X-Ramanujan Graphs

04/06/2019 ∙ by Sidhanth Mohanty, et al. ∙ Carnegie Mellon University berkeley college 0

Let X be an infinite graph of bounded degree; e.g., the Cayley graph of a free product of finite groups. If G is a finite graph covered by X, it is said to be X-Ramanujan if its second-largest eigenvalue λ_2(G) is at most the spectral radius ρ(X) of X, and more generally k-quasi-X-Ramanujan if λ_k(G) is at most ρ(X). In case X is the infinite Δ-regular tree, this reduces to the well known notion of a finite Δ-regular graph being Ramanujan. Inspired by the Interlacing Polynomials method of Marcus, Spielman, and Srivastava, we show the existence of infinitely many k-quasi-X-Ramanujan graphs for a variety of infinite X. In particular, X need not be a tree; our analysis is applicable whenever X is what we call an additive product graph. This additive product is a new construction of an infinite graph AddProd(A_1, ..., A_c) from finite 'atom' graphs A_1, ..., A_c over a common vertex set. It generalizes the notion of the free product graph A_1 * ... * A_c when the atoms A_j are vertex-transitive, and it generalizes the notion of the universal covering tree when the atoms A_j are single-edge graphs. Key to our analysis is a new graph polynomial α(A_1, ..., A_c;x) that we call the additive characteristic polynomial. It generalizes the well known matching polynomial μ(G;x) in case the atoms A_j are the single edges of G, and it generalizes the r-characteristic polynomial introduced in [Ravichandran'16, Leake-Ravichandran'18]. We show that α(A_1, ..., A_c;x) is real-rooted, and all of its roots have magnitude at most ρ(AddProd(A_1, ..., A_c)). This last fact is proven by generalizing Godsil's notion of treelike walks on a graph G to a notion of freelike walks on a collection of atoms A_1, ..., A_c.

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1 Introduction

Let be an infinite graph, such as one of the eight partially sketched below.

Figure 1: Some infinite graphs (with only a finite portion sketched, repeating in the obvious way).

For each such , we are interested in finite graphs  that “locally resemble” . More precisely we are interested in , all the finite graphs (up to isomorphism) that are covered by ; essentially, this means that there is a graph homomorphism that is a local isomorphism (see Section 2). For example, if is the bottom-left graph in Figure 1, then consists of the finite graphs that are collections of ’s, with each vertex participating in three ’s. Or, if is the top-right graph in Figure 1, namely the infinite regular tree , the set is just the -regular finite graphs.

For , a well known problem is to understand the possible expansion properties of graphs in the set; in particular, to understand the possible eigenvalues of -regular graphs. Every -regular graph has a “trivial” eigenvalue of . As for the next largest eigenvalue, , a theorem of Alon and Boppana [Alo86] implies that for any , almost all have . This special quantity, , is precisely the spectral radius of the infinite graph . (In general, the spectral radius of the infinite -regular tree is .) The “Ramanjuan question” is to ask whether the Alon–Boppana bound is tight; that is, whether there are infinitely many -regular graphs  with . Such “optimal spectral expanders” are called -regular Ramanujan graphs.

In this paper, we investigate the same question for other infinite graphs , beyond just and other infinite trees. Take again the bottom-left in Figure 1, which happens to be the “free product graph” . Every is -regular, so Alon–Boppana implies that almost all these have . But in fact, an extension of Alon–Boppana due to Grigorchuk and Żuk [GZ99] shows that almost all must have , where is the spectral radius of . The Ramanujan question for then becomes: Are there infinitely many -Ramanujan graphs, meaning graphs with ? We will show the answer is positive.

We may repeat the same question for, say, the sixth graph in Figure 1, , the Cayley graph of the modular group : We will show there are infinitely many with at most the spectral radius of , namely .

Or again, consider the top-left graph in Figure 1, , a graph determined by Paschke [Pas93] to have spectral radius , the largest root of . Paschke showed this is the infinite graph of smallest spectral radius that is vertex-transitive, -regular, and contains a triangle. Using Paschke’s work, Mohar [Moh10, Proof of Theorem 6.2] showed that almost all finite -regular graphs in which every vertex participates in a triangle must have . We will show that, conversely, there are infinitely many such graphs  with .

Motivations.

We discuss here some motivations from theoretical computer science and other areas. The traditional motivation for Ramanujan graphs is their optimal (spectral) expansion property. The usual -Ramanujan graphs have either few or no short cycles. However one may conceive of situations requiring graphs with both strong spectral expansion properties and plenty of short cycles. Our -Ramanujan graphs (for example) have this property. Ramanujan graphs with special additional structures have indeed played a role in areas such as quantum computation [PS18] and cryptography [CFL18]. Another recent application comes from a work by Kollár–Fitzpatrick–Sarnak–Houck [KFSH19] on circuit quantum electrodynamics, where finite -regular graphs with carefully controlled eigenvalue intervals (and in particular, -Ramanujan graphs) play a key role.

Recently, there has been a lot of interest in high-dimensional expanders (see the survey [Lub17]), which are expander graphs with certain constrained local structure — for example a 2-dimensional expander is a graph composed of triangles where the neighborhood of every vertex is an expander. We speculate that the tools introduced in this work might be useful in constructing high dimensional expanders.

Our original motivation for investigating these questions came from the algorithmic theory of random constraint satisfaction problems (CSPs). There are certain predicates (“constraints”) on a small number of Boolean variables that are well modeled by (possibly edge-signed) graphs. Besides the “cut” predicate (modeled by a single edge), examples include: the (Not-All-Equals) predicate on three Boolean variables, which is modeled by the triangle graph ; and, the the predicate on four Boolean variables, which is modeled by the graph (with three edges “negated”). It is of considerable interest to understand how well efficient algorithms (e.g., eigenvalue/SDP-based algorithms) can solve large CSPs. The most challenging instances tend to be large random CSPs, and this motivates studying the eigenvalues of large random regular graphs composed of, say, triangles (for random CSPs) or ’s (for random ). The methods we introduce in this paper provide a natural way (“additive lifts”) to produce such large random instances, as well as to understand their eigenvalues. See [DMO19] for more details on the CSP, [Moh18] for more details on the CSP, and recent followup work of the authors and Paredes [MOP19] on more general CSPs, including “Friedman’s Theorem”-style results for some random CSPs.

2 -Ramanujan graphs

In this section, (and similarly , , etc.) will denote a connected undirected graph on at most countably many vertices, with uniformly bounded vertex degrees, and with multiple edges and self-loops allowed. We also identify with its associated adjacency matrix operator, acting on . We recall some basic facts concerning spectral properties of ; see, e.g., [MW89, Moh82]. The spectrum of is the set of all complex such that is not invertible. In fact, is a subset of since we’re assuming is undirected; it is also a compact set. When is finite, we extend to be a multiset — namely, the roots of the characteristic polynomial , taken with multiplicity. In this case, we write the spectrum (eigenvalues) of  as .

The spectral radius of  is

where denotes the number of walks of length  in  from vertex to vertex ; it is not hard to show this is independent of the particular choice of [Woe00, Lemma 1.7]. The spectral radius is also equal to the operator norm of acting on , and to . We also have . Finally, holds for every and . If is bipartite then its spectrum is symmetric about ; i.e., if and only if . In particular, .

For graphs , , we say that is a cover of (and that is a quotient of ) if there exists where and are surjections satisfying , and bijects the edges incident to in with the edges incident to in . The map is also called a covering. As an example, one may check that the infinite graph depicted on the left in Figure 2 is a cover of the finite graph depicted on the right.

Figure 2: The infinite graph on the left covers the finite graph on the right.

If is a cover of , then . Furthermore, if is finite then . The first statement in the above fact is an easy and well known consequence of Section 2, since distinct closed walks in map to distinct closed walks in . The second statement may be considered folklore, and appears in Greenberg’s thesis [Gre95]. We now review some definitions and results from that thesis. (See also [LN98].) Given , we write to denote the family of finite graphs111Up to isomorphism. covered by . (For this definition, we are generally interested in infinite graphs .) The set may be empty; but otherwise, by a combination of the fact that and have the same universal covering tree, the fact that any two graphs with the same universal covering tree have a common finite cover (due to [Lei82]), and Section 2, for all . When we write for the common spectral radius of all . If is -regular and then (because all are -regular). We also remark that it is not particularly easy to decide whether or not , given . In case is an infinite tree, it is known that is nonempty if and only if is the universal cover tree of some finite connected graph [BK90].

2.1 Alon–Boppana-type theorems

When is nonempty, we are interested in the spectrum of graphs in . Each such will have , so we consider the remaining eigenvalues , particularly . In the case that is -regular, the smallness of (and ) controls the expansion of . If is bipartite, then and the bipartite expansion is controlled just by .

A notable theorem of Alon and Boppana [Alo86] is that for all and all , there are only finitely many -regular graphs  with . It is well known that is the spectral radius of the infinite -regular tree, . Since is the set of all finite connected -regular graphs, the Alon–Boppana Theorem may also be stated as

(Notice that if we restrict attention to bipartite -regular graphs  on vertices, we’ll have and we can conclude that there are only finitely many such graphs that have .) Thus the Alon–Boppana gives a limitation on the spectral expansion quality of -regular graphs and bipartite graphs.

There are several known extensions and strengthenings of the Alon–Boppana Theorem. One strengthening (usually attributed to Serre [Ser97]) says that for any , there are only finitely many -regular with . Indeed, in an -vertex -regular graph , at least eigenvalues from must be at least , for some .

Another interesting direction, due to Feng and Li [FL96], concerns the same questions for -biregular graphs. (This includes the case of -regular bipartite graphs, by taking .) These are precisely the graphs when is the infinite -biregular tree . It holds that and . Feng and Li showed the Alon–Boppana Theorem analogue in this setting,

they also showed the Serre-style strengthening. Mohar gave certain generalizations of these results to multipartite graphs [Moh10].

In these Alon–Boppana(–Serre)-type results for quotients of infinite trees , the lower bound on second eigenvalues for graphs in has been . In fact, it turns out that this phenomenon holds even when is not a tree. The following theorem was first proved by Greenberg [Gre95] (except that he considered the absolute values of eigenvalues); see [Li96, Chap. 9, Thm. 13] and Grigorchuk–Żuk [GZ99] for proofs: For any and any , there is a constant such that for all -vertex it holds that at least eigenvalues from are at least . (In particular, — and indeed, for any — is at least for all but finitely many .)

2.2 Ramanujan and -Ramanujan graphs

The original Alon–Boppana Theorem showed that, for any , one cannot have infinitely many (distinct) -regular graphs  with . But can one have infinitely many -regular graphs  with ? Such graphs are called (-regular) Ramanujan graphs, and infinite families of them were first constructed (for prime) by Lubotzky–Phillips–Sarnak [LPS88] and by Margulis [Mar88]. Let us clarify here the several slightly different definitions of Ramanujan graphs. We shall call an -vertex, -regular graph (with ) (one-sided) Ramanujan if , and two-sided Ramanujan if in addition . If is bipartite and (one-sided) Ramanujan, we will call it bipartite Ramanujan. (A bipartite graph cannot be two-sided Ramanujan, as will always be .) The Lubotzky–Phillips–Sarnak–Margulis constructions give infinitely many -regular two-sided Ramanujan graphs whenever is a prime congruent to  mod , and also infinitely many -regular bipartite Ramanujan graphs in the same case. By 1994, Morgenstern [Mor94] had extended their methods to obtain the same results only assuming that is a prime power. It should be noted that these constructions produced Ramanujan graphs only for certain numbers of vertices  (namely all , for certain fixed polynomials ).

To summarize these results: for all with a prime power, there are infinitely many with . (We will later discuss the improvement to these results by Marcus, Spielman, and Srivastava [MSS15b, MSS15d].) Based on Greenberg’s Section 2.1, it is very natural to ask if analogous results are true not just for other , but for other infinite . This was apparently first asked for the infinite biregular tree by Hashimoto [Has89] (later also by Li and Solé [LS96]).222In fact, they defined a notion of Ramanujancy for -biregular graphs that has a flavor even stronger than that of two-sidedness; they required ; in other words, for all with , not only do we have , but also . This definition is related to the Riemann Hypothesis for the zeta function of graphs. More generally, Grigorchuk and Żuk [GZ99] made (essentially333Actually, they worked with normalized adjacency matrices rather than unnormalized ones; the definitions are equivalent in the case of regular graphs. They also only defined one-sided Ramanujancy.) the following definition: Let be an infinite graph and let . We say that is (one-sided) -Ramanujan if . The notions of two-sided -Ramanujan and bipartite -Ramanujan (for bipartite ) are defined analogously. Note that here the notion of Ramanujancy is tied to the infinite covering graph ; see Clark [Cla07, Sec. 5] for further advocacy of this viewpoint (albeit only for the case that is a tree). Starting with Greenberg and Lubotzky [Lub94], a number of authors defined a fixed finite (irregular) graph  to be “Ramanujan” if , where denotes the universal cover tree of  [Ter10]. However Footnote 3 takes a more general approach, allowing us to ask the usual Ramanujan question: Given an infinite , are there infinitely many -Ramanujan graphs?

Given , an obvious necessary condition for a positive answer to this question is ; one should at least have the existence of infinitely many finite graphs covered by . However this is known to be an insufficient condition, even when is a tree: (Lubotzky–Nagnibeda [LN98].) There exists an infinite tree  such that is infinite but contains no -Ramanujan graphs. Based on this result, Clark [Cla06] proposed the following definition and question (though he only discussed the case of being a tree): An infinite graph  is said to be Ramanujan if contains infinitely many -Ramanujan graphs. It is said to be weakly Ramanujan if contains at least one -Ramanujan graph. (One can apply here the usual additional adjectives two-sided/bipartite.) If is weakly Ramanujan, must it be Ramanujan? Clark [Cla06] also made the following philosophical point. The proof of the Lubotzky–Nagnibeda Theorem only explicitly establishes that there is an infinite tree  with infinite but no having . But as long as we’re excluding , why not also exclude , or constantly many exceptional eigenvalues? Clark suggested the following definition: An infinite graph  is said to be -quasi-Ramanujan () if there are infinitely many with . Clark observed that a positive answer to the following question (weaker than Section 2.2) is consistent with Greenberg’s Section 2.1: If is weakly Ramanujan, must it be -quasi-Ramanujan for some ?

Later, Clark [Cla07] directly conjectured the following related statement: (Clark.) For every finite , there are infinitely many lifts of  such that every “new” eigenvalue has . (The notion of an -lift of a graph will be reviewed in Section 3.3.) This effectively generalizes an earlier well known conjecture by Bilu and Linial [BL06]: (Bilu–Linial.) For every regular finite , there is a -lift of  such that every “new” eigenvalue has . As a consequence, one can obtain a tower of infinitely many such lifts of . (Actually, in the above, Bilu–Linial and Clark had the stronger conjecture .) As pointed out in [Cla07], one can easily check that the complete bipartite graph is -Ramanujan, and hence a positive resolution of Section 2.2 would show not only that is quasi-Ramanujan, but that there are in fact infinitely many -Ramanujan graphs.

Finally, Mohar [Moh10] studied these questions in the case of multi-partite trees. Suppose one has a -partite finite graph where every vertex in the th part has exactly neighbors in the th part. Call the degree matrix of such a graph. There are very simple conditions under which a matrix is the degree-matrix of some -partite graph, and in this case, there is a unique infinite tree with as its degree matrix. As an example, the )-biregular infinite tree corresponds to

(1)

Mohar conjectured that for all degree matrices , the answer to Section 2.2 is positive for . In fact he made a slightly more refined conjecture. Given , he defined . (As an example, for the in Equation 1.) He then conjectured: (Mohar.) For a degree matrix :

  1. [Moh10, Conj. 4.2] If is weakly Ramanujan, then it is -quasi-Ramanujan.

  2. [Moh10, Conj. 4.4] If , then is Ramanujan.

Mohar mentions that Item 1 might be true simply as the statement “ is -quasi-Ramanujan for every ”, but was unwilling to explicitly conjecture this.

2.3 Interlacing polynomials

Several of the conjectures mentioned in the previous section were proven by Marcus, Spielman, and Srivastava using the method of “interlacing polynomials” [MSS15b, MSS15c, MSS17, MSS15d]. Particularly, in [MSS15b] they proved Clark’s Section 2.2 and the Bilu–Linial Conjecture (for one-sided/bipartite Ramanujancy). One can also show their work implies Item 1 of Mohar’s Equation 1.

Their first work [MSS15b] in particular shows that for all degree , if one starts with the complete graph or the complete bipartite graph , one can perform a sequence of -lifts, obtaining larger and larger finite graphs each of which has ; i.e., each is one-sided/bipartite -Ramanujan.

In a subsequent work [MSS15d], they showed the existence of -regular bipartite Ramanujan on vertices for each and each , as well as -regular one-sided Ramanujan graphs on vertices for each . The bipartite graphs in this case are -lifts of the (non-simple) -vertex graph with edges, which one might denote by . However, Marcus–Spielman–Srivastava don’t really analyze them as lifts of per se. Rather, they analyze them as sums of random (bipartition-respecting) permutations of , where denotes the perfect matching on vertices. In general, [MSS15d] can be thought of as analyzing the eigenvalues of sums of random permutations of a few large graphs (each of which itself might be the union of many disjoint copies of a fixed small graph). As such, as we will discuss in Appendix B, its techniques can be used to construct infinitely many -Ramanujan graphs for vertex-transitive free product graphs .

Finally, in a followup work, Hall, Puder, and Sawin [HPS18] reinterpret [MSS15d] from the random lift perspective, and generalize it to work for -lifts of any base graph. Specifically, they prove Clark’s Section 2.2 in a strong way: (Hall–Puder–Sawin.) Let be a connected finite graph without loops (but possibly with parallel edges). Then for every , there is an -lift of such that every new eigenvalue has .

2.4 Our work and technical overview

Suppose is an additive product graph such that . Then it is -quasi-Ramanujan for some . (Recall here that is the common spectral radius of all .)

In order to show that all additive product graphs are -quasi-Ramanujan for some , we show the existence of an infinite family of finite quotients of , which each have at most eigenvalues that exceed . To this end, we start with a base graph that is a quotient of and show that for each there is an -lift of that is (i) a quotient of , and (ii) at most eigenvalues of exceed . To show the existence of an appropriate lift, we pick a uniformly random lift from a restricted class of lifts called additive lifts

that satisfy (i), and show that such a random lift satisfies (ii) with positive probability.

We achieve this with the method of interlacing polynomials, which one should think of as the “polynomial probabilistic method”, introduced in [MSS15b, MSS15c]

. Let’s start with a very simple true statement ‘for any random variable

, there is a positive probability that . For a polynomial valued random variable , we could attempt to make the statement ‘there is positive probability that ’ where is the polynomial obtained from coefficient-wise expectations of . While this statement is not true in general, it is when is drawn from a well structured family of polynomials known as an ‘interlacing family’, described in Section 5. We take to be the polynomial whose roots are the new eigenvalues introduced by the random lift, and a key fact we use to get a handle on this polynomial is that it can be seen as the characteristic polynomial of the matrix one obtains by replacing every edge in with the standard representation of the permutation labeling it. This helps us prove that is indeed drawn from an interlacing family in Section 6 and establish bounds on the roots of . We begin by studying the case , where is always the characteristic polynomial of a signing of and , which we call the additive characteristic polynomial, generalizes the well-known matching polynomial and equals it whenever is a tree. We show that the roots of the additive characteristic polynomial lie in

by proving a generalization of Godsil’s result that the root moments of the matching polynomial of a graph count the number of treelike walks in the graph. In particular, we prove that the

-th root moment of the additive characteristic polynomial counts the number of length- ‘freelike walks’ in , which are defined in Section Section 4.1 and obtain the required root bound by upper bounding the number of freelike walks. Our final ingredient is showing that the same root bounds apply to for general , which we show follows from the case when in Section Section 7 using representation theoretic machinery.

The main avenue where our techniques differ from those in previous work (i.e. [MSS15b] and [HPS18]) are in how the root bounds are proven on the expected characteristic polynomial. The proof that the root moments of the matching polynomial count closed treelike walks from [God81] exploits combinatorial structure of the matching polynomial that is not shared by the additive characteristic polynomial. Instead, we prove the analogous statement to Godsil’s result that we need by relating certain combinatorial objects resembling matchings with a particular kind of walk using Newton’s identities and Viennot’s theory of heaps.

3 Additive Products and Additive Lifts

3.1 Elementary definitions

Let be an matrix with entries in a commutative ring. We identify  with a directed, weighted graph on vertex set  (with self-loops allowed, but no parallel arcs); arc is present if and only if . In the general case, the entries are taken to be distinct formal variables. In the unweighted case, each entry is either  or ; in this case,  is the adjacency matrix of the underlying directed graph. The plain case is defined to be when  is unweighted, symmetric, and with diagonal entries ; in this case, is the adjacency matrix of a simple undirected graph (with each undirected edge considered to be two opposing directed arcs).

Throughout this work we will consider finite sequences of matrices over the same set of vertices ; we call each an atom, and the index  its color. We use the terms general / unweighted / plain whenever all ’s have the associated property; we will also use the term monochromatic when . When the ’s are thought of as graphs, we call the associated sum graph; note that even if all the ’s are unweighted, may not be (it may have parallel edges). In the plain case, we use the notation for and to denote an edge that occurs in . A common sum graph case will be when  is a simple undirected graph with  (undirected) edges, and are the associated single-edge graphs on ’s vertex set ; we call the edge atoms for . Note that this is an instance of the plain case.

3.2 The additive product

In this section we introduce the definition of the additive product of atoms. This is a “quasi-transitive” infinite graph, meaning one whose automorphism group has only finitely many orbits. For simplicity, we work in the plain, connected case.

Let be plain atoms on common vertex set . Assume that the sum graph is connected; letting denote with isolated vertices removed, we also assume that each is nonempty and connected. We now define the (typically infinite) additive product graph where and are constructed as follows.

Let be a fixed vertex in ; let be the set of strings of the form for such that:

  1. [label=()]

  2. each is in and each is in ,

  3. for all ,

  4. and are both in for all ;

and, let be the set of edges on vertex set such that for each string ,

  1. [label=()]

  2. we let be in if is an edge in ,

  3. we let be in if is an edge in , and

  4. we let be in if is an edge in .

Two examples are given in Figure 3.

Figure 3: Two examples of the additive product; of course, only a part of each infinite graph can be shown.

is well defined up to graph isomorphism, independent of choice of .

Proof.

Let be the additive product graph generated by selecting to be some and let be the additive product graph generated by selecting to be . To establish the proposition, we will show that and are isomorphic. First define

Let be a string such that and are both in (identifying with and with ). We claim that that maps to is an isomorphism. Define as the string obtained by reversing , i.e., define as . To see is a bijection, consider the map that maps to . It can be verified that is the inverse of , and thus is bijective. It can also be verified that if and in share an edge, then so do and . ∎

covers .

Proof.

Define . For any edge in , without loss of generality assume that the string corresponding to is at least as long as the string corresponding to . And now define where is the last color that appears in . It can be verified that is a valid covering map. ∎

Now, we will go over some common infinite graphs and see how they are realized as additive products.

When is a connected -edge graph with edge atoms , the additive product coincides with the universal cover tree of .

Proof.

Indeed, the additive product of edge atoms is a tree, which by Section 3.2 covers . It coincides with the universal cover tree since all trees that cover are isomorphic. ∎

When each atom is a (nonempty) vertex-transitive graph on vertex set , the additive product coincides with the free product , as defined for vertex-transitive graphs by Znoĭko [Zno75], and for general rooted graphs by Quenell [Que94].

In fact, given vertex-transitive graphs on vertex sets of possibly different sizes (e.g., Cayley graphs of finite groups), we can also realize their free product as an additive product, as follows. Let . Consider the following atom graphs on vertices: For each and let be a copy of placed on vertices . Then the graph

is isomorphic to the free product . Figure 4 illustrates Section 3.2 in the case of the free product (the Cayley graph of the modular group, the sixth graph in Figure 1).

Figure 4: Realizing as an additive product of five atoms.

All of the graphs in Figure 1 are additive products. The first and last are from Figure 3. The second through fourth (namely, , , and the biregular ) are all universal cover trees, and hence additive products by Section 3.2. The fifth and seventh are additive products as from Section 3.2; they are and , respectively. Finally, the sixth graph is an additive product as illustrated in Figure 4.

Let be the second additive product graph in Figure 3. Our techniques show the existence of an infinite family of -Ramanujan graphs — is a notable example of a graph that is neither a tree nor a free product of vertex transitive graphs that is also an additive product.

3.3 Lifts and balanced lifts

In this section, a graph will mean a (possibly infinite) undirected graph, with parallel edges allowed, but loops disallowed. Thus should be thought of as a multiset (with its elements being sets of cardinality ). Given an (undirected) graph , its directed version is the directed graph , where the multiset is formed replacing each edge with a corresponding dart — i.e., pair of directed edges , . Given one edge  in such a dart, we write for the other edge. A warning: if has copies of an edge , then will contain pairs , , and the notation refers to a fixed perfect matching on those pairs.

Given an -vertex graph , we identify the vertices with an orthonormal basis of

; the vector for vertex 

is denoted .444We are using the Dirac bra-ket notation, in which denotes a column vector and denotes its conjugate-transpose . A directed edge may be associated with the matrix . The adjacency matrix of is the Hermitian matrix defined by

Let be a graph and let be a labeling of directed edges by permutations of  satisfying . The associated -lift graph is defined as follows. The lifted vertex set is ; we may sometimes identify these vertices with vectors . The lifted oriented edge set consists of a dart for each dart and each . Given an -vertex graph and a we introduce the -extended adjacency matrix , a Hermitian matrix in defined by

where denotes the identity operator. When this is the usual adjacency matrix. In general, may be thought of as the adjacency matrix of the trivial -lift of , the one where all directed edges are labeled by the identity permutation. This graph consists of disjoint copies of .

Given a graph and a group , a -potential is simply an element  of the direct product group . We think of as assigning a group element, written or , to each vertex of . We will be concerned almost exclusively with the case , the symmetric group.

Let denote the -dimensional unitary matrices. In this work we will assume that all group representations are unitary. Recall that if is a group with -dimensional representation , and is a set of cardinality , then the associated

outer tensor product representation

of the group is defined by

Given an -vertex graph , a -potential , and a -dimensional unitary representation , we introduce the notation for the -dimensional -lifted adjacency matrix

In the setting of Section 3.3, consider the directed edge-labeling defined by . Then is the matrix

This matrix was introduced by Hall, Puder, and Sawin [HPS18] under the notation . In fact, they studied such matrices for general edge-labelings satisfying , not just the so-called balanced ones arising as from a potential . We will recover their level of generality shortly, when we consider sum graphs. For many representations — e.g., the standard representation of the symmetric group — it is not natural to pick one particular unitary representation among all the isomorphic ones. However if is isomorphic to  via the unitary , then is conjugate to  via the unitary , and the same is true of and . Hence these two matrices have the same spectrum and characteristic polynomial, which is what we mainly care to study anyway.

Another representation of is the -dimensional sign representation, . When , the and representations coincide, and we obtain the well-known correspondence between -lifts and edge-signings of the adjacency matrix of .

Suppose is a graph. When and is the usual permutation representation, the matrix is the adjacency matrix of a certain -lift of , which we call a balanced -lift. We write  for this lifted graph, the one obtained from the edge-labeling discussed in Section 3.3. The vertex set of is , and the directed edge set is formed as follows: for each dart and each , we include dart if and only if . We remark that a balanced -lift conists of disjoint copies of . In general, not all lifts of  are balanced lifts. This is true, though, if is an acyclic graph; indeed, it’s not hard to check that for each connected component of an acyclic graph, the balanced lifts are in -to- correspondence with general lifts. As mentioned earlier, we will recover the full generality of lifts shortly when we consider sum graphs.

3.4 Additive lifts

When is a sum graph on vertex set , is a sequence of -potentials , and is a representation, we introduce the notation

In case and , this is the adjacency matrix of a new sum graph

that we call the additive -lift of  by . Here a balanced -lift is performed on each atom, and the results are summed together. Suppose we regard an ordinary graph as a sum graph where the atoms are single edges, as in Section 3.1. In this case, every -lift of is a balanced -lift (indeed, in different ways, as noted in Section 3.3). Thus the additive -lifts of  — when viewed again as ordinary graphs — recover all ordinary -lifts of .

As is well known, the spectrum of — i.e., of — is the multiset-union of the “old” spectrum of (of cardinality ), as well as “new” spectrum (of cardinality ). As observed in [HPS18], this “new” spectrum is precisely the spectrum of , where is the standard representation of .

The following fact is important for the proof of our main theorem. If is connected, then it is a quotient of .

Proof.

Recall that can be written as . Expressing each as a sum graph of disjoint copies of , we obtain an expression of as a sum graph of atoms.

We show that the additive product is isomorphic to , and our proposition then follows from Section 3.2.

We note the following:

  1. For any in such that