Open problems in the spectral theory of signed graphs

Signed graphs are graphs whose edges get a sign +1 or -1 (the signature). Signed graphs can be studied by means of graph matrices extended to signed graphs in a natural way. Recently, the spectra of signed graphs have attracted much attention from graph spectra specialists. One motivation is that the spectral theory of signed graphs elegantly generalizes the spectral theories of unsigned graphs. On the other hand, unsigned graphs do not disappear completely, since their role can be taken by the special case of balanced signed graphs. Therefore, spectral problems defined and studied for unsigned graphs can be considered in terms of signed graphs, and sometimes such generalization shows nice properties which cannot be appreciated in terms of (unsigned) graphs. Here, we survey some general results on the adjacency spectra of signed graphs, and we consider some spectral problems which are inspired from the spectral theory of (unsigned) graphs.

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1 Introduction

A signed graph is a graph , with vertex set and edge set , together with a function assigning a positive or negative sign to each edge. The (unsigned) graph is said to be the underlying graph of , while the function is called the signature of . Edge signs are usually interpreted as . In this way, the adjacency matrix of is naturally defined following that of unsigned graphs, that is by putting or whenever the corresponding edge is either positive or negative, respectively. One could think about signed graphs as weighted graphs with edges of weights in , however the two theories are very different. In fact, in signed graphs the product of signs has a prominent role, while in weighted graphs it is the sum of weights that is relevant. A walk is positive or negative if the product of corresponding weights is positive or negative, respectively. Since cycles are special kinds of walks, this definition applies to them as well and we have the notions of positive and negative cycles.

Many familiar notions related to unsigned graphs directly extend to signed graphs. For example, the degree of a vertex in is simply its degree in . A vertex of degree one is said to be a pendant vertex. The diameter of is the diameter of its underlying graph , namely, the maximum distance between any two vertices in . Some other definitions depend on the signature, for example, the positive (resp., negative) degree of a vertex is the number of positive (negative) edges incident to the vertex, or the already mentioned sign of a walk or cycle. A signed graph is balanced if all its cycles are positive, otherwise it is unbalanced. Unsigned graphs are treated as (balanced) signed graphs where all edges get a positive sign, that is, the all-positive signature.

An important feature of signed graphs is the concept of switching the signature. Given a signed graph and a subset , let be the signed graph obtained from by reversing the signs of the edges in the cut , namely for any edge between and , and otherwise. The signed graph is said to be (switching) equivalent to and to , and we write or . It is not difficult to see that each cycle in maintains its sign after a switching. Hence, and have the same positive and negative cycles. Therefore, the signature is determined up to equivalence by the set of positive cycles (see zas2 ). Signatures equivalent to the all-positive one (the edges get just the positive sign) lead to balanced signed graphs: all cycles are positive. By we mean that the signature is equivalent to the all-positive signature, and the corresponding signed graph is equivalent to its underlying graph. Hence, all signed trees on the same underlying graph are switching equivalent to the all-positive signature. In fact, signs are only relevant in cycles, while the edge signs of bridges are irrelevant.

Note that (unsigned) graph invariants are preserved under switching, but also by vertex permutation, so we can consider the isomorphism class of the underlying graph. If we combine switching equivalence and vertex permutation, we have the more general concept of switching isomorphism of signed graphs. For basic results in the theory of signed graphs, the reader is referred to Zaslavsky zas2 (see also the dynamic survey dynamic1 ).

We next consider matrices associated to signed graphs. For a signed graph and a graph matrix , the -polynomial is . The spectrum of is called the -spectrum of the signed graph . Usually, is the adjacency matrix or the Laplacian matrix , but in the literature one can find their normalized variants or other matrices. In the remainder, we shall mostly restrict to being the adjacency matrix . The adjacency matrix is the symmetric -matrix such that whenever the vertices and are adjacent, and otherwise. As with unsigned graphs, the Laplacian matrix is defined as , where is the diagonal matrix of vertices degrees (of the underlying graph ). In the sequel we will mostly restrict to the adjacency matrix.

Switching has a matrix counterpart. In fact, let and be two switching equivalent graphs. Consider the matrix such that

The matrix is the switching matrix. It is easy to check that

Hence, signed graphs from the same switching class share similar graph matrices by means of signature matrices (signature similarity). If we also allow permutation of vertices, we have signed permutation matrices, and we can speak of (switching) isomorphic signed graphs. Switching isomorphic signed graphs are cospectral, and their matrices are signed-permutationally similar. From the eigenspace viewpoint, the eigenvector components are also switched in signs and permuted. Evidently, for each eigenvector, there exists a suitable switching such that all components become nonnegative.

In the sequel, let denote the eigenvalues of the adjacency matrix of the signed graph of order ; they are all real since is a real symmetric matrix. The largest eigenvalue is sometimes called the index of . If contains at least one edge, then since the sum of the eigenvalues is . Note that in general, the index does not equal the spectral radius because the Perron–Frobenius Theorem is valid only for the all-positive signature (and those equivalent to it). For example, an all signing (all-negative signature) of the complete graph on vertices will have eigenvalues and .

We would like to end this introduction by mentioning what may be the first paper on signed graph spectra zas2 . In that paper, Zaslavsky showed that appears as an -eigenvalue in connected signed graphs if and only if the signature is equivalent to the all-positive one, that is, is a balanced signed graph.

For notation not given here and basic results on graph spectra, the reader is referred to CvDS ; cve , for some basic results on the spectra of signed graphs, to zas1 , and for some applications of spectra of signed graphs, to Gallier .

In Section 2, we survey some important results on graph spectra which are valid in terms of the spectra of signed graphs. In Section 3 we collect some open problems and conjectures which are open at the writing of this note.

2 What do we lose with signed edges?

From the matrix viewpoint, when we deal with signed graphs we have symmetric -matrices instead of just symmetric -matrices. Clearly, the results coming from the theory of nonnegative matrices can not be applied directly to signed graphs. Perhaps the most important result that no longer holds for adjacency matrices of signed graphs is the Perron–Frobenius theorem. We saw one instance in the introduction and we will see some other consequences of the absence of Perron–Frobenius in the next section. Also, the loss of non-negativity has other consequences related to counting walks and the diameter of the graph (Theorem 3.10). On the other hand, all results based on the symmetry of the matrix will be still valid in the context of signed graphs with suitable modifications. In this section, we briefly describe how some well-known results are (possibly) changed when dealing with matrices of signed graphs.

We start with the famous Coefficient Theorem, also known as Sachs Formula. This formula, perhaps better than others, describes the connection between the eigenvalues and the combinatorial structure of the signed graph. It was given for unsigned graphs in the 1960s independently by several researchers (with different notation), but possibly first stated by Sachs (cf. (CvDS, , Theorem 1.2) and the subsequent remark). The signed-graph variant can be easily given as follows. An elementary figure is the graph or (). A basic figure (or linear subgraph, or sesquilinear subgraph) is the disjoint union of elementary figures. If is a basic figure, then denote by the class of cycles in , with , and by the number of components of and define . Let be the set of basic figures on vertices.

Theorem 2.1 (Coefficient Theorem).

Let be a signed graph and let be its adjacency characteristic polynomial. Then

Another important connection between the eigenvalues and the combinatorial structure of a signed graph is given by the forthcoming theorem. If we consider unsigned graphs, it is well known that the

-th spectral moment gives the number of closed walks of length

(cf. (cve, , Theorem 3.1.1)). Zaslavsky zas1 observed that a signed variant holds for signed graphs as well, and from his observation we can give the subsequent result.

Theorem 2.2 (Spectral Moments).

Let be a signed graph with eigenvalues . If denotes the difference between the number of positive and negative closed walks of length , then

Next, we recall another famous result for the spectra of graphs, that is, the Cauchy Interlacing Theorem. Its general form holds for principal submatrices of real symmetric matrices (see (cve, , Theorem 1.3.11)). It is valid in signed graphs without any modification to the formula. For a signed graph and a subset of vertices , then is the signed graph obtained from by deleting the vertices in and the edges incident to them. For , we also write instead of . Similar notation applies when deleting subsets of edges.

Theorem 2.3 (Interlacing Theorem for Signed Graphs).

Let be a signed graph. For any vertex of ,

In the context of subgraphs, there is another famous result which is valid in the theory of signed graphs. In fact, it is possible to give the characteristic polynomial as a linear combination of vertex- or edge-deleted subgraphs. Such formulas are known as Schwenk’s Formulas (cf. (cve, , Theorem 2.3.4), see also belisi ). As above, () stands for the signed graph obtained from in which the vertex (resp., edge ) is deleted. Also, to make the formulas consistent, we set .

Theorem 2.4 (Schwenk’s Formulas).

Let be a signed graph and (resp., ) one of its vertices (resp., edges). Then

where denotes the set of cycles passing through .

Finally, a natural question is the following: if we fix the underlying graph, how much can the eigenvalues change when changing the signature? Given a graph with cyclomatic number , then there are at most nonequivalent signatures as for each independent cycle one can assign either a positive or a negative sign. However, among the signatures, some of them might lead to switching isomorphic graphs, as we see later. In general, the eigenvalues coming from each signature cannot exceed in modulus the spectral radius of the underlying graph, as is shown in the last theorem of this section.

Theorem 2.5 (Eigenvalue Spread).

For a signed graph , let be its spectral radius. Then .

Proof.

Clearly, equals or . Let be the adjacency matrix of , and be the adjacency matrix of

. For a vector

, let .

If is a unit eigenvector corresponding to , by the Rayleigh quotient we get

Similarly, if is a unit eigenvector corresponding to the least eigenvalue , by the Rayleigh quotient we get

By gluing together the two inequalities, we get the assertion. ∎

It is evident from the preceding results that the spectral theory of signed graphs well encapsulates and extends the spectral theory of unsigned graphs. Perhaps, we can say that adding signs to the edges just gives more variety to the spectral theory of graphs. This fact was already observed with the Laplacian of signed graphs, which nicely generalizes the results coming from the Laplacian and signless Laplacian theories of unsigned graphs. It is worth mentioning that thanks to the spectral theory it was possible to give matrix-wise definitions of the signed graph products gehaza , line graphs besi ; zas1 and subdivision graphs besi .

3 Some open problems and conjectures

In this section we consider some open problems and conjectures which are inspired from the corresponding results in the spectral theory of unsigned graphs. We begin with the intriguing concept of “sign-symmetric graph” which is a natural signed generalization of the concept of bipartite graph.

3.1 Symmetric spectrum and sign-symmetric graphs

One of the most celebrated results in the adjacency spectral theory of (unsigned) graphs is the following.

Theorem 3.1.
  1. A graph is bipartite if and only if its adjacency spectrum is symmetric with respect to the origin.

  2. A connected graph is bipartite if and only if its smallest eigenvalue equals the negative of its spectral radius.

For the first part, one does not need Perron–Frobenius theorem. To the best of our knowledge, Perron–Frobenius is crucial for the second part (see (BH, , Section 3.4) or (GR, , Section 8.8) or (VLW, , Ch. 31)).

On the other hand, in the larger context of signed graphs the symmetry of the spectrum is not a privilege of bipartite and balanced graphs. A signed graph is said to be sign-symmetric if is switching isomorphic to its negation, that is,

. It is not difficult to observe that the signature-reversal changes the sign of odd cycles but leaves unaffected the sign of even cycles. Since bipartite (unsigned) graphs are odd-cycle free, it happens that bipartite graphs are a special case of sign-symmetric signed graphs, or better to say, if a signed graph

has a bipartite underlying graph , then and are switching equivalent. In Fig. 1 we depict an example of a sign-symmetric graph. Here and in the remaining pictures as well negative edges are represented by heavy lines and positive edges by thin lines.

Figure 1: A sign-symmetric signed graph.

If is switching isomorphic to , then and are similar and we immediately get:

Theorem 3.2.

Let be a sign-symmetric graph. Then its adjacency spectrum is symmetric with respect to the origin.

The converse of the above theorem is not true, and counterexamples arise from the theory of Seidel matrices. The Seidel matrix of a (simple and unsigned) graph is , so that adjacent vertices get the value and non-adjacent vertices the value . Hence, the Seidel matrix of an unsigned graph can be interpreted as the adjacency matrix of a signed complete graph. The signature similarity becomes the famous Seidel switching. The graph in Fig. 2 belongs to a triplet of simple graphs on 8 vertices sharing the same symmetric Seidel spectrum but not being pairwise (Seidel-)switching isomorphic. In (ssc, , p. 253), they are denoted as , its complement and (note, and its complement are Seidel switching isomorphic). In fact, and its complement are cospectral but not Seidel switching isomorphic. In terms of signed graphs, the signed graph whose adjacency matrix is has symmetric spectrum but it is not sign-symmetric.

Figure 2: The graph .

Note that the disjoint union of sign-symmetric graphs is again sign-symmetric. Since the above counterexamples involve Seidel matrices which are the same as signed complete graphs, the following is a natural question.

Problem 3.3.

Are there non-complete connected signed graphs whose spectrum is symmetric with respect to the origin but they are not sign-symmetric?

Observe that signed graphs with symmetric spectrum have odd-indexed coefficients of the characteristic polynomial equal to zero and all spectral moments of odd order are also zero. A simple application of Theorems 2.1 and 2.2 for or , respectively, leads to equal numbers of positive and negative triangles in the graph. When we consider or , we cannot say that the numbers of positive and negative pentagons are the same. The following corollary is an obvious consequence of the latter discussion (cf. also (ssc, , Theorem 1)).

Corollary 3.4.

A signed graph containing an odd number of triangles cannot be sign-symmetric.

Remark 3.5.

As we mentioned in Section 2, a signed graph with cyclomatic number has exactly not equivalent signatures (see also mal ). On the other hand, the symmetries, if any, in the structure of the underlying graph can make several of those signatures lead to isomorphic signed graphs.

3.2 Signed graphs with few eigenvalues

There is a well-known relation between the diameter and the number of distinct eigenvalues of an unsigned graph (cf. (cve, , Theorem 3.3.5)). In fact, the number of distinct eigenvalues cannot be less than the diameter plus 1. With signed graphs, the usual proof based on the minimal polynomial does not hold anymore. Indeed, the result is not true with signed graphs. As we can see later, it is possible to build signed graphs of any diameter having exactly two distinct eigenvalues.

For unsigned graphs, the identification of graphs with a small number of eigenvalues is a well-known problem. The unique connected graph having just two distinct eigenvalues is the complete graph . If a graph is connected and regular, then it has three distinct eigenvalues if and only if it is strongly regular (see (cve, , Theorem 3.6.4)). At the 1995 British Combinatorial Conference, Haemers posed the problem of finding connected graphs with three eigenvalues which are neither strongly regular nor complete bipartite. Answering Haemers’ question, Van Dam vanDam3 and Muzychuk and Klin Muz described some constructions of such graphs. Other constructions were found by De Caen, van Dam and Spence CaDaSp who also noticed that the first infinite family nonregular graphs with three eigenvalues already appeared in the work of Bridges and Mena BrMe . The literature on this topic contains many interesting results and open problems. For example, the answer to the following intriguing problem posed by De Caen (see (vanDamDom, , Problem 9)) is still unknown.

Problem 3.6.

Does a graph with three distinct eigenvalues have at most three distinct degrees?

Recent progress was made recently by Van Dam, Koolen and Jia DamKooJia who constructed connected graphs with four or five distinct eigenvalues and arbitrarily many distinct degrees. These authors posed the following bipartite version of De Caen’s problem above.

Problem 3.7.

Are there connected bipartite graphs with four distinct eigenvalues and more than four distinct valencies ?

For signed graphs there are also some results. In 2007, McKee and Smyth smyth considered symmetric integral matrices whose spectral radius does not exceed 2. In their nice paper, they characterized all such matrices and they further gave a combinatorial interpretation in terms of signed graphs. They defined a signed graph to be cyclotomic if its spectrum is in the interval . The maximal cyclotomic signed graphs have exactly two distinct eigenvalues. The graphs appearing in the following theorem are depicted in Fig. 3.

Theorem 3.8.

Every maximal connected cyclotomic signed graph is switching equivalent to one of the following:

  • For some , the -vertex toroidal tessellation ;

  • The -vertex signed graph ;

  • The -vertex signed hypercube .

Further, every connected cyclotomic signed graph is contained in a maximal one.

Figure 3: Maximal cyclotomic signed graphs.

It is not difficult to check that all maximal cyclotomic graphs are sign-symmetric. Note that for even has a bipartite underlying graph, while for odd has not bipartite underlying graph but it is sign-symmetric, as well. The characteristic polynomial to is , so is an example of a signed graph with two distinct eigenvalues and diameter .

Problem 3.9 (Signed graphs with exactly 2 distinct eigenvalues).

Characterize all connected signed graphs whose spectrum consists of two distinct eigenvalues.

In the above category we find the complete graphs with homogeneous signatures and , the maximal cyclotomic signed graphs , and , and that list is not complete (for example, the unbalanced -cycle and the 3-dimensional cube whose cycles are all negative must be included). There is already some literature on this problem, and we refer the readers to see gha ; Ram . All such graphs have in common the property that positive and negative walks of length greater than or equal to 2 between two different and non-adjacent vertices are equal in number. In this way we can consider a signed variant of the diameter. In a connected signed graph, two vertices are at signed distance if they are at distance and the difference between the numbers of positive and negative walks of length among them is nonzero, otherwise the signed distance is set to . The signed diameter of , denoted by , is the largest signed distance in . Recall that the -entry of equals the difference between the numbers of positive and negative walks of length among the vertices indexed by and . Then we have the following result (cf. (cve, , Theorem 3.3.5)):

Theorem 3.10.

Let be a connected signed graph with distinct eigenvalues. Then .

Proof.

Assume the contrary, so that has vertices, say and , at signed distance . The adjacency matrix of has minimal polynomial of degree , and so we may write . This yields the required contradiction because the -entry on the right is zero, while the -entry on the left is non-zero. ∎

Recently, Huang Huang constructed a signed adjacency matrix of the -dimensional hypercube whose eigenvalues are , each with multiplicity . Using eigenvalue interlacing, Huang proceeds to show that the spectral radius (and therefore, the maximum degree) of any induced subgraph on vertices of the -dimensional hypercube, is at least . This led Huang to a breakthrough proof of the Sensitivity Conjecture from theoretical computer science. We will return to Huang’s construction after Theorem 3.23.

3.3 The largest eigenvalue of signed graphs

In the adjacency spectral theory of unsigned graphs the spectral radius is the largest eigenvalue and it has a prominent role because of its algebraic features, its connections to combinatorial parameters such as the chromatic number, the independence number or the clique number and for its relevance in applications. There is a large literature on this subject, see BrNe ; CveRo ; Hoff1 ; Hoff2 ; Nik1 ; Ste ; WCWF for example.

As already observed, the presence of negative edges leads invalidates of the Perron–Frobenius theorem, and we lose some nice features of the largest eigenvalue:

  • The largest eigenvalue may not be the spectral radius although by possibly changing the signature to its negative, this can be achieved.

  • The largest eigenvalue may not be a simple eigenvalue.

  • Adding edges might reduce the largest eigenvalue.

Therefore one might say that it not relevant to study signed graphs in terms of the magnitude of the spectral radius. In this respect, Theorem 2.3 and Theorem 2.5 are helpful because the spectral radius does not decrease under the addition of vertices (together with some incident edges), and the spectral radius of the underlying graph naturally limits the magnitude of the eigenvalues of the corresponding signed graph. For the same reason, the theory of limit points for the spectral radii of graph sequences studied by Hoffman in Hoff1 ; Hoff2 is still valid in the context of signed graphs.

The Hoffman program is the identification of connected graphs whose spectral radii do not exceed some special limit points established by A.J. Hoffman Hoff2 . The smallest limit point for the spectral radius is (the limit point of the paths of increasing order), so the first step would be to identify all connected signed graphs whose spectral radius does not exceed . The careful reader notices that the latter question has already been completely solved by Theorem 3.8. Therefore, the problem jumps to the next significant limit point, which is , where is the golden mean. This limit point is approached from above (resp., below) by the sequence of positive (resp., negative) cycles with exactly one pendant vertex and increasing girth.

In BrNe ; CvDoGu , the authors identified all connected unsigned graphs whose spectral radius does not exceed . Their structure is fairly simple: they mostly consist of paths with one or two additional pendant vertices. Regarding signed graphs, we expect that the family is quite a bit larger than that of unsigned graphs. A taste of this prediction can be seen by comparing the family of Smith Graphs (the unsigned graphs whose spectral radius is , cf. Fig. 2.4 in CvDS ) with the graphs depicted in Fig. 3. On the other hand, the graphs identified by Cvetković et al. acts as a “skeleton” (that is, appear as subgraphs) of the signed graphs with the same bound on the spectral radius.

Problem 3.11 (Hoffman Program for Signed Graphs).

Characterize all connected signed graphs whose spectral radius does not exceed .

3.4 The smallest eigenvalue of signed graphs

Unsigned graphs with smallest eigenvalue at least have been characterized in a veritable tour de force by several researchers. We mention here Cameron, Goethals, Seidel and Shult CGSS , Bussemaker and Neumaier BN who among other things, determined a complete list of minimal forbidden subgraphs for the class of graphs with smallest eigenvalue at least . A monograph devoted to this topic is cve-2 whose Chapter 1.4 tells the history about the characterization of graphs with smallest eigenvalue at least .

Theorem 3.12.

If is a connected graph with smallest eigenvalue at least , then is a generalized line graph or has at most vertices.

In the case of unsigned graphs, their work was extended, under some minimum degree condition, from to by Hoffman Hoff77 and Woo and Neumaier WN and more recently, to by Koolen, Yang and Yang KYY .

For signed graphs, some of the above results were extended by Vijayakumar V who showed that any connected signed graph with smallest eigenvalue less than has an induced signed subgraph with at most vertices and smallest eigenvalue less than . Chawathe and Vijayakumar ChVi determined all minimal forbidden signed graphs for the class of signed graphs whose smallest eigenvalue is at least . Vijayakumar’s result (V, , Theorem 4.2) was further extended by Koolen, Yang and Yang (KYY, , Theorem 4.2) to signed matrices whose diagonal entries can be or . These authors introduced the notion of -integrable graphs. For an unsigned graph with smallest eigenvalue and adjacency matrix , the matrix is positive semidefinite. For a natural number , is called -integrable if there exists an integer matrix such that . Note that generalized line graphs are exactly the -integrable graphs with smallest eigenvalue at least . In a straightforward way, the notion of s-integrabilty can be extended to signed graphs. Now we can extend Theorem 3.12 to the class of signed graphs with essentially the same proof.

Theorem 3.13.

Let be a connected signed graph with smallest eigenvalue at least . Then is -integrable. Moreover, if has at least vertices, then is -integrable.

As has vectors of (squared) norm , one can take from each pair of such a vector and its negative exactly one to obtain a signed graph on vertices with smallest eigenvalue that is not 1-integrable. Many of these signed graphs are connected.

Koolen, Yang and Yang KYY proved that if a connected unsigned graph has smallest eigenvalue at least and valency large enough, then is -integrable. An interesting direction would be to prove a similar result for signed graphs.

Problem 3.14.

Extend (KYY, , Theorem 1.3) to signed graphs.

An interesting related conjecture was posed by Koolen and Yang KY .

Conjecture 3.15.

There exists a constant such that if is an unsigned graph with smallest eigenvalue at least , then is -integrable.

Koolen, Yang and Yang KYY also introduced -maximal graphs or maximal graphs with smallest eigenvalue . These are connected graphs with smallest eigenvalue at least such any proper connected supergraph has smallest eigenvalue less than . Koolen and Munemasa KM proved that the join between a clique on three vertices and the complement of the McLaughlin graph (see Goethals and Seidel GoeSei or Inoue Inoue for a description) is -maximal.

Problem 3.16.

Construct maximal signed graphs with smallest eigenvalue at least .

Woo and Neumaier WN introduced the notion of Hoffman graphs, which has proved an essential tool in many results involving the smallest eigenvalue of unsigned graphs (see KYY ). Perhaps a theory of signed Hoffman graphs is possible as well.

Problem 3.17.

Extend the theory of Hoffman graphs to signed graphs.

3.5 Signatures minimizing the spectral radius

As observed in Section 2, an unsigned graph with cyclomatic number gives rise to at most switching non-isomorphic signed graphs. In view of Theorem 2.5, we know that, up to switching equivalency, the signature leading to the maximal spectral radius is the all-positive one. A natural question is to identify which signature leads to the minimum spectral radius.

Problem 3.18 (Signature minimizing the spectral radius).

Let be a simple and connected unsigned graph. Determine the signature(s) such that for any signature of , we have .

This problem has important connections and consequences in the theory of expander graphs. Informally, an expander is a sparse and highly connected graph. Given an integer and a real number, a -expander is a connected -regular graph whose (unsigned) eigenvalues (except and possibly if the graph is bipartite) have absolute value at most . It is an important problem in mathematics and computer science to construct, for fixed , infinite families of -expanders for small (see BL ; HLW ; MSS15 for example). From work of Alon-Boppana (see CKNV16 ; HLW ; N ), we know that is the best bound we can hope for and graphs attaining this bound are called Ramanujan graphs.

Bilu and Linial BL proposed the following combinatorial way of constructing infinite families of -regular Ramanujan graphs. A double cover (sometimes called -lift or -cover) of a graph is the (unsigned) graph with vertex set such that is adjacent to for . It is easy to see that if is -regular, then is -regular. A crucial fact is that the spectrum of the unsigned adjacency matrix of is the union of the spectrum of the unsigned adjacency matrix and the spectrum of signed adjacency matrix , where for any edge of and otherwise (see BL for a short proof). Note that this result can be deduced using the method of equitable partitions (see (BH, , Section 2.3)), appears in the mathematical chemistry literature in the work of Fowler Fowler and was extended to other matrices and directed graphs by Butler Butler .

The spectral radius of a signing is the spectral radius of the signed adjacency matrix . Bilu and Linial BL proved the important result

Theorem 3.19 (Bilu-Linial Bl ).

Every connected -regular graph has a signing with spectral radius at most , where is some absolute constant.

and made the following conjecture.

Conjecture 3.20 (Bilu-Linial Bl ).

Every connected -regular graph has a signature with spectral radius at most .

If true, this conjecture would provide a way to construct or show the existence of an infinite family of -regular Ramanujan graphs. One would start with a base graph that is -regular Ramanujan (complete graph or complete bipartite graph for example) and then repeatedly apply the result of the conjecture above. Recently, Marcus, Spielman and Srivastava MSS15 made significant progress towards solving the Bilu-Linial conjecture.

Theorem 3.21.

Let be a connected -regular graph. Then there exists a signature of such that the largest eigenvalue of is at most .

As mentioned before, may have negative entries and one cannot apply the Perron–Frobenius theorem for it. Therefore, the spectral radius of is not always the same as the largest eigenvalue of . In more informal terms, the Bilu-Linial conjecture is about bounding all the eigenvalues of by and while the Marcus-Spielman-Srivastava result shows the existence of a signing where all the eigenvalues of are at most . By taking the negative of the signing guaranteed by Marcus-Spielman-Srivastava, one gets a signed adjacency matrix where all eigenvalues are at least , of course.

There are several interesting ingredients in the Marcus-Spielman-Srivastava result. The first goes back to Godsil and Gutman GG who proved the remarkable result that the average of the characteristic polynomials of the all the signed adjacency matrices of a graph equals the matching polynomial of . This is defined as follows. Define and for , let denote the number of matchings of consisting of exactly edges. The matching polynomial of is defined as

(1)

where is the number of vertices of . Heilmann and Lieb HL proved the following results regarding the matching polynomial of a graph. See Godsil’s book Gac for a nice, self-contained exposition of these results.

Theorem 3.22.

Let be a graph.

  1. Every root of the matching polynomial is real.

  2. If is -regular, then every root of has absolute value at most .

If is a -regular graph, then the average of the characteristic polynomials of its signed adjacency matrices equals its matching polynomial whose roots are in the desired interval . As Marcus-Spielman-Srivastava point out, just because the average of certain polynomials has roots in a certain interval, does not imply that one of the polynomials has roots in that interval. However, in this situation, the characteristic polynomials of the signed adjacency matrices form an interlacing family of polynomials (this is a term coined by Marcus-Spielman-Srivastava in MSS15 ). The theory of such polynomials is developed in MSS15 and it leads to an existence proof that one of the signed adjacency matrices of has the largest eigenvalue at most . As mentioned in MSS15 ,

The difference between our result and the original conjecture is that we do not control the smallest new eigenvalue. This is why we consider bipartite graphs.

Note that the result of Marcus, Spielman and Srivastava MSS15 implies the existence of an infinite family of -regular bipartite Ramanujan graphs, but it does not provide a recipe for constructing such family. As an amusing exercise, we challenge the readers to solve Problem 3.18 by finding a signature of the Petersen graph (try it without reading ZasPet ) or of their favorite graph that minimizes the spectral radius.

A weighing matrix of weight and order is a square matrix with entries satisfying . When , this is the same as a Hadamard matrix and when , this is called a conference matrix. Weighing matrices have been well studied in design and coding theory (see GerSeb for example). Examining the trace of the square of the signed adjacency matrix, Gregory DAG proved the following.

Theorem 3.23.

If is any signature of , then

(2)

where is the average degree of . Equality happens if and only if is -regular and is a symmetric weighing matrix of weight .

This result implies that for any signature with equality if and only if a conference matrix of order exists. By a similar argument, one gets that with equality if and only if there is a Hadamard matrix of order . Note also that when , the graphs attaining equality in the previous result are known from McKee and Smyth’s work smyth (see Theorem 3.8 above). Using McKee and Smyth characterization and the argument below, we can show that the only -regular graph attaining equality in Theorem 3.23 is the -dimensional cube.

Let denote the -dimensional hypercube. Huang Huang constructed a signed adjacency matrix of recursively as follows:

for . It is not too hard to show that for any and thus, attains equality in Theorem 3.23. We remark that Huang’s method can be also used to produce infinite families of regular graphs and signed adjacency matrices attaining equality in Theorem 3.23. If is a -regular graph of order with signed adjacency matrix such that , then define the -regular graph by taking two disjoint copies of and adding a perfect matching between them and a signed adjacency matrix for as . Because , we can get that . Thus, using any -regular graph from McKee and Smyth smyth (see again Theorem 3.8) with a signed adjacency matrix satisfying , one can construct a -regular graph with signed adjacency matrix such that . The following is a natural question.

Problem 3.24.

Are there any other -regular graphs attaining equality in Theorem 3.23 ?

If the regularity assumption on is dropped, Gregory considered a the following variant of Conjecture 3.20.

Conjecture 3.25 (Dag ).

If is the largest vertex degree of a nontrivial graph , then there exists a signature such that .

Gregory came to the above conjecture by observing that in view of Theorem 3.22 the bound in the above conjecture holds for the matching polynomial of and by noticing that

where is the set of subgraphs of consisting of cycles and is the number of cycles of . Since the matching polynomial of is the average of polynomials of signed graphs on , one could expect that there is at least one signature such that does not exceed the spectral radius of . As observed in DAG , for odd unicyclic signed graphs the spectral radius of the matching polynomial is always less than the spectral radius of the corresponding adjacency polynomial, but the conjecture still remains valid. We ask the following question whose affirmative answer would imply Conjecture DAG .

Problem 3.26.

If is the spectral radius of a connected graph , then is there a signature such that ?

In view of the above facts, we expect that the signature minimizing the spectral radius is the one balancing the contributions of cycles so that the resulting polynomial is as close as possible to the matching polynomial. For example, we can have signatures whose corresponding polynomial equals the matching polynomial, as in the following proposition.

Proposition 3.27.

Let be a signed graph consisting of odd cycles of pairwise equal length and opposite signs. Then

Is the signature in Proposition 3.27 the one minimizing the spectral radius? We leave this as an open problem (see also zascycle ).

We conclude this section by observing that for a general graph, it is not known whether Problem 3.18 is NP-hard or not. However, progress is made in CCCKK where the latter mentioned problem is shown to be NP-hard when restricted to arbitrary symmetric matrices. Furthermore, the problems described in this subsection can be considered in terms of the largest eigenvalue , instead of the spectral radius.

3.6 Spectral determination problems for signed graphs

A graph is said to be determined by its (adjacency) spectrum if cospectral graphs are isomorphic graphs. It is well-known that in general the spectrum does not determine the graph, and this problem has pushed a lot of research in spectral graph theory, also with respect to other graph matrices. In general, we can say that there are three kinds of research lines: 1) Identify, if any, cospectral non-isomorphic graphs for a given class of graphs; 2) Routines to build cospectral non isomorphic graphs (e.g., Godsil-McKay switching); 3) Find conditions such that the corresponding graphs are determined by their spectrum.

Evidently, the same problems can be considered for signed graphs with respect to switching isomorphism. On the other hand, when considering signed graphs, there are many more possibilities for getting pairs of switching non-isomorphic cospectral signed graphs. For example, the paths and the cycles are examples of graphs determined by their spectrum, but the same graphs as signed ones are no longer determined by their spectrum since they admit cospectral but non-isomorphic mates cycle ; path .

Hence, the spectrum of the adjacency matrix of signed graphs has less control on the graph invariants. In view of the spectral moments we get the following proposition:

Proposition 3.28.

From the eigenvalues of a signed graph we obtain the following invariants:

  • number of vertices and edges;

  • the difference between the number of positive and negative triangles ();

  • the difference between the number of positive and negative closed walks of length ()

Contrarily to unsigned graphs, from the spectrum we cannot decide any more whether the graph has some kind of signed regularity, or it is sign-symmetric. For the former, we note that the co-regular signed graph (it is a regular graph with net regular signature) is cospectral with (cf. Fig. 4). For the latter, we observe that the signed graphs and are cospectral but is not sign-symmetric while is sign-symmetric.

Figure 4: The cospectral pair and .

3.7 Operations on signed graphs

In graph theory we can find several operations and operators acting on graphs. For example, we have the complement of a graph, the line graph, the subdivision graph and several kind of products as the cartesian product, and so on. Most of them have been ported to the level of signed graph, in a way that the resulting underlying graph is the same obtained from the theory of unsigned graphs, while the signatures are given in order to preserve the balance property, signed regularities, and in many cases also the corresponding spectra. However, there are a few operations and operators which do not yet have a, satisfactory, ’signed’ variant.

One operator that is missing in the signed graph theory is the complement of a signed graph. The complement of signed graph should be a signed graph whose underlying graph is the usual complement, however the signature has not been defined in a satisfactory way yet. What we can ask from the signature of the complement of a signed graph? One could expect some nice features on the spectrum, as for the Laplacian, so that the spectra of the two signed graphs and are complementary to the spectrum of the obtained complete graph.

Problem 3.29.

Given a graph , define the complement such that there are nice (spectral) properties derived from the complete signed graph .

In terms of operators, in the literature we have nice definitions for subdivision and line graphs of signed graphs besi ; zas1 , but for example a satisfactory definition of signed total graph of a signed graph has not yet been provided (but it is considered in the forthcoming paper bestaza ).

From the product viewpoint, most standard signed graph products have been defined and considered in gehaza and the more general NEPS (or, Cvetković product) of signed graphs have been there considered. In germa66 the lexicographic product was also considered, but the given definition is not stable under the equivalence switching classes.

However, there are some graph products which do not have a signed variant yet. As an example, we mention here the wreath product and the co-normal product.

3.8 Seidel matrices

The Seidel matrix of a graph on vertices is the adjacency matrix of a signed complete graph in which the edges of are negative and the edges not in are positive . More formally, the Seidel matrix equals . Zaslavsky zas1 confesses that

This fact inspired my work on adjacency matrices of signed graphs.

Seidel matrices were introduced by Van Lint and Seidel VLS and studied by many people due to their interesting properties and connections to equiangular lines, two-graphs, strongly regular graphs, mutually unbiased bases and so on (see (BH, , Section 10.6) and BDKS ; GKMS ; Sei1 for example). The connection between Seidel matrices and equiangular lines is perhaps best summarized in (BH, , p.161):

To find large sets of equiangular lines, one has to find large graphs where the smallest Seidel eigenvalue has large multiplicity.

Let be a natural number and denote the Euclidean -dimensional space with the usual inner product . A set of lines (represented by unit vectors) is called equiangular if there is a constant such that for any . For given , let be the maximum with this property. The Gram matrix of the vectors is the matrix whose -th entry equals . The matrix is a symmetric matrix with diagonal and entries off-diagonal. It is therefore the Seidel matrix of some graph and contains all the relevant parameters of the equiangular line system. The multiplicity of the smallest eigenvalue of is the smallest dimension where the line system can be embedded into .

Lemmens and Seidel LS (see also BDKS ; GKMS ; JP ; LY ; Neumaier for more details) showed that for sufficiently large and made the following conjecture.

Conjecture 3.30.

If , . If , then .

The fact that for sufficiently large was proved by Neumaier Neumaier and Greaves, Koolen, Munemasa and Szöllősi GKMS . Recently, Lin and Yu LY made progress in this conjecture by proving some claims from Lemmens and Seidel LS . Note that these results can be reformulated in terms of Seidel matrices with smallest eigenvalue . Seidel and Tsaranov SeiTsa classified the Seidel matrices with smallest eigenvalue .

Neumann (cf. (LS, , Theorem 3.4)) proved that if , then is an odd integer. Bukh Bukh proved that , where is a constant depending only on . Balla, Dräxler, Keevash and Sudakov BDKS improved this bound and showed that for sufficiently large and , . Jiang and Polyanskii JP further improved these results and showed that if , then for sufficiently large. When is an odd integer, Glazyrin and Yu GY obtained a general bound for all .

Bukh Bukh and also, Balla, Dräxler, Keevash and Sudakov BDKS conjecture the following.

Conjecture 3.31.

If is an integer, then for sufficiently large.

When is not a totally real algebraic integer, then . Jiang and Polyanskii JP studied the set and showed that the closure of contains the closed interval using results of Shearer Shearer on the spectral radius of unsigned graphs.

Seidel matrices with two distinct eigenvalues are equivalent to regular two-graphs and correspond to equality in the relative bound (see (BH, , Section 10.3) or GKMS for example). It is natural to study the combinatorial and spectral properties of Seidel matrices with three distinct eigenvalues, especially since for various large systems of equiangular lines, the respective Seidel matrices have this property. Recent work in this direction has been done by Greaves, Koolen, Munemasa and Szöllősi GKMS who determined several properties of such Seidel matrices and raised the following interesting problem.

Problem 3.32.

Find a combinatorial interpretation of Seidel matrices with three distinct eigenvalues.

A classification for the class of Seidel matrices with exactly three distinct eigenvalues of order less than was obtained by Szöllősi and Östergård SO . Several parameter sets for which existence is not known were also compiled in GKMS . Greaves Gre18 studied Seidel matrices with three distinct eigenvalues, observed that there is only one Seidel matrix of order at most having three distinct eigenvalues, but its switching class does not contain any regular graphs. In Gre18 , he also showed that if the Seidel matrix of a graph has three distinct eigenvalues of which at least one is simple, then the switching class of contains a strongly regular graph. The following question was posed in Gre18 .

Problem 3.33.

Do there exist any Seidel matrices of order at least with precisely three distinct eigenvalues whose switching class does not contain a regular graph ?

The switching class of conference graph and isolated vertex has two distinct eigenvalues. If these two eigenvalues are not rational, then the switching class does not contain a regular graph. So we suspect that there must be infinitely many graphs whose Seidel matrix has exactly three distinct eigenvalues and its switching graph does not contain a regular graph. A related problem also appears in Gre18 .

Problem 3.34.

Does every Seidel matrix with precisely three distinct rational eigenvalues contain a regular graph in its switching class ?

The Seidel energy of a graph is the sum of absolute values of the eigenvalues of the Seidel matrix of . This parameter was introduced by Haemers Hae2 who proved that for any graph of order with equality if and only is a conference matrix. Haemers Hae2 also conjectured that the complete graphs on vertices (and the graphs switching equivalent to them) minimize the Seidel energy.

Conjecture 3.35.

If is a graph on vertices, then .

Ghorbani Gho proved the Haemers’ conjecture in the case and very recently, Akbari, Einollahzadeh, Karkhaneei and Nematollah Akbari1 finished the proof of the conjecture. Ghorbani (Gho, , p.194) also conjectured that the fraction of graphs on vertices with goes to as tends to infinity. This conjecture was also recently proved by Rizzolo Rizzolo .

It is known that if has even order, then its Seidel matrix is full-rank. If a graph has odd order , then . There are examples such for example where . Haemers Hae posed the following problem which is still open to our knowledge.

Problem 3.36.

If , then there exists an eigenvector of corresponding to that has only entries ?

Recently, Van Dam and Koolen DamKoo19 determined an infinitely family of graphs on vertices whose Seidel matrix has rank and their switching class does not contain a regular graph.

4 Conclusions

Spectral graph theory is a research field which has been very much investigated in the last 30–40 years. Our impression is that the study of the spectra of signed graphs is very far from the level of knowledge obtained with unsigned graphs. So the scope of the present note is to promote investigations on the spectra of signed graphs. Of course, there are many more problems which can be borrowed from the underlying spectral theory of (unsigned) graphs. Here we just give a few of them, but we have barely scratched the surface of the iceberg.

Acknowledgements

The authors are indebted to Thomas Zaslavsky for reading and revising with many comments an earlier draft of this note. The research of the first author was supported by the University of Naples Federico II under the project “SGTACSMC”. The research of the second author was supported by the grants NSF DMS-1600768 and CIF-1815922. The research of the third author is partially supported by the National Natural Science Foundation of China (Grant No. 11471009 and Grant No. 11671376) and by Anhui Initiative in Quantum Information Technologies (Grant No. AHY150200). The research of the fourth author was supported by the National Natural Science Foundation of China (No. 11461054).

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