (I (G\v, x ⊕ t)). R and e = (0, . . . , 0, 1). By Lemma 3.3 the polynomials s → I (G, se − x ⊕ t) and s → I (G\v, se − x ⊕ t) are both real-rooted. Denote their roots by α 1 , . . . , α 2n and β 1 , . . . , β 2n−2 , respectively, where n = |V (G)|. We claim that by induction on the number of vertices of G. Indeed, the claim is vacuously true if |V (G)| = 1. Suppose therefore |V (G)| > 1. If G is not connected, then G = G 1 G 2 for some non-empty graphs G 1 , G 2 . Without loss assume v ∈ G 1 . Then, G\v = (G 1 \v) G 2 . By induction, the claim holds for the pair G 1 and G 1 \v. This implies the claim for G and G\v since I (G, x ⊕ t) is multiplicative with respect to disjoint union. We may therefore assume G is connected. Thus, G\N [v] is of strictly smaller size than G\v. We have By induction, the maximal root γ of I (G\N [v], se−x⊕t) is less than the maximal root β of I (G\v, se − x ⊕ t). Since I (G\N [v], se − x ⊕ t) is an even degree polynomial with positive leading coefficient we have that I (G\N [v], se − x ⊕ t) ≥ 0 for all s ≥ γ . By (3), this implies that Since each of the terms involved in the polynomials I (G, se − x ⊕ t) and I (G\v, se − x ⊕ t) has even degree in s − t, their respective roots are symmetric about s = t. Hence, min i α i ≤ min i β i proving the claim. Finally, if x 0 ⊕ t 0 ∈ Λ ++ (I (G, x ⊕ t)), then min i α i > 0 so by the claim min i β i > 0 showing that x 0 ⊕ t 0 ∈ Λ ++ (I (G\v, x ⊕ t)). This proves the lemma.

Remark 3.8
Since we see by Lemma 3.7 that setting vertex variables equal to zero relaxes the hyperbolicity cone.

Theorem 3.9
If G is a simplicial graph, then the hyperbolicity cone of I (G, x ⊕ t) is spectrahedral.
Proof Let K be a simplicial clique of G. Arguing by induction as in Theorem 2.16, using the clique tree T (G, K ) instead of the path tree T (G, u), and invoking Theorem 3.6 we get a factorization where v ∈ K is fixed, K w = N [w]\K and for w ∈ K . Repeated application of Lemma 3.7 gives By factorization (4) and induction, we hence get the desired cone inclusion Since L(T ∠ (G, K )) ∼ = T (G, K ) by Lemma 3.5 we see that Hence, I (T (G, K ), x⊕t) has a definite determinantal representation by Lemma 2.14 proving the theorem.

Convolutions
If G is a simple undirected graph with adjacency matrix A = (a i j ), then we may associate a signing s = (s i j ) ∈ {±1} E(G) to its edges. The symmetric adjacency matrix A s = (a s i j ) of the resulting graph is given by a s i j = s i j a i j for i j ∈ E(G) and a s i j = 0 otherwise. Godsil and Gutman [13] proved that In other words, the expected characteristic polynomial of an independent random signing of the adjacency matrix of a graph is equal to its matching polynomial. Therefore, the expected characteristic polynomial is real-rooted. This was one of the facts used by Marcus, Spielman and Srivastava [23] in proving that there exist infinite families of regular bipartite Ramanujan graphs. Since then, several other families of characteristic polynomials have been identified with real-rooted expectation (see, e.g. [15,25]). Such families are called interlacing families, based on the fact that there exists a common root interlacing polynomial if and only if every convex combination of the family is real-rooted. The method of interlacing families has been successfully applied to other contexts, in particular to the affirmative resolution of the Kadison-Singer problem [24]. In this section, we define a convolution of multivariate determinant polynomials and show that it is hyperbolic as a direct consequence of a more general theorem by Brändén [6]. In particular, this convolution can be viewed as a generalization of the fact that the expectation in (5) is real-rooted. Namely, we show that the expected characteristic polynomial over any finite set of independent random edge weightings is real-rooted barring certain adjustments to the weights of the loop edges.
Recall that every symmetric matrix may be identified with the adjacency matrix of an undirected weighted graph (with loops). Definition 4.1 Let W ⊆ R be a finite set. Given a real symmetric matrix A and a vector w ∈ W ( n 2 ) , define a weighting of A to be a symmetric matrix A w = (a w i j ) given by j=1 and Y = (y i j ) n i, j=1 be symmetric matrices in variables x = (x i j ) i≤ j and y = (y i j ) i≤ j , respectively. Let W ⊆ R be a finite set. We define the convolution We have the following general fact about hyperbolic polynomials. Theorem 4.3 (Brändén [6]) Let h(x) be a hyperbolic polynomial with respect to e ∈ R n ; let V 1 , . . . , V m be finite sets of vectors of rank at most one in where u ∈ R n and (α 1 , . . . , α m ) ∈ R m . Then, E We note that h(e) = 1 = 0. Let δ 1 , . . . , δ n denote the standard basis of R n . Put where v i jw = (δ i + wδ j )(δ i + wδ j ) T for i < j and w ∈ W . Note that v i jw is a rank one matrix belonging to the hyperbolicity cone of positive semidefinite matrices (with nonzero eigenvalue w 2 + 1). Letting u = 0 and α X i j = x i j , α Y i j = y i j for i < j, we see that where the right hand side is a real-rooted polynomial in t by Theorem 4.3. Hence, det(X ) * W det(Y ) is hyperbolic with respect to e.
is hyperbolic, where the expectation is taken over independent random signings of the matrices X and Y as in (5) without weighting the diagonal. This shows in particular that the expectation in (5) is real-rooted.

Corollary 4.6 Let W ⊆ R be a finite subset and A a real symmetric n × n matrix.
Then, is real-rooted.
Proof By Corollary 4.4, the polynomial det(Y ) * W det(X ) is hyperbolic, so in particular t → E w det(t I − A w ) is real-rooted with X = 0 and Y = A.
Next, we see that convolution (6) over independent random signings can be realized as a convolution of multivariate matching polynomials. The proof is similar to that of the univariate identity (5) (cf [13]). Let G X and G Y denote the weighted graphs corresponding to the symmetric matrices X and Y .
Proposition 4.7 Let X = (x i j ) n i, j=1 and Y = (y i j ) n i, j=1 be symmetric matrices in variables x = (x i j ) i≤ j and y = (y i j ) i≤ j . Then E s (1) ,s (2) det(X s (1) + Y s (2) ) =