The negative tetrahedron and the first infinite family of connected digraphs that are strongly determined by the Hermitian spectrum
Pepijn Wissing, Edwin R. van Dam
TL;DR
This paper introduces the first infinite family of connected digraphs that are uniquely determined by their Hermitian spectrum, constructed through a novel approach involving negative tetrahedra and twin vertices.
Contribution
It constructs the first infinite family of connected digraphs that are strongly determined by their Hermitian spectra, advancing spectral graph theory.
Findings
Identifies the rarity of SHDS digraphs.
Constructs an infinite family of SHDS digraphs.
Analyzes digraphs with exactly one negative eigenvalue.
Abstract
Thus far, digraphs that are uniquely determined by their Hermitian spectra have proven elusive. Instead, researchers have turned to spectral determination of classes of switching equivalent digraphs, rather than individual digraphs. In the present paper, we consider the traditional notion: a digraph (or mixed graph) is said to be strongly determined by its Hermitian spectrum (abbreviated SHDS) if it is isomorphic to each digraph to which it is cospectral. Convincing numerical evidence to support the claim that this property is extremely rare is provided. Nonetheless, the first infinite family of connected digraphs that is SHDS is constructed. This family is obtained via the introduction of twin vertices into a structure that is named negative tetrahedron. This special digraph, that exhibits extreme spectral behavior, is contained in the surprisingly small collection of all digraphs with…
| 3 | 4 | 5 | 6 | 7 | 8 | |
|---|---|---|---|---|---|---|
| 6.25 | 3.21 | 7.36 | 9.87 | 6.16 | 2.20 | |
| 9 | 10 | 11 | 12 | 13 | 14 | |
| 3.89 | 3.79 | 1.85 | 4.89 | 6.50 | 4.58 | |
| 15 | 16 | 17 | 18 | 19 | 20 | |
| 1.63 | 3.06 | 2.90 | 1.43 | 3.59 | 4.64 |
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The negative tetrahedron and the first infinite family of connected digraphs that are strongly determined by the Hermitian spectrum
Pepijn Wissing, Edwin R. van Dam
CentER and Department of Econometrics and Operations Research, Tilburg University Corresponding author: [email protected]
1 Introduction
The relation between the eigenvalues of a graph and its structural characteristics has been studied by many. Finding use in various fields such as combinatorics [3], optimization [22, 24] and computer science [25], the literature on graph spectra is vast. Of particular interest to the authors is the question: can we determine a graph by its spectrum? This question has received considerable attention; the known results have been surveyed in [9, 10].
Nevertheless, the same question has yielded far fewer results in the directed graph (or mixed graph, see Sec. 1.1) paradigm. This is in part due to the absence of a consensus as to which matrix best reflects the characteristics of a directed graph (henceforth abbreviated as digraph) in its eigenvalues. A natural choice is the well-known adjacency matrix. However, as this matrix is in general not symmetric, its eigenvalues are not necessarily real. Other alternatives include the skew-symmetric adjacency matrix (see [5]), which only works for digraphs that contain no bidirected edges, and the normalized Laplacian [8], which has a particularly impractical definition.
The introduction of the Hermitian adjacency matrix (”Hermitian” or ””, for short), independently by Liu and Li [21] and Guo and Mohar [14], has provided us with an interesting new candidate. A number of important pieces of machinery from undirected graph theory have been shown to hold with respect to the Hermitian. These include, but are not limited to, eigenvalue interlacing and the quotient matrix [17, 14] and the concept of computing the coefficients of the characteristic polynomial by considering elementary subdigraphs [21].
One of the main drawbacks of the Hermitian, from a spectral analysis point of view, is that a large number of digraphs with the same underlying graph have an identical -spectrum. Suppose, for example, that a digraph has a cut-edge. Then this edge may be directed in either direction, or even be made bidirected, all the while leaving the corresponding -spectrum unchanged. This illustration is a special case of an operation uncovered by Guo and Mohar [14], called four-way switching, which partitions the digraph in four (possibly empty) parts and performs a series of operations that either change the direction or nature (from arc to digon or vice versa) of the arcs between the parts or take the converse of the digraph as a whole, without affecting the -spectrum.
In recent days, the Hermitian spectrum has been the subject of several publications. In addition to co-authoring one of the two fundamental papers [21, 14], mentioned above, Mohar [23] has characterized all digraphs whose Hermitian adjacency matrix have rank 2, and shown that there are infinitely many digraphs that have cospectral mates, which are not members of the same switching equivalence class. Wang et al. [27] extends the research in [23] to the digraphs of rank 3; their main result is that any pair of weakly connected, cospectral rank 3 digraphs is switching equivalent. Although using a different approach, Tian and Wong [26] obtain similar results as in [27].
Further recent research that is concerned with the Hermitian spectrum but less relevant to the current paper includes Greaves et al. [13], Guo and Mohar [15], Greaves [12], Hu et al. [20], Chen et al. [6] and Chen et al. [7].
In his article [23], Mohar investigates which digraphs of rank 2 are determined by their -spectrum. (Abbreviated HDS hereafter.) Due to the before-mentioned characteristics of the Hermitian, Mohar defines a digraph to be HDS if it is cospectral only to those digraphs that are obtained from the digraph by a switching operation, possibly followed by the reversal of all edges. This definition is, however, much weaker than that of the similarly named concept in undirected graph context; if a graph is said to be determined by its adjacency spectrum, then one is able to uniquely (up to isomorphism) construct said graph when one is given its spectrum.
As such, the authors set out to classify digraphs which are strongly determined by their -spectrum; that is, digraphs whose spectra occur uniquely. Two prominent examples of such digraphs are shown in Figure 1. These digraphs are extremely rare, as any such digraph must be self-converse. We observe that the fraction of digraphs that satisfies this property rapidly goes to zero as the number of vertices grows. While no formal proof was found in the literature, numerical evaluation of the counting polynomials by Harary [18] and Harary and Palmer [19] provides convincing evidence to support this claim.
In the present paper, the authors were inspired by a result first encountered in [16] and later in [14], which occurs here as Lemma 3.1. In particular, by said lemma, there is exactly one kind of induced subdigraph of order 3 that may contribute negatively to , where is the Hermitian of a digraph . This order digraph is named negative triangle and shown in Figure 1(a). Furthermore, eigenvalue interlacing is used extensively.
The main results of this paper are as follows. We construct the first infinite family of connected, strongly HDS digraphs, in Theorem 5.3. This family is obtained by twin expansion (see Def. 2.10) of a key digraph, which is named the negative tetrahedron. This peculiar digraph is a tetrahedron, whose four faces are all negative triangles, as is shown in Figure 1(b). Moreover, it is the only reduced111A digraph is said to be reduced if it contains no two vertices whose neighborhoods exactly coincide, and no isolated vertices. See Def. 2.8. digraph that has rank 4 and exactly one negative eigenvalue. Additionally, we determine all digraphs that have precisely one negative eigenvalue in Theorem 4.8, and show that any pair of connected rank members of this class is switching equivalent if they are cospectral in Proposition 6.4.
2 Preliminaries
Let us first thoroughly discuss the definitions and known results. Most of the concepts below are well-known and will not always be explicitly referenced. For more detail, the reader is referred to e.g., Bondy and Murty [2] or any other recent book on graph theory.
2.1 Terminology and notation
Throughout this work, we will adopt the following terminology and notation. A directed graph or digraph of order is denoted . Here, we denote the vertex set of as (sometimes specified as ) and edge set (or ), where . An arc or directed edge is an ordered pair of vertices; is called the initial vertex and is called the terminal vertex. If both and , we say that the unordered pair is a digon in . A loop is an arc of which the terminal vertex equals the initial vertex. Throughout, we will not allow digraphs that are considered in this work to contain loops.
Some of the cited literature considers mixed graphs, which are defined to be an ordered triple where is the vertex set, is the undirected edge set and is the directed edge set. Since a single bidirected edge is, for our purposes throughout this paper, equivalent to two arcs whose directions are reversed, we consider the class of mixed graphs equivalent to the class of digraphs as described above. In the interest of clearness, we will only use the word ”graph” when it concerns an undirected graph.
The Hermitian adjacency matrix, introduced independently by Guo and Mohar [14] and Liu and Li [21], is formally defined as follows.
Definition 2.1**.**
[14, 21] Let be a digraph of order . Define the Hermitian as the matrix with entries
[TABLE]
The Hermitian is diagonalizable [3] and one may apply eigenvalue interlacing (see Lemma 2.1) with respect to , as shown in [16, 14]. Throughout, we will use the Hermitian to characterize digraphs; the rank of and the spectrum of are said to be respectively the rank and the spectrum of the Hermitian of . Accordingly, the characteristic polynomial of is defined as . The spectrum of is denoted , where are the distinct eigenvalues, whose multiplicities are . As is convention, denotes the spectral radius of , i.e., its largest eigenvalue in absolute value.
We may sometimes be interested in the underlying graph of a given digraph . Let be the operator that transforms a digraph into its underlying graph. That is, given , the graph is obtained by replacing every arc in with the corresponding digon. Two vertices are said to be adjacent in if they are adjacent in . Moreover, note that we may determine the number of edges in the underlying graph of a given digraph from its spectrum by using that .
The converse of a digraph is denoted , and is obtained by reversing the direction of all arcs. Note that , and thus any digraph is inherently cospectral to its converse. Furthermore, a digraph that is isomorphic to its converse is called self-converse; this rare property is necessary for strong spectral determination.
The last mentioned piece of notation is concerned with induced subdigraphs. If and let , then we denote the induced subgraph that is obtained by removing any vertices in and removing any edges that are incident to a vertex in as .
As was briefly mentioned before, Mohar [23] pioneered spectral characterization with respect to the Hermitian adjacency matrix. We include the details of Mohar’s definitions below.
Definition 2.2**.**
[14] A four-way switching is the operation of changing a digraph into the digraph by choosing an appropriate diagonal matrix with and setting Informally, is appropriate when is a Hermitian adjacency matrix.
Definition 2.3**.**
[23] Two digraphs and are said to be switching equivalent if may be obtained from by a four-way switching, possibly followed by a reversal of all arcs.
Definition 2.4**.**
[23] A digraph is said to be (weakly) determined by its Hermitian spectrum (WHDS) if it is switching equivalent to every digraph to which it is -cospectral.
However, the following definition of spectral determination is in a sense more loyal to its undirected graph analogue.
Definition 2.5**.**
A digraph is said to be strongly determined by its Hermitian spectrum (SHDS) if it is isomorphic to every digraph to which it is -cospectral.
The distinction between Definitions 2.4 and 2.5 summarizes what sets this work apart from previous articles regarding Hermitian spectral characterization. To distinguish between the two definitions, the authors have added the word ”weakly” to the former and ”strongly” to the latter. The terminology is justified by the observation that any SHDS digraph is implicitly WHDS.
We end this section with a few words of warning, regarding a frequent mistake surrounding the term (W)HDS. Since neither four-way switching nor taking the converse changes a digraph’s underlying graph, two digraphs of which just one is connected cannot be switching equivalent. Hence, if a connected digraph is cospectral to a digraph that contains isolated vertices, neither may be said to be (W)HDS. While [27, 26] show that any pair of cospectral, connected rank 3 digraphs is switching equivalent, cases in which such a connected digraph is cospectral to a disconnected digraph may still be found. Therefore, the phrasing of the final theorems of both [27] and [26], in which it is claimed that any rank 3 digraph is (W)HDS, is flawed.
2.2 Known results
Here, we will list some of the results that are vital to the discussion in this paper. Likely the single most used tool throughout is known as eigenvalue interlacing (Haemers [17]), which is a particularly powerful tool, adopted from graph theory. In [11], this concept has been shown to be valid with respect to Hermitian matrices.
Lemma 2.1**.**
[17, 11]** Suppose is a Hermitian matrix with eigenvalues . Then the eigenvalues of a principal submatrix of size satisfy for .
In particular, Lemma 2.1 will be one of the tools to determine the collection of all digraphs with a single negative eigenvalue. To explicitly write the spectra of these families, the concepts known as quotient matrix and equitable partition are used. Both originate from [17], in graph context, and have been published in [14] for Hermitian context.
Let be a digraph and let be a partition of . One may order the vertices of such that induces a partition of into block matrices as
[TABLE]
The quotient matrix of a Hermitian with respect to is the matrix , where is the average row sum of block . The partition is said to be equitable if every block has constant row sum. One then has the following result.
Lemma 2.2**.**
[14]** Let be a digraph with Hermitian , and let be an equitable partition of its vertices. Moreover, let be the quotient matrix of with respect to If is an eigenvalue of with multiplicity , then it is an eigenvalue of with multiplicity at least .
In [23], Mohar works with digraphs of rank 2, particularly those that are complete bipartite or complete tripartite. With regard to the former, the following unproven claim is made.
Conjecture 2.3**.**
[23]** There are only finitely many integers and for which the complete bipartite graph is WHDS.
Regarding complete tripartite digraphs, Mohar claims that there are many instances that are WHDS for number theoretic reasons. In particular, if we denote by the complete tripartite digraph with parts , , , where , and , with all arcs from to , to , and to , then the following claims hold.
Proposition 2.4**.**
[23]** , and are WHDS for every .
Corollary 2.4.1**.**
[23]** Suppose that and are integers such that Then is WHDS if and only if is not divisible by a prime that is congruent to modulo .
This line of research was extended to rank 3 independently by Wang et al. [27] and Tian and Wong [26]. Of most relevance to this work is the following result.222Originally formulated as ”All connected digraphs of order with rank are WHDS,” which is not entirely accurate, as discussed at the end of Sec. 2.1. The authors took the liberty of slightly rephrasing to avoid confusion.
Proposition 2.5**.**
[27, 26]** Any two connected, cospectral, rank digraphs are switching equivalent.
However, Wang et al. [27] also show that, if the assumption on connectedness is omitted, one may construct infinite families of rank 3 digraphs that are not WHDS.
Proposition 2.6**.**
[27]** There are infinitely many digraphs with rank that are not WHDS.
2.3 Twins
The first half of the discussion in this paper will concern ’small’ digraphs with exactly one negative eigenvalue; the second half will extend this discussion to ’large’ digraphs. That said, the discussed digraphs remain largely similar, from a structural point of view. Specifically, in order to increase the size of the considered digraphs without compromising the structural arguments made in the former part, twins are introduced into the small digraphs. Since there have been several authors (e.g., [1, 4]) to have introduced a similarly named concept, each with subtle differences, we provide the formal definition as it is used throughout this paper.
Definition 2.6**.**
Two vertices in , whose Hermitian is , are called twins if for every
A simple but important observation is that and are implicitly not twins if , as loops are not allowed throughout. Moreover, if are vertices in , is said to distinguish and if . Naturally, if such a vertex exists in , then and are not twins in , which justifies the terminology.
We will often want to consider the structure that is in a sense fundamental to a large digraph that contains a set of equivalent vertices. To this end, we define the twin reduction operation, which reduces such a collection of twin vertices down to a single vertex; this may significantly reduce the order of a digraph, while retaining its general structure and rank. Moreover, using said operation, we define a property that characterizes digraphs we consider to be ’small’.
Definition 2.7**.**
We define to be the twin reduction operator, which removes vertices and edges from a digraph in such a way that exactly one of every collection of twins is kept and no isolated vertices remain.
Definition 2.8**.**
A digraph is called reduced if .
Naturally, we may also want to reverse this operation, to increase the size of the digraph without compromising the fundamental structure. The formal definition of the corresponding operation is given below.
Definition 2.9**.**
A vector is called an expansion vector for a digraph if and .
Definition 2.10**.**
Let be a digraph with an ordered set of vertices, and let be an appropriate expansion vector. The twin expansion of with respect to is denoted and is obtained by replacing each vertex in by twins, and adding isolated vertices. Formally, if , let , where are mutually disjoint sets, with . In , is a set of isolated vertices, and for every , , , where and are the Hermitians of and , respectively.
Note that each entry of the expansion vector thus corresponds to a specific vertex in the digraph that is to be expanded. As a direct consequence, the vertex ordering does matter, in the sense that permuting the expansion vector does not change the resulting (expanded) digraph if the vertex order of the source digraph is permuted accordingly. Thus, we will fix the vertex orderings of the relevant digraphs to ensure that the above does not occur, when permutations of given expansion vectors are discussed. Throughout, the northmost vertex of any digraph defined through an illustration is assigned ’’, after which vertices are subsequently labeled in counterclockwise order.
For the sake of clarity, we include the following example that shows the working of the twin expansion operator.
Example 1. Let Then the vertices of the negative triangle and its twin expansion may be labeled such that their Hermitians are
[TABLE]
We conclude this section with two observations that will be quite obvious to the experienced graph theorist, though the ideas are in a sense key to the presented discussion. As such, their proofs are omitted.
Lemma 2.7**.**
Let be a digraph of order and let be an expansion vector for . Then
[TABLE]
Lemma 2.8**.**
Let be a digraph of order and let be an expansion vector for . Suppose that has positive and negative eigenvalues. Then and also have positive and negative eigenvalues.
3 The negative tetrahedron
In the present paper, we are interested in families of digraphs that contain many copies of a given substructure, which is in a sense counted by the spectrum. In this section, we will provide a thorough introduction of these families and the elementary observations upon which many of the later results are built.
3.1 Digraphs related to the negative triangle
Upon investigation of properties that may be inferred from the Hermitian spectrum of a digraph, we are inspired by the following lemma by Guo [16], that ties in closely to a similar, well-known result for undirected graphs.
Lemma 3.1**.**
[16]** Let be a digraph with Hermitian . Then , where denotes the number of copies of that occur as induced subdigraphs of . The structures are shown in Figure 2.
The main observation to take away from Lemma 3.1 is that apparently, the structure , above, is the only order 3 substructure that has a negative impact on , which in turn may be computed directly from the spectrum of a digraph. Thus, we may be able to identify (or even determine) digraphs that have many such substructures. In the interest of clearness, we name the following two structures, which occurred before in Figure 1 and that are in a sense fundamental to the discussion in this paper.
Definition 3.1**.**
Figure 2(d) is called the negative triangle and is denoted .
Definition 3.2**.**
Figure 3 is called the negative tetrahedron and is denoted .
The negative tetrahedron is an interesting digraph for a number of reasons, and has come up in the before mentioned works. One might first notice its extreme degree of structural symmetry; is, in fact, vertex-transitive. A second interesting fact is that and are exactly the two digraphs with rank more than that are antispectral333A pair of digraphs and is said to be antispectral to one another if . to a complete graph. and , whose spectra are and , respectively, are antispectral to respectively and . Guo and Mohar [14] have shown that there are no higher rank digraphs that admit to this property. Lastly, it is mentioned in [14] that exhibits extreme spectral behavior, in the sense that it attains the bound , where is the largest eigenvalue of .
In addition to and , there are two more digraphs that play a prominent role throughout this paper. In the interest of structure, we include their definitions here.
Definition 3.3**.**
The digraphs and are shown in Figures 4(a) and 4(b), respectively.
We note that both digraphs are reduced and have rank . Furthermore, we observe that and are closely related to , from a spectral point of view. In fact, if one expands a single vertex of once (to obtain, say, ), then are all cospectral and switching equivalent. More detail concerning this relation is provided at the end of Section 3.3.
3.2 The first families of SHDS digraphs
Using only the tools we have thus far, we are already able to construct some infinite SHDS families. Probably the first, most trivial SHDS digraph that comes to mind is the empty graph of order , denoted . Indeed, using that , the all-zero spectrum certainly determines the empty graph.
It is easy to check that is the smallest non-empty digraph that is strongly determined by its Hermitian spectrum. In fact, it is a simple exercise to show the following result,444A similar result occurs as Prop. 5.2 in [14], which concerns digraphs antispectral to . While the obtained collection of digraphs is almost identical, the authors chose to include a proof as the claim requirements and the argument are signifficantly different. that signifies the essential role and play in the proposed discussion. As a first step towards a less trivial infinite family of SHDS digraphs, we classify all digraphs with largest eigenvalue .
Lemma 3.2**.**
Let be a connected digraph with largest eigenvalue . Then is either or , where is the oriented .
Proof.
By interlacing, it follows that there is no such that since every digraph whose underlying graph is has where is the largest eigenvalue. Hence, , for . If , both digraphs with have , and are therefore valid options. Note that , only if ; other potential digraphs of order have an eigenvalue or . Moreover, one should also observe that if , each order- induced subdigraph of should again be , or else the claim is false by interlacing. Thus, if , . Moreover, as before, if , any order induced subdigraph of must be .
Finally, suppose that , and suppose that . (Note that we are not imposing any extra assumptions; if these induced subdigraphs are not , certainly has .) Then we have , as . It follows that , by which we have a contradiction and thus . ∎
By the result above, we are able to draw an interesting conclusion with regard to the spectral determination of the class of digraphs with largest eigenvalue .
Corollary 3.2.1**.**
Let be a digraph with largest eigenvalue . Then is WHDS.
Proof.
By Lemma 3.2, every connected component must be or . It should be clear that every eigenvalue belongs to a copy of , every eigenvalue belongs to a , and every eigenvalue [math] belongs to an isolated node. Likewise, every eigenvalue belongs to or . Since is clearly switching equivalent to the claim follows. ∎
Moreover, if one excludes and , such a digraph is SHDS, by the same argument.
Corollary 3.2.2**.**
Let be a digraph with largest eigenvalue and no eigenvalues . Then is SHDS.
Thus, we have obtained an infinite family of SHDS digraphs, that may consist of arbitrarily many disjoint copies of and , as well as isolated vertices.
3.3 The spectra of expansions of and related digraphs
In the discussion leading up to our main theorem, we will be interested in twin expansions (recall Def. 2.10) of , in particular. The following Lemma is added for completeness; its correctness should be evident from a brief look at Figure 5.
Lemma 3.3**.**
Let and let . Then contains vertices, edges, and copies of .
Note that a permutation of the coefficients would not necessarily yield an isomorphic digraph, as is illustrated in Example 5, (Sec. 5) while the spectrum is invariant under such a permutation, as we will see shortly.
As we set out to show that particular twin expansions of are SHDS, it seems fitting to include the explicit spectra of this interesting family of diraphs. In the below, we write their characteristic polynomials explicitly by employing Lemma 2.2. While one could have used a before-mentioned result from [21] that counts elementary subdigraphs to obtain the coefficients in (3), the authors found the approach below to be significantly more comprehensible.
Lemma 3.4**.**
Let be an expansion vector and let . Then has characteristic polynomial
[TABLE]
Proof.
By construction, we may write the Hermitian of as the block matrix
[TABLE]
where the diagonal blocks have sizes , respectively. Note that all of the blocks in (4) are constant, and thus (4) is a so-called equitable partition. We may then write the quotient matrix [17, 14] as
[TABLE]
One may compute to find
[TABLE]
Now, observe that . Hence, has four nonzero roots, which are the (not necessarily distinct) eigenvalues of . Since (4) is an equitable partition, each of the also occur as an eigenvalue of . Moreover, since by construction , it is clear that we have
[TABLE]
∎
Moreover, by plugging in , one may readily show the following results.
Corollary 3.4.1**.**
Let and . For the following special cases of expansion vector , we may write the spectrum of explicitly. In the below, is the sum of the entries of .
- (i)
If then
[TABLE] 2. (ii)
If then
[TABLE] 3. (iii)
If then
[TABLE]
Proof.
Follows directly by plugging into Lemma 3.4. ∎
Corollary 3.4.2**.**
Let and . If then has an eigenvalue .
Proof.
Plug in (3) to find
[TABLE]
which clearly has a root at ∎
Now, one would like to conclude that the reverse is also true; that from the occurrence of an integer eigenvalue it follows that an expansion vector contains twice. This is in general not true, as shown by the following example.
Example 2. Suppose that . Then, by Lemma 3.4, has characteristic polynomial
[TABLE]
and thus has an eigenvalue , while none of the equals
That said, if an integer eigenvalue occurs at least twice, then we are able to conclude the reverse, as we will discuss in the proof of Theorem 5.3.
We conclude this section with some brief notes regarding the spectral similarity of , , and , and their respective expanded versions. As before, we are able to compute their spectra explicitly by employing the quotient matrix.
Lemma 3.5**.**
Let be an expansion vector and let . Then has characteristic polynomial
[TABLE]
Lemma 3.6**.**
Let and be an expansion vector. Let and . Then
[TABLE]
and
[TABLE]
Thus, it follows that , and are all cospectral. Lastly, note that these digraphs are also all pairwise switching equivalent.
4 Classification of digraphs with one negative eigenvalue
In order to construct the desired infinite family of SHDS digraphs, we first set out to find its smallest members. It turns out that the members of the family in which we are interested share the interesting property of only having a single negative eigenvalue; a property that is satisfied by very few reduced digraphs. We note the following useful obseravation with regard to such digraphs.
Lemma 4.1**.**
Let be reduced with exactly one negative eigenvalue. Then is connected.
Proof.
Note that the spectrum of any connected component of order at least contains at least one negative eigenvalue, since the sum of the eigenvalues of a Hermitian matrix must sum up to its trace, which is zero for the Hermitian adjacency matrix of a digraph without loops. Moreover, recall that if consists of two disjoint, connected components and , then and thus contains at least two negative elements. Lastly, note that no isolated vertices are allowed by definition of reducedness. ∎
We also impose a minor assumption on the rank of the considered digraphs, in order to exclude cases that are in a sense trivial. Specifically, we require digraphs to have rank larger than . Recall that there are no digraphs with rank less than besides the empty graph, and that any nonempty rank 2 digraph trivially has precisely one positive and one negative eigenvalue, by the observation above. However, no such digraph is interesting for the present paper, as any rank digraph is cospectral to its underlying graph [23] and such digraphs are in general not WDHS.555By employing twin expansion on e.g., and , one easily finds cospectral classes whose members have at least two distinct underlying graphs. As such, we exclude this class of digraphs; the interested reader is referred to [23], in which this class is researched in considerable detail.
If one requires the considered digraphs to have rank larger than in addition to being reduced, one finds just four digraphs. The main result of this section, which is the following theorem, shows exactly that.
Theorem 4.2**.**
Let be a reduced digraph of rank larger than and with exactly one negative eigenvalue. Then is one of , , , or .
In order to prove Theorem 4.2, we first show several intermediate results. First, we will provide a few crucial observations that are somewhat obvious, but that are added for the sake of completeness. In Proposition 4.4, we will see that there are exactly three reduced digraphs on four vertices that have the required single negative eigenvalue. The remainder of the section is concerned with bounding the size of a reduced digraph with exactly one negative eigenvalue. In particular, we will find that such a digraph may contain at most four vertices; the correctness of Theorem 4.2 then follows.
As was mentioned before, the negative triangle plays an essential role throughout. The simple, but useful fact that any digraph of sufficient rank must contain such triangles if it has a single negative eigenvalue, is shown below.
Lemma 4.3**.**
Let be a digraph with rank larger than . If has a single negative eigenvalue, then for some
Proof.
Let be the positive eigenvalues of and let be the negative eigenvalue. We have , so and thus contains at least one negative triangle by Lemma 3.1. ∎
Corollary 4.3.1**.**
If is a digraph with order , rank larger than , and exactly one negative eigenvalue, then .
We are now ready to show the first major result, in which we obtain the collection of order four digraphs that meets our requirements. We would remark here that, while the collection of all order four digraphs is small enough to simply apply full enumeration by computer, the authors opt for a constructive argument that may be used in similar fashion when the order is increased.
Proposition 4.4**.**
Let be a reduced digraph with order , rank larger than , and exactly one negative eigenvalue Then is one of , , or .
Proof.
Let be a digraph of order , with exactly one negative eigenvalue, and rank larger than . By Lemma 4.3, contains . Hence, we may write as
[TABLE]
where , and Note that the variable entries of are put in this form to make (5) symmetric. One then readily obtains that
[TABLE]
Now, we make the following observation. Since contains , it has at least two positive eigenvalues, by interlacing. Moreover, from it follows that if and only if has an even number of negative eigenvalues. Hence, satisfies the requirements of the claim if and only if the corresponding is such that . Note that if exactly one is nonzero, one may plug in (5) to obtain and thus has more than one negative eigenvalue. Therefore, at least two elements of must be nonzero.
Suppose that two elements of are nonzero. Then (by (5)) and thus we are only interested in the case that Suppose that Then Hence, either or . Similarly, if then or and if then or . It is readily verified that out of these six possible , three correspond to a digraph that contains a twin and therefore do not meet the requirements of the claim; the remaining three correspond to either or .
Lastly, suppose that no element of is zero. It is easily observed from (5) that since and has no solution for . Thus, any that meets the requirements of the claim has Hence, , with at most one pair such that . W.l.o.g., assume that . Then and , and thus which corresponds to exactly . ∎
In order for us to prove Theorem 4.2, we should consider the reduced digraphs of larger order. As we will shortly show, we find that no digraph of order larger than satisfies both required properties, i.e., being reduced and having exactly one negative eigenvalue. In the interest of structure, the discussion to support this claim is split up into Lemmas 4.5, 4.6 and 4.7.
The approach below is, for the most part, based on the idea of taking some small substructure that is certainly contained in any of the candidates, and attempting to build a digraph that meets all requirements by adding vertices and edges to it. In particular, all of the digraphs we encounter contain at least one copy of . Moreover, from the results at the top of this section, we know that there are scarcely any ways to extend with vertices and edges without invalidating assumptions. Using these facts, we will show in Lemmas 4.5, 4.6 and 4.7 that the order of the digraphs that have so far been shown to satisfy our requirements cannot be extended without introducing a twin vertex.
Lemma 4.5**.**
Suppose that is a digraph of order , rank larger than , that does not contain , , or . Moreover, suppose that has exactly one negative eigenvalue. Then is not reduced.
Proof.
By contradiction. We will show, through combinatorial reasoning, that a digraph that admits to the assumptions in the claim must contain twins. This reasoning is illustrated with an example in Figure 6. We note that while the exact nature of the edges in Figure 6 may differ depending on , the reasoning below remains valid.666In fact, one could even disregard the arc orientations in Figure 6 altogether and apply the reasoning in the proof of Lemma 4.5 to the underlying graph of the considered digraph, without compromising the proof. The authors have opted for an example that best illustrates the situation at hand.
Suppose that is reduced. By Lemma 4.3, contains as an induced subdigraph and is connected by Lemma 4.1. Let be such that . As , there must be a vertex that is adjacent to in . We now observe that must contain a twin. Indeed, if we suppose that is reduced, then by Proposition 4.4, is either , or . However, since none of these digraphs are allowed to be contained as induced subdigraphs, by the requirements of the claim, we have a contradiction. Thus, is not reduced, and is the twin of a vertex in . (At this point, may look like Figure 6(a).)
But since is assumed to be reduced, there must be some vertex that distinguishes from in . Let us call this vertex and assume without loss of generality that is adjacent to . Then, consider and use the same argument as above to obtain that must be the twin of some vertex (since and are adjacent) in Label the final unlabeled vertex in with ; we then obtain Figure 6(b). Note that, again, by the same argument, contains and should thus contain a twin. But as is not adjacent to (since is the twin of in ), must be adjacent to and implicitly is the twin of in as well, (Figure 6(c)) and hence also in . Finally, if we consider the full structure (Figure 6(d)) it then follows that does not distinguish between and , which is a contradiction. ∎
Let us now consider digraphs that do contain , , or .
Lemma 4.6**.**
Suppose that is a digraph of order , that contains , , or . Moreover, suppose that has exactly one negative eigenvalue. Then is not reduced.
Proof.
First, suppose that is such that . Let be the fifth vertex in . We may assume that is not an isolated vertex, otherwise would not be reduced. We make the following observations from Proposition 4.4: cannot have valency 1, and if has valency , then the subdigraph of induced by and its neighbors is isomorphic to . It then follows that is connected to at least two out of every three vertices in . This, in turn, implies that has valency at least .
Suppose that has valency , let be non-adjacent to and let . Then , and thus it follows that and are twins. If has valency , one may apply the same argument twice to obtain that should be the twin of two distinct vertices in , which is impossible. Hence, has valency and is not reduced.
Next, suppose that Then one may write
[TABLE]
where (), , and Since a positive determinant implies an even (and thus larger than ) number of negative eigenvalues, it follows that . Note that the proof of Proposition 4.4 gives us all possible solutions for . We remove those with , since is not allowed. Besides , we obtain , , , . The solution makes the fifth vertex an isolated vertex, whereas each of the other solutions makes it a twin of one of the four other vertices.
The proof for is analogous to the above and therefore omitted.
∎
Using Lemma 4.6, we are able to extend its claim to arbitrarily large .
Lemma 4.7**.**
Suppose that is a digraph of order , that contains , or . Moreover, suppose that has exactly one negative eigenvalue. Then is not reduced.
Proof.
By contradiction. Suppose that is reduced and let be such that is either , or . Fix some such that induces a weakly connected subdigraph . By Lemma 4.6, there is a vertex in , say , such that and are twins in If we let then
Because is assumed to be reduced, there is some vertex that distinguishes from Let and and observe that, by Lemma 4.6, is twin to a member of and in and , respectively. Now, let be a vertex in with valency in . Note that regardless of the choice of , such a vertex exists for each of the cases for . Moreover, the relation to (in-neighbor, out-neighbor, undirected neighbor or no neighbor) is different for each of the four vertices in . This implies that the relation of to determines which of the vertices in is the twin of in .
Suppose that the twin of in is . Then the twin of in is , because and have the same relation to . But then does not distinguish from , since it is not adjacent to either of the two. This means that the twin of in must be a member of . Specifically, it is the same vertex as the twin of in , as a consequence of the unique relation to . But this twin does not distinguish from (as ), so neither does , and we have our final contradiction. Thus, is not reduced. ∎
By Lemmas 4.5 through 4.7, we now have all the necessary tools to prove Theorem 4.2.
Proof.
(Of Theorem 4.2.) Suppose is a reduced digraph of order with rank larger than and exactly one negative eigenvalue. Since any digraph of order at most has rank at most , it follows that Next, we distinguish two cases: either contains at least one of , , or it does not. In the former case, we have by Lemmas 4.6 and 4.7 that otherwise we would lose the reducedness of . In the latter case, the same conclusion follows unless by Lemma 4.5.
We are thus left with two possibilities: either or . If , we have that , by Corollary 4.3.1. Finally, if then by Proposition 4.4 it holds that ∎
We are, in fact, able to conclude much more. Using that the numbers of positive and negative eigenvalues do not change when twin reduction is applied, the results of Theorem 4.2 extend to the underlying structure of any digraph with a single negative eigenvalue. This key observation is formalized in Theorem 4.8.
Theorem 4.8**.**
Let be a digraph of order , rank larger than , with exactly one negative eigenvalue. Then one of the following cases is true.
- (i)
* and either , or ,* 2. (ii)
* and .*
Proof.
Let . Then is reduced and has exactly one negative eigenvalue, which by Theorem 4.2 implies that The claim clearly follows. ∎
In particular, we observe that if one is given a spectrum that contains three positive, one negative, and arbitrarily many zero eigenvalues, one could say with certainty that the underying structure of the corresponding digraph is exactly . In other words, this digraph is a twin expansion of Inspired by this fact, the authors were convinced that many SHDS digraphs were within reach. Consider, for example, a spectrum of the form
[TABLE]
It is now not hard to show that this spectrum belongs to by considering , and Theorem 4.8. In the next section, we show a more general result, based on these principles.
As mentioned before, any rank digraph trivially has precisely one negative eigenvalue. For completeness, we recall that a digraph has rank 2 if and only if is either , , or , where in the latter case it must additionally be required that contains an odd number of arcs. For more detail, the interested reader is referred to the full characterization in [23].
5 An infinite family of connected SHDS digraphs
We have so far restricted ourselves to reduced digraphs, as digraphs that admit to this assumption are in a sense the fundamental structure to the digraphs that may be obtained by introducing twin vertices. In this section, we will be relaxing said assumption and consider twin expansions of the digraphs from Theorem 4.2, to further inquire into the class of digraphs with exactly one negative eigenvalue. In particular, we use that there is exactly one of those digraphs with rank four, to arrive at a remarkable conclusion.
The main result of this section is Theorem 5.3, in which we show that any digraph , where is an expansion vector, is strongly determined by its Hermitian spectrum. In other words, we obtain an infinite family of digraphs that is SHDS, which includes the first such connected infinite family.
As was briefly touched on in the introduction, such digraphs are incredibly rare, as there is an extreme degree of similarity within the collection of Hermitian adjacency matrices of a given order, informally speaking. Even when one just considers a digraph and its converse, which are clearly cospectral but in general not isomorphic, and hence in general not SHDS. Indeed, one is easily convinced that any SHDS digraph is necessarily self-converse, which is by itself an extremely rare property. While no formal proof has been found in the literature, one is easily convinced that the claim ”the fraction of self-converse digraphs of order goes to zero as goes to infinity” should be true. Indeed, this conclusion seems certain if one numerically evaluates the counting polynomials by Harary [18] and Harary and Palmer [19], which count the number of digraphs and the number of self-converse digraphs on vertices, respectively. This evaluation is included in Appendix A.
First, we observe that, regretfully, there are still many twin expansions of that we may not determine uniquely from their spectra, as the following example illustrates.
Example 3. Let and Then and are -cospectral by Lemma 3.4. However, they are clearly not isomorphic, as is visible in Figure 7: contains digons, whereas contains only .
Example 5 clearly illustrates the main obstacle in this part of our quest to construct SHDS digraphs; expansion vectors that are closely related, but which do not quite yield isomorphic digraphs when used to expand . Specifically, permutations of the same expansion vector yield cospectral, but not necessarily isomorphic, digraphs. This observation is formalized in the following lemmas.
Lemma 5.1**.**
Let , and let be a permutation of If and , then and are cospectral.
Proof.
Immediate from Lemma 3.4. ∎
Lemma 5.2**.**
Let and let be such that fewer than three entries of are equal. Then there exists a , obtained as a permutation of such that if and , then .
Proof.
By contradiction. Recall that and are digons in , and that two digraphs may only be isomorphic if they contain equal numbers of digons. Let , and assume that no three ’s are equal. (.) Let and . Suppose that all three construct isomorphic expansions. Then by counting digons
[TABLE]
and thus , , , or , which is a contradiction. ∎
From Lemma 5.2, we find the following necessary condition for SHDS expansions of .
Corollary 5.2.1**.**
, , , is SHDS only if at least three entries of are equal.
Intuitively, one might also say that if the condition in Corollary 5.2.1 is satisfied, then any permutation of would yield digraphs that are isomorphic to one another. In other words, that the condition above is not just necessary, but also sufficient. In the theorem below, which the authors consider to be the main contribution of this paper, we show exactly that.
Theorem 5.3**.**
Let be an expansion vector. Then is SHDS if and only if , where and is a permutation of , .
Proof.
Necessity was addressed in Corollary 5.2.1, so we will only show sufficiency. Let be a digraph and suppose that . If we can show , the claim is true. Since has rank 4 with three positive eigenvalues, we know by Theorem 4.8 that for some expansion vector . Moreover, by Corollary 3.4.1, has eigenvalue with multiplicity 2. Since , the eigenvectors corresponding to eigenvalue are orthogonal to the nullspace of . Therefore, the eigenvalues are constant on the four nontrivial parts of and 0 on the isolated vertices in . That is, the eigenvectors corresponding to satisfy
[TABLE]
Recall that Hermitian matrices are diagonalizable [3] and that diagonalizable matrices have geometric multiplicities equal to their algebraic multiplicities. Hence, there are two independent eigenvectors and that correspond to and satisfy (7). Moreover, any linear combination of and is again an eigenvector for . Hence, there is an eigenvector that is obtained as such a linear combination, which is zero for any one of the nontrivial parts of . Suppose that at least one of is nonzero and
[TABLE]
where
[TABLE]
for blocks of appropriate dimensions. Working out the latter equality in (8), we obtain
\savestack\systemA\setstackEOL
,\setstackTAB \left\{\ensurestackMath{\tabbedCenterstack[r]{{b_{2}}x_{2}+{ib_{3}}x_{3}-{ib_{4}}x_{4}=0,{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}4x_{2}}{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}+}{-ib_{3}}x_{3}+{ib_{4}}x_{4}={t_{1}}x_{2},{ib_{2}}x_{2}{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}+}{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}0x_{3}}+x_{4}={t_{1}}x_{3},{-ib_{2}}x_{2}+{b_{3}}x_{3}{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}+}{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}0x_{4}}={t_{1}}x_{4}}}\right. \systemA
from which it follows that , and . Moreover, at most one of can be zero and therefore at least two of are equal to . But similarly, this implies that at least two among each three of equals , and thus at least three of are equal to Thus, since the last four numbers may be permuted to any order without changing the spectrum, we may assume without loss of generality that .
Finally, by comparing the number of edges in the underlying graph, we obtain , and since , we have . Hence, if then and thus . ∎
Thus, by Theorem 5.3, we may be certain that we may uniquely determine each digraph whose Hermitian spectrum is of the form
[TABLE]
6 Closing remarks regarding WHDS digraphs
While most of this paper has been concerned with strong determination by the Hermitian spectrum, we conclude with some remarks on its weaker counterpart. It stands to reason that the digraphs that are not SHDS due to the problem illustrated in Example 5 might be WHDS, as most of the machinery that was used to show Theorem 5.3 is still in place. We first observe that it is in general not true that any expansion of is WHDS, after which we prove an analogue to Proposition 2.5.
As mentioned before, relaxing the connectivity assumption may yield cospectral digraphs with distinct underlying graphs, which implies that said digraphs are not switching equivalent. Recall the following example by Wang et al. [27].
Example 4. [27] Let and , for some . Then (plug in (3)) regardless of . Since and do not contain an equal number of isolated vertices, they cannot be switching equivalent. Thus, there are infinitely many pairs of cospectral mates. Most notably, suppose that . Then is connected, but not switching equivalent to , while they are cospectral. Thus, is not WHDS, even when .
As we have seen throughout this paper, there are many parallels between and . This has caused us to believe that a similar phenomemon occurs for the negative tetrahedron. As we will see shortly, there are indeed pairs of expansion vectors for such that and are both connected and have the same nonzero eigenvalues, but not the same number of vertices, which allows us to formulate an analogue of Proposition 2.6.
Proposition 6.1**.**
There are infinitely many rank digraphs with exactly one negative eigenvalue that are not WHDS.
The correctness of Proposition 6.1 follows immediately from the following example.
Example 5. Let and let and . Then and both have characteristic polynomial
[TABLE]
Moreover, as the two contain distinct numbers of isolated vertices, they are clearly not switching equivalent.
However, it should also be noted that such examples have proven to be extremely rare. If one enumerates all vectors with777Note that this does not restrict the investigation, as we are not interested in switching equivalent pairs. and computes the characteristic polynomials corresponding to via Lemma 3.4, we find as few as five characteristic polynomials whose nonzero roots do not occur uniquely. Out of these five polynomials, as in (9) is of smallest order; the corresponding digraphs contain at least 107 vertices. For the remaining digraphs, we obtain strong evidence that they are WHDS. In particular, we may thus conclude the following.
Proposition 6.2**.**
Any digraph of order less than that has rank and exactly one negative eigenvalue is WHDS.
Proof.
It is clear that any pair of digraphs , with and equal nonzero eigenvalues would have occurred in the performed enumeration. Since they did not, and the number of isolated vertices that may be added is bounded by the assumption on the order of the digraphs, the claim follows. ∎
Proposition 6.3**.**
Let with , . Then is WHDS.
Proof.
Let with , , and where contains at least one element larger than 100. Then
[TABLE]
Thus, if there were expansion vectors that would have yielded the same characteristic polynomial coefficients as , they would have occurred in the performed enumeration. ∎
Moreover, if we assume connectivity, we obtain a result similar to Proposition 2.5.
Proposition 6.4**.**
Any two connected, cospectral rank digraphs with exactly one negative eigenvalue are switching equivalent.
Proof.
Let and be connected, cospectral, rank 4 digraphs with exactly one negative eigenvalue. By Theorem 4.8, . Let be such that and . (Note that is implied.)
We first observe that may be obtained as a permutation of . Indeed, consider equations (10a)-(10d), which should all hold simultaneously by Lemma 3.4. (Note that (10a) is implied by .) Then, if these elementary symmetric polynomials in and are the same, is indeed distinct from by at most a reordering of its entries.
[TABLE]
Now, observe that any permutation of might be obtained by a sequence of pairwise interchanges of elements. Thus, if we can show that every such pairwise exchange yields a switching equivalent digraph, we are done. Suppose that is obtained from by interchanging elements and , and let and . We distinguish two cases: either is a digon in or it is not. Note that we may assume without loss of generality that in the former case, and in the latter.
Suppose that . Then , and thus is switching equivalent to . Finally, suppose that and let . If we let be the digraph determined by the Hermitian , then . Since was obtained from by a four-way switching, is again switching equivalent to , which completes the proof. ∎
Appendix A Fraction of self-converse digraphs
In the interest of completeness, we include the following numerical evidence, that illustrates the scarceness of SHDS digraphs, which necessarily must be self-converse.
Conjecture A.1**.**
Let be the number of self-converse digraphs on vertices, and let be the number of digraphs on vertices. Then:
[TABLE]
Numerical evidence for Conjecture A.1. Define to be , where and are the counting polynomials that count respectively the number of self-converse digraphs on vertices and the number of digraphs on vertices, as defined in [19, 18]. Conjecture A.1 claims that as . Since explicit formulas are available, one may numerically evaluate them; the results are shown in Table 1. By the shear speed at which the fraction of self-converse graphs converges to zero, one is easily convinced that the claim is true.
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