1 Introduction
Let be an infinite graph, such as one of the eight partially sketched below.
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 bottomleft 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 topright 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 bottomleft 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 topleft 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 vertextransitive, 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 highdimensional expanders (see the survey [Lub17]), which are expander graphs with certain constrained local structure — for example a 2dimensional 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 edgesigned) graphs. Besides the “cut” predicate (modeled by a single edge), examples include: the (NotAllEquals) 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/SDPbased 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 selfloops 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.
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 graphs^{1}^{1}1Up 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–Boppanatype 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 Serrestyle 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 ) (onesided) Ramanujan if , and twosided Ramanujan if in addition . If is bipartite and (onesided) Ramanujan, we will call it bipartite Ramanujan. (A bipartite graph cannot be twosided Ramanujan, as will always be .) The Lubotzky–Phillips–Sarnak–Margulis constructions give infinitely many regular twosided 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]).^{2}^{2}2In fact, they defined a notion of Ramanujancy for biregular graphs that has a flavor even stronger than that of twosidedness; 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 (essentially^{3}^{3}3Actually, they worked with normalized adjacency matrices rather than unnormalized ones; the definitions are equivalent in the case of regular graphs. They also only defined onesided Ramanujancy.) the following definition: Let be an infinite graph and let . We say that is (onesided) Ramanujan if . The notions of twosided 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 twosided/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 quasiRamanujan () 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 quasiRamanujan 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 quasiRamanujan, but that there are in fact infinitely many Ramanujan graphs.
Finally, Mohar [Moh10] studied these questions in the case of multipartite 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 degreematrix 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 :

[Moh10, Conj. 4.2] If is weakly Ramanujan, then it is quasiRamanujan.

[Moh10, Conj. 4.4] If , then is Ramanujan.
Mohar mentions that Item 1 might be true simply as the statement “ is quasiRamanujan 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 onesided/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 onesided/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 onesided Ramanujan graphs on vertices for each . The bipartite graphs in this case are lifts of the (nonsimple) 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 (bipartitionrespecting) 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 vertextransitive 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 quasiRamanujan for some . (Recall here that is the common spectral radius of all .)
In order to show that all additive product graphs are quasiRamanujan 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 coefficientwise 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 wellknown matching polynomial and equals it whenever is a tree. We show that the roots of the additive characteristic polynomial lie inby 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 selfloops 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 singleedge 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 “quasitransitive” 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:

[label=()]

each is in and each is in ,

for all ,

and are both in for all ;
and, let be the set of edges on vertex set such that for each string ,

[label=()]

we let be in if is an edge in ,

we let be in if is an edge in , and

we let be in if is an edge in .
Two examples are given in Figure 3.
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) vertextransitive graph on vertex set , the additive product coincides with the free product , as defined for vertextransitive graphs by Znoĭko [Zno75], and for general rooted graphs by Quenell [Que94].
In fact, given vertextransitive 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).
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 .^{4}^{4}4We are using the Dirac braket notation, in which denotes a column vector and denotes its conjugatetranspose . A directed edge may be associated with the matrix . The adjacency matrix of is the Hermitian matrix defined byLet 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 byGiven 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 edgelabeling 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 edgelabelings satisfying , not just the socalled 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 wellknown correspondence between lifts and edgesignings 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 edgelabeling 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 multisetunion 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:

For any in such that
Comments
There are no comments yet.