Stable multivariate generalizations of matching polynomials
The first part of this note concerns stable averages of multivariate matching polynomials. In proving the existence of infinite families of bipartite Ramanujan -coverings, Hall, Puder and Sawin introduced the -matching polynomial of a graph , defined as the uniform average of matching polynomials over the set of -sheeted covering graphs of . We prove that a natural multivariate version of the -matching polynomial is stable, consequently giving a short direct proof of the real-rootedness of the -matching polynomial. Our theorem also includes graphs with loops, thus answering a question of said authors. Furthermore we define a weaker notion of matchings for hypergraphs and prove that a family of natural polynomials associated to such matchings are stable. In particular this provides a hypergraphic generalization of the classical Heilmann-Lieb theorem.
The real-rootedness of the matching polynomial of a graph is a well-known result in algebraic graph theory due to Heilmann and Lieb . Slightly less quoted is its stronger multivariate counterpart (see ) which proclaims that the multivariate matching polynomial is non-vanishing when its variables are restricted to the upper complex half-plane, a property known as stability. Other stable polynomials occurring in combinatorics include e.g. multivariate Eulerian polynomials , several bases generating polynomials of matroids (including multivariate spanning tree polynomials)  and certain multivariate subgraph polynomials . In the present note we consider several different stable generalizations of multivariate matching polynomials. Hall, Puder and Sawin prove in  that every connected bipartite graph has a Ramanujan -covering of every degree for each , generalizing seminal work of Marcus, Spielman and Srivastava [16, 18] for the case . An important object in their proof is a certain generalization of the matching polynomial of a graph , called the -matching polynomial, defined by taking averages of matching polynomials over the set of -sheeted covering graphs of . The authors prove (via an indirect method) that the -matching polynomial of a multigraph is real-rooted provided that the graph contains no loops. We prove in Theorem 3.7 that the latter hypothesis is redundant by establishing a stronger result, namely that the multivariate -matching polynomial is stable for any multigraph (possibly with loops). In the final section we consider a hypergraphic generalization of the Heilmann-Lieb theorem. The hypergraphic matching polynomial is not real-rooted in general (see ) so it does not admit a natural stable multivariate refinement. However by relaxing the notion of matchings in hypergraphs we prove in Theorem 5.6 that an associated “relaxed” multivariate matching polynomial is stable.
2.1. Graph coverings and group labelings
In this subsection we outline relevant definitions from . Let be a finite, connected, undirected graph on . In particular we allow to have multiple edges between vertices and contain edges from a vertex to itself, i.e., is a multigraph with loops.
A graph homomorphism is called a local isomorphism if for each vertex in , the restriction of to the neighbours of in is an isomorphism onto the neighbours in . We call a covering map if it is a surjective local isomorphism, in which case we say that covers . If the image of under the covering map is connected, then each fiber of is an independent set of vertices in of the same size . If so, we call a -sheeted covering (or -covering for short) of .
Although is undirected we shall dually view it as an oriented graph, containing two edges with opposite orientation for each undirected edge. We denote the edges with positive (resp. negative) orientation by (resp. ) and identify with the disjoint union . If , then we write for the corresponding edge in with opposite orientation. Moreover we denote by and , the head and tail of the edge respectively. A -covering of a graph can be constructed via the following model, introduced in [1, 7]. The vertices of are defined by . The edges of are determined, as described below, by a labeling (see Figure 1) satisfying . For notational purposes we write . For every positively oriented edge we introduce (undirected) edges in connecting to for , that is, we replace each undirected edge in by the perfect matching induced by , see Figure 2. We shall interchangeably refer to the map and the covering graph which it determines, as a covering of . Let denote the probability space of all -coverings of endowed with the uniform distribution.
Instead of labeling each edge in by a permutation in we may label the edges with elements coming from an arbitrary finite group . A -labeling of a graph is a function satisfying . Let denote the probability space of all -labelings of endowed with the uniform distribution. Let be a representation of . For any -labeling of , let denote the matrix obtained from the adjacency matrix of by replacing the entry in with the block (where the sum runs over all oriented edges from to ) and by a zero block if there are no edges between and . The matrix is called a -covering of .
Consider the -dimensional representation of the symmetric group mapping every to its corresponding permutation matrix. The representation is reducible since the -dimensional space , where , is invariant under the action of . The action of on the orthogonal complement is an irreducible -dimensional representation called the standard representation, denoted . As outlined in , every -covering of corresponds uniquely to a -covering of .
2.2. Stable polynomials
A polynomial is said to be stable if whenever for all . By convention we also regard the zero polynomial to be stable. A stable polynomial with only real coefficients is said to be real stable. Note that univariate real stable polynomials are precisely the real-rooted polynomials (i.e. real polynomials in one variable with all zeros in ). Thus stability may be regarded as a multivariate generalization of real-rootedness. Below we collect a few facts about stable polynomials which are relevant for the forthcoming sections. For a more comprehensive background we refer to the survey by Wagner  and references therein.
A common technique for proving that a polynomial is stable is to realize as the image of a known stable polynomial under a stability preserving linear transformation. Stable polynomials satisfy several basic closure properties, among them are diagonalization for and differentiation where . The following theorem by Lieb and Sokal provides the construction for a large family of stability preserving linear transformations.
Theorem 2.1 (Lieb-Sokal ).
If is a stable polynomial, then is a stability preserving linear operator.
Borcea and Brändén  gave a complete characterization of the linear operators preserving stability. The following is the transcendental characterization of stability preservers on infinite-dimensional complex polynomial spaces. Define the complex Laguerre-Pólya class to be the class of entire functions in variables that are limits, uniformly on compact sets of stable polynomials in variables. Throughout we will use the following multi-index notation
where and .
Theorem 2.2 (Borcea-Brändén ).
Let be a linear operator. Then preserves stability if and only if either
has range of dimension at most one and is of the form
where is a linear functional on and is a stable polynomial, or
belongs to the Laguerre-Pólya class.
A polynomial is said to be multiaffine if each variable occurs with degree at most one in , and is called symmetric if for all . The Grace-Walsh-Szegö coincidence theorem is a cornerstone in the theory of stable polynomials frequently used to depolarize symmetries before checking stability. One version of it is stated below, see [3, 21] for modern references and alternative proofs.
3. Stability of multivariate -matching polynomials
A matching of an undirected graph is a subset such that no two edges in share a common vertex. Let denote the set of vertices in the matching . For , the -matching polynomial of is defined by
and denotes the number of matchings in of size with . In particular if , then coincides with the conventional matching polynomial . The following results are proved in .
Theorem 3.1 (Hall-Puder-Sawin ).
Let be a finite group and be an irreducible representation such that is a complex reflection group, i.e., is generated by pseudo-reflections. If is a finite connected multigraph, then
Remarkably the expected characteristic polynomial in (3.1) depends only on the dimension of the irreducible representation and not on the particular choice of group , nor the specifics of the map . Real-rooted expected characteristic polynomials have seen a surge of interest recently in light of the Kadison-Singer problem and Ramanujan coverings, see e.g. [2, 11, 16, 17, 18, 19].
A classical result due to Godsil and Gutman  states that if is the adjacency matrix of a finite simple undirected graph , then
where for all and . In other words, the expected characteristic polynomial over all signings of equals the matching polynomial of . In the language of Hall, Puder and Sawin this corresponds to taking and to be the sign representation in Theorem 3.1.
Generalizing and extending the following theorem will be the main focus of this section.
Theorem 3.4 (Hall-Puder-Sawin ).
If is a finite multigraph with no loops, then is real-rooted.
Hall, Puder and Sawin also showed that the roots of are contained inside the Ramanujan interval of (see ).
Define the multivariate -matching polynomial of by
and the sum runs over all matchings in .
The real-rootedness of was proved indirectly in  by considering a limit of interlacing families converging to the left-hand side in Theorem 3.1. In this section we use a more direct approach for proving the real-rootedness of . In fact we prove something stronger, namely that is stable. Our proof also holds for graphs with loop edges, thus removing the redundant hypothesis in Theorem 3.4.
Coverings of graphs with loop edges have interesting properties. In particular, consider the -dimensional regular representation sending an element to the permutation matrix afforded by acting on through left translation . The bouquet graph is the graph consisting of a single vertex with loop edges. A -covering of is equivalent to the Cayley graph of with respect to the set . In this sense -coverings of finite multigraphs with loops generalize Cayley graphs of finite groups.
and note that it is a stability preserving linear operator. Indeed the symbol
is stable being a product of stable polynomials. Since the range of MAP has dimension greater than one, it follows that MAP preserves stability by Theorem 2.2. Given the identity
where is the subgraph of induced by and denotes the degree of in , we have that
and hence that is stable being the image of a stable polynomial under MAP. This result is also known as the Heilmann-Lieb theorem .
By using Theorem 2.3 and the stability preserving linear operator MAP we will show below that is stable.
If is a finite multigraph (possibly with loops), then is stable for all .
For a -covering , let
where denotes the set of positively oriented loops in . Since no matching may contain loops we have excluded the factors from the subgraph generating polynomial in (3.2) where is the covering graph corresponding to . This explains the form of . It follows that
For the polynomials
are symmetric and multiaffine polynomials in the two sets of variables
respectively. By Theorem 2.3 we have that is stable if and only if
is stable, the latter of which is clear. Similary if , then is symmetric and multiaffine in the set of variables , so checking stability of reduces by Theorem 2.3 to checking the stability of , which is again clear. Hence is stable being a product of stable polynomials. Finally we have
Hence is stable. ∎
If is a finite multigraph (possibly with loops), then is real-rooted for all .
Follows by putting in Theorem 3.7 ∎
4. Stable expected matching polynomials over induced subgraphs
In the previous section we considered stable averages of multivariate matching polynomials over the set of -sheeted covering graphs of . In this section we consider stable averages over (vertex-) induced subgraphs of . To this end, if , let denote the subgraph of induced by the vertices in . Let be a probability distribution on the power set . The polynomial
is called the partition function of . The probability distribution is called Rayleigh if
for all , and is called strong Rayleigh if (4.1) holds for all , .
Theorem 4.1 (Brändén ).
A probability distribution is strong Rayleigh if and only if is stable.
Let be a finite undirected graph on and let be a probability distribution on . If is strong Rayleigh, then is stable.
If is a strong Rayleigh probability distribution, then is real-rooted.
The following example demonstrates that the converse to Proposition 4.2 is false. Consider the graph on two vertices and one edge. If is a probability distribution with , , and , then
which is stable if and only of . On the other hand
is stable if and only if . Hence there exists probability distributions which are not strong Rayleigh for which is stable. An interesting question would be to characterize the probability distributions for which is stable.
A natural probability distribution on the set of induced subgraphs of is the Bernoulli distribution where a vertex is selected independently with probability and not selected with probability . Note that is a strong Rayleigh probability distribution since
is stable. Hence is stable by Proposition 4.2.
Next we shall provide bounds for the real roots of .
Let and define a graph with vertex set being the set of all non-backtracking walks in starting from , i.e., sequences such that , and are adjacent and . Two such walks are connected by an edge in if one walk extends the other by one vertex, i.e., is adjacent to . The graph thus constructed is a tree that covers . It is called the universal covering tree of . The universal covering tree of is unique up to isomorphism and has the property that it covers every other covering of . Thus we henceforth remove reference to the root and write for the universal covering tree of . The tree is countably infinite, unless is a finite tree, in which case .
The spectral radius of a finite graph is the largest absolute eigenvalue of the adjacency matrix of . By a theorem of Mohar  the spectral radius of an infinite graph can be defined as follows,
If is a finite undirected graph, then let denote the spectral radius of its universal covering tree. Say that a probability distribution on has constant parity if the set consists of numbers with the same parity (i.e. are either all odd or all even).
Let be a finite undirected graph with vertices and a probability distribution on . Then the real roots of are bounded above by . Moreover if has constant parity, then the real roots of are contained in .
Let . There is a clear injective embedding of into such that any finite induced subgraph of is an induced subgraph of . Therefore . Heilmann and Lieb  showed that for any finite graph , the roots of are contained in . Therefore for all , being a convex combination of monic polynomials with the same property. Hence the real roots of the expectation are bounded above by . If also has constant parity, then is a convex combination of monic polynomials with same degree parity and are therefore, by above, strictly positive or strictly negative on the interval . Hence the real roots are contained in . ∎
Let be a finite undirected graph on vertices and . Then the uniform average of all matching polynomials over the set of induced size -subgraphs of is a real-rooted polynomial with all roots contained in the interval .
Let be the probability distribution on with uniform support on . Then
where denotes the elementary symmetric polynomial of degree . The polynomial is stable, e.g. by Theorem 2.3. Therefore is a strong Rayleigh probability distribution by Theorem 4.1, so the statement follows by Corollary 4.3 and Proposition 4.6. ∎
5. Stable relaxed matching polynomials
A hypergraph is a set of vertices together with a family of subsets of called hyperedges (or edges for short). The degree of a vertex is defined as . In analogy with graph matchings, a matching in a hypergraph consists of a subset of edges with empty pairwise intersection. Although the matching polynomial of a graph is real-rooted, the analogous polynomial for hypergraphs is not real-rooted in general, see e.g. . From the point of view of real-rootedness we consider a weaker notion of matchings that provide a natural generalization of the real-rootedness property of to hypergraphs.
Let be a hypergraph. Define a relaxed matching in to be a collection of edge subsets such that , , and for all pairwise distinct (see Figure 4).
If is a graph then the concept of relaxed matching coincides with the conventional notion of graph matching. Note also that a conventional hypergraph matching is a relaxed matching for which for all .
The subsets in the relaxed matching are labeled by the edge they are chosen from in order to avoid ambiguity. However if is a linear hypergraph, that is, the edges pairwise intersect in at most one vertex, then the subsets uniquely determine the edges they belong to and therefore no labeling is necessary. Graphs and finite projective geometries (viewed as hypergraphs) are examples of linear hypergraphs.
Let denote the set of vertices in the relaxed matching. Moreover let denote the number of subsets in the relaxed matching of size . Define the multivariate relaxed matching polynomial of by
where the sum runs over all relaxed matchings of and
Let denote the univariate relaxed matching polynomial.
Note that if is a graph.
Our aim is to prove that is a stable polynomial. In fact we shall prove the stability of a more general polynomial accommodating for arbitrary degree restrictions on each vertex.
Define a relaxed subgraph of to be a hypergraph with edges such that , and for with . Again if is a graph, then the notion of a relaxed subgraph coincides with the conventional notion of a (edge-induced) subgraph of . Let . Define a relaxed -subgraph of to be a relaxed subgraph of such that for . Let and let denote the Pochhammer symbol.
Define the multivariate relaxed -subgraph polynomial of by
where the sum runs over all relaxed -subgraphs of and
Note that a relaxed matching in is the same as a relaxed -subgraph of and that .
In the rest of this section we will adopt the following notation,
where and .
With abuse of notation we shall let the multiaffine part operator MAP act analogously on polynomial spaces of differential operators as follows,
The following lemma follows from Theorem 2.1.
If is a linear operator such that is stable, then preserves stability.
Write . Since is a stability preserver we have that is stable and hence by Theorem 2.1 that is a stability preserving linear operator. ∎
Let be a hypergraph and . Then the multivariate relaxed -subgraph polynomial is stable with
Let . Then