The minimum rank problem for circulants
The minimum rank problem is to determine for a graph the smallest rank of a Hermitian (or real symmetric) matrix whose off-diagonal zero-nonzero pattern is that of the adjacency matrix of . Here is taken to be a circulant graph, and only circulant matrices are considered. The resulting graph parameter is termed the minimum circulant rank of the graph. This value is determined for every circulant graph in which a vertex neighborhood forms a consecutive set, and in this case is shown to coincide with the usual minimum rank. Under the additional restriction to positive semidefinite matrices, the resulting parameter is shown to be equal to the smallest number of dimensions in which the graph has an orthogonal representation with a certain symmetry property, and also to the smallest number of terms appearing among a certain family of polynomials determined by the graph. This value is then determined when the number of vertices is prime. The analogous parameter over is also investigated.
Key words and phrases:circulant graphs; circulant matrices; minimum rank problem; minimum semidefinite rank
2010 Mathematics Subject Classification:Primary: 05C50. Secondary: 15A03.
The location of the off-diagonal nonzero entries of a Hermitian or real symmetric matrix can naturally be specified by a graph. More formally, we have the following.
Let be an Hermitian matrix and be a simple graph on vertices, say with . We say that is the graph of if it is the case that if and only if , for all with .
A problem of interest in combinatorial matrix theory is to determine particular ways in which the graph of a matrix constrains its rank. Because the diagonal entries of the matrix play no role in Definition 1.1, every graph allows a diagonally dominant matrix, so the question of how large the rank may be is not interesting. On the other hand, to determine the smallest rank among all matrices with a given graph is an interesting problem, known as the minimum rank problem for graphs. More formally, the problem is to determine the value of the graph parameter defined as follows.
Let be a graph. We write for the set of all complex Hermitian matrices with graph .
Let be a graph. The minimum rank of is
The present work focuses on the case in which is a circulant graph. We may then consider the smallest rank among all Hermitian (or real symmetric) circulant matrices whose off-diagonal nonzero entries occur according to the edges of .
A question that naturally arises is: When is a circulant graph, under what conditions is the smallest rank among all Hermitian (or real symmetric) matrices with graph attained by a circulant matrix? In Section 5 we show that this in fact does occur for at least one broad class of circulant graphs, namely those in which each vertex neighborhood comprises a consecutive set of vertices. We also give examples of circulants for which this does not occur, however the problem of providing a complete characterization of such circulants remains open.
We also investigate the problem in the positive semidefinite setting. First, in Section 3, we show that the problem of determining the smallest rank among all positive semidefinite circulant matrices with a given graph is equivalent to determining the smallest number of dimensions admitting an orthogonal representation for the graph with a specific symmetry property. Then, in Section 4, this problem in turn is shown to be equivalent to determining the smallest number of terms in a real polynomial with nonnegative coefficients whose zeros intersect a precise subset, determined by the graph, of the complex roots of unity. In Section 5, this value is determined for two broad classes of circulants. Finally, Section 6 develops analogous results over .
We begin with some fundamental definitions. In particular, we need to set out what is meant by a circulant, in both the sense of a graph and of a matrix, and then establish appropriate connections between the two notions.
2.1. Circulant graphs
Intuitively, is a circulant graph precisely when its vertices may be arranged around a circle such that the presence of an edge between any two vertices is determined entirely by the distance “around the circle” from one to the other. More precisely, we have the following definition.
A graph on vertices is said to be a circulant graph if its vertices may be labeled as such that there exists a set with
When , we write for the neighborhood of , i.e., the set of all vertices adjacent to . Note that when is a circulant graph, its entire edge set is determined by the neighborhood of any single vertex. Hence, it is convenient to specify a circulant graph by giving its number of vertices together with the neighborhood of just one vertex.
Let be a positive integer and be closed under taking the additive inverse modulo . By we denote the unique circulant graph on vertex set such that . For the sake of convenience, when writing the elements of a specific set , we do not restrict ourselves to integers in , but may instead write other integers that represent the same residues modulo .
Note that, by definition, is necessarily regular of degree .
A graph is a consecutive circulant if there exist integers and with such that for .
Note that Definition 2.3 precludes a graph with no edges; we do not consider the empty graph to be a consecutive circulant.
The circulant graphs and are shown in Figure 2.1. The former is a consecutive circulant.
2.2. Circulant matrices
In what follows, the rows and columns of every matrix, as well as the coordinates of every vector, are taken to be zero-indexed. That is, the coordinates of each -dimensional vector, as well as the rows and the columns of each matrix will be indexed by the integers . We write for the entry residing in row and column of matrix .
Accordingly, given subsets and of the set , we write for the submatrix of that lies on the rows of with indices in and the columns with indices in . When , we denote this submatrix simply by .
An matrix is a circulant matrix if there exist values for such that
Note that is a circulant matrix if and only if the value of is a function of the residue of modulo . For a general reference on circulant matrices and their properties, see the monograph  or the more recent .
In any context in which is understood to be a positive integer, we write for the complex th root of unity .
For every positive integer , the Fourier matrix is defined by . At times we wish to neglect the scaling factor of , and so for convenience of notation we let . Explicitly, we have
Note that is the Vandermonde matrix of the complex th roots of unity. Note also that , so that is unitary. The significance of the Fourier matrix in the context of circulant matrices stems from the fact that diagonalizes every circulant matrix.
Theorem 2.8 (see, e.g., [9, Theorem 3.2.2]).
If is an circulant matrix, then is diagonal.
2.3. Minimum circulant rank
As established above, the term “circulant” has a distinct meaning as applied to a matrix as opposed to a graph. When that distinction is clear from the context, however, we will sometimes refer to an object simply as a “circulant” for the sake of brevity.
The classical minimum rank problem seeks the smallest rank among all Hermitian (or real symmetric) matrices with the zero-nonzero pattern specified by a given graph. We wish to consider this problem with its scope restricted to the circulant matrices. Hence, for a given circulant graph, we seek the smallest rank of a Hermitian or real symmetric circulant matrix whose zero-nonzero pattern is that given by the graph. That is, we wish to study the following graph parameter analogous to that set out in Definition 1.3.
Let be a circulant graph. The minimum circulant rank of is
The following easy observation shows that is well-defined.
A graph is a circulant graph if and only if the adjacency matrix of can be taken to be a circulant matrix.
Occasionally, we will consider some previously-defined minimum rank parameter over a specific field other than . To be precise, we establish the following notation.
When denotes a field, we write as a superscript to denote an analogous minimum rank parameter defined such that the matrices in question are taken to be symmetric with entries in . For instance, denotes the minimum rank over all real symmetric circulant matrices with graph .
We refer to the problem of determining the value of for a graph as the circulant minimum rank problem. The following question thus arises naturally.
What conditions on a circulant graph are sufficient to ensure that ?
2.4. The positive semidefinite case
A variant of the minimum rank problem which has been well-studied (see, e.g., , , [13, Section 46.3] and ) is that in which only positive semidefinite matrices are considered. For convenience in this context, we set up the following notation, analogous to Notation 1.2.
Let be a graph. We write for the set of all positive semidefinite Hermitian matrices with graph .
Let be a graph on vertices. The minimum semidefinite rank of is
We wish to consider the effect of restricting our attention to circulants in the positive semidefinite setting as well. Hence, we define the following graph parameter.
Let be a circulant graph. The minimum semidefinite circulant rank of is
The following observation shows that this parameter is well-defined.
Given a circulant graph on vertices, let be its adjacency matrix. Since is a circulant, it is -regular for some . In particular, the graph Laplacian matrix of , namely , is well-known to be positive semidefinite of rank at most , and is clearly a circulant matrix. Hence, is defined and .
Consider the graph , shown in Figure 2.1. In particular, note that is the complement of the -cycle. The real symmetric matrix
has graph and is a circulant matrix with rank . Meanwhile, from [4, Theorem 14] we have that . Hence, for this graph we have
Note that is not positive semidefinite; its nonzero eigenvalues are (with multiplicity 2) and . In fact, it is shown in Example 6.21 that .
The analog of Question 2.12 is again of interest. In particular, for which circulant graphs do and differ? Example 6.21 gives one such graph, showing that the two notions are in fact distinct. More broadly, certain inequalities between the various minimum rank parameters are inherent from the sets over which their respective minima are defined. These inequalities are illustrated in Figure 2.2, which is annotated with references showing separation between pairs of parameters, when available.
Although Definition 2.15 may seem restrictive, Theorem 5.7 will show that there is at least one natural class of circulant graphs for which and coincide, meaning that the smallest rank among all Hermitian matrices with graph is realized by a positive semidefinite circulant matrix. Of course, as the following observation notes, all of the minimum rank parameters mentioned here do coincide for the complete graph.
The matrix with every entry equal to witnesses that
3. Orthogonal representations and symmetry
A critical tool in studying the minimum semidefinite rank of a graph is the notion of an orthogonal representation of the graph. This is an assignment of a single vector to each vertex of the graph such that the adjacency relation of the graph is captured by the orthogonality relation on the corresponding vectors. More precisely, we make the following definition.
Let be a graph and be a field. An orthogonal representation for in is a function such that, whenever are distinct, if and only if . We also refer to such a function as an orthogonal representation for over . The rank of the orthogonal representation is the dimension of the subspace spanned by .
Less formally, given a graph with vertices, we say that a sequence of vectors forms an orthogonal representation for when some correspondence between the vectors and the vertices of the graph gives an orthogonal representation.
The following theorem gives the significance of the notion of an orthogonal representation in the context of the minimum rank problem. Although this result is well-known, we include a proof for the sake of completeness, and because it serves as a prototype for the proofs of some analogous results to follow.
Theorem 3.2 (well-known).
Let be a graph. The following are equivalent.
There exists a positive semidefinite Hermitian matrix with rank and graph .
There exists an orthogonal representation for over with rank .
Let be an positive semidefinite Hermitian matrix with rank and graph . Then for some matrix of rank by [14, Theorem 7.2.7]. Since has graph , the columns of form an orthogonal representation for with rank .
Conversely, let be an orthogonal representation for over with rank . Then the Gram matrix of the vectors in the image of is a positive semidefinite Hermitian matrix of rank by [14, Theorem 7.2.10]. Since those vectors form an orthogonal representation for , this matrix has graph . ∎
Note that the proof of Theorem 3.2 actually shows that if a graph has an orthogonal representation over with rank , then has an orthogonal representation in . Combining this observation with Theorem 3.2 gives the following corollary.
Let be a graph and let be the smallest integer such that has an orthogonal representation in . Then .
Hence, the problem of determining for a particular graph is equivalent to determining the smallest such that has an orthogonal representation in . An entirely similar argument shows that is identical with the smallest such that has an orthogonal representation in .
The first major goal of the present work is to show that the connection established by Theorem 3.2 has a natural analog for the minimum semidefinite circulant rank of Definition 2.15. In particular, the restriction to circulant matrices corresponds to a requirement that the orthogonal representations considered possess symmetry in the sense set out precisely as follows.
Let be a graph with . An orthogonal representation for in is said to be cyclically symmetric if there exists some unitary matrix with and some such that for each .
The following example serves to illustrate Definition 3.5, and represents essentially the same construction used by Lovász for [18, Theorem 2]. The discussion of Example 6.19 details how the theory of Section 6 was applied to construct this example.
Consider the -cycle, . Figure 3.1 shows five vectors in that form an orthogonal representation for this graph; the endpoints of the vectors are at a distance from the -plane such that and meet at an angle of precisely when . Explicitly, these vectors are given by
for . In particular, each is the image of under a rotation about the -axis by an angle of , where the subscripts are computed modulo . Hence, taking to be any of the five vectors and to be the unitary matrix whose action on is the aforementioned rotation shows that Definition 3.5 is satisfied. That is, the orthogonal representation depicted is cyclically symmetric.
The following proposition shows that when Definition 3.5 is met by an orthogonal representation for in , the associated unitary matrix can be taken to be real orthogonal.
Let be a graph with . If is a cyclically symmetric orthogonal representation for , then there exists some real orthogonal matrix such that and some such that for each .
Suppose is a cyclically symmetric orthogonal representation for . Then, by Definition 3.5, there exists some and some unitary matrix such that and for each . In particular, letting be the -span of , we have that is invariant under . Since is a unitary transformation, the action of on this subspace of induces a real isometry. Hence, there is some real orthogonal matrix such that for every , so that, in particular, for every . ∎
A fact that will be important to what follows is that, whenever a cyclically symmetric orthogonal representation exists, one can be found in a certain very simple canonical form; with a view toward expressing this form, we introduce the following notation.
When is understood to be a positive integer, we write for the diagonal matrix whose th diagonal entry is , i.e., . Note that .
For any integer ,
so that, for any vector ,
and hence, for any integer ,
Unsurprisingly, the matrix has a close connection with the Fourier matrix given in Definition 2.7. The following lemma gives one view of this connection.
Given , let . Then the Gram matrix of the vectors is .
Let and note that . It follows from (3.1) that column of is given by for each . Hence, letting be the Gram matrix of the vectors , we have that is the inner product of the th and th columns of for every . Thus,
Given any , the vectors by definition form a cyclically symmetric orthogonal representation for the graph of their Gram matrix. Lemma 3.10 shows how the eigenvalues of this matrix arise directly from the coordinates of . A very useful consequence is that the rank of the Gram matrix, and hence of the associated orthogonal representation, becomes transparent.
If has support of size , then the vectors span a -dimensional subspace.
By Lemma 3.10, the Gram matrix of the vectors has eigenvalues , and exactly of these are nonzero. ∎
Lemma 3.11 will be especially useful once we have shown that cyclically symmetric orthogonal representations by vectors of the form are essentially the only ones which must be considered. This fact is anticipated by the following result, which uses Theorem 2.8 to give Lemma 3.10 a natural interpretation in terms of circulant matrices.
An positive semidefinite Hermitian matrix is a circulant matrix if and only if it is the Gram matrix of the vectors for some nonnegative .
Let be an positive semidefinite Hermitian matrix. Suppose first that is a circulant matrix. Then, by Theorem 2.8,
where are the eigenvalues of . Since these are nonnegative, we may take . Then is nonnegative and, by Lemma 3.10, is the Gram matrix of the vectors .
Now suppose is the Gram matrix of the vectors for some nonnegative . Then is positive semidefinite by [14, Theorem 7.2.10]. It remains to show only that is a circulant matrix, and this follows from the fact that, since ,
for every with . ∎
The following theorem serves to establish the canonical form for a cyclically symmetric orthogonal representation that was alluded to earlier. However, its primary significance is seen by analogy with Theorem 3.2. Just as that theorem showed the existence of a positive semidefinite matrix with a given graph to be equivalent to the existence of an orthogonal representation for that graph with the same rank, the following result shows that a positive semidefinite circulant matrix with a given graph exists precisely when there is a corresponding cyclically symmetric orthogonal representation for that graph with the same rank.
Let . The following are equivalent.
There exists a positive semidefinite Hermitian circulant matrix with rank and graph .
There exists a cyclically symmetric orthogonal representation for over with rank .
There exists a nonnegative with support of size such that is an orthogonal representation for .
Suppose first that (2) holds. Then there exist a vector and a unitary matrix with such that gives an orthogonal representation for of rank . Let be the Gram matrix of the vectors . As in the proof of Theorem 3.2, is a positive semidefinite Hermitian matrix with rank and graph . By an argument identical to that used in the proof of Proposition 3.12, is also a circulant matrix. Hence, (1) holds.
Finally, assume that (3) holds, i.e., that there exists some nonnegative with support of size such that gives an orthogonal representation for . Let be the vector formed from the nonzero coordinates of and let be the unitary matrix that results from deleting the th row and column from precisely when , i.e., let
Using Observation 3.9, we have that for every ,
As noted in Observation 2.16, whenever is a circulant graph, there exists a positive semidefinite Hermitian circulant matrix with graph . By Theorem 3.13, then, every circulant graph has a cyclically symmetric orthogonal representation over . It is also clear that no such orthogonal representation may exist unless is a circulant graph, so that, although Definition 3.5 made no stipulations on the graph, we have the following.
A graph is a circulant graph if and only if there exists a cyclically symmetric orthogonal representation for over .
Let be a circulant graph and let be the smallest integer such that has a cyclically symmetric orthogonal representation in . Then .
Hence, given a circulant graph , the problem of finding a positive semidefinite matrix with graph of smallest rank that is a circulant matrix is equivalent to the problem of finding an orthogonal representation for in the smallest number of dimensions that is cyclically symmetric.
4. Connection with polynomials
Let be a circulant graph on vertices. We next show that finding a cyclically symmetric orthogonal representation for over is equivalent to finding a polynomial satisfying certain combinatorial conditions determined by . These conditions are entirely in terms of the values of the polynomial on the complex th roots of unity. We may therefore bound the degree of the polynomials we consider.
Let and . Then there exists a unique polynomial with such that for all .
Letting be the sequence of real numbers with for such that
Then, since for any , the desired property holds. Uniqueness follows since the values of are prescribed for distinct values of . ∎
We ultimately wish to show that an appropriate polynomial gives rise to an orthogonal representation for by vectors in . To this end, we establish a correspondence between polynomials and vectors.
We write for the subset of comprising those polynomials with every coefficient nonnegative.
Let . Lemma 4.1 gives the existence of a unique with degree at most whose value coincides with that of on every complex th root of unity. Say
with for . Then the normalized coefficient vector of is the vector
Given a nonnegative vector , the polynomial corresponding to v is
Definitions 4.3 and 4.4 together give a bijective correspondence between polynomials in of degree at most and nonnegative vectors in . The motivation for setting up this correspondence is made clear by the following lemma.
Suppose and is its normalized coefficient vector. Then, for every ,
Suppose and is its normalized coefficient vector. Then is a cyclically symmetric orthogonal representation for if and only if satisfies
Note that is an orthogonal representation for if and only if
for every , where the final equality holds by Lemma 4.5. ∎
Let . The following are equivalent.
There exists a positive semidefinite Hermitian circulant matrix with rank and graph .
There exists a cyclically symmetric orthogonal representation for over with rank .
There exists a polynomial whose normalized coefficient vector has support of size such that
There exists a polynomial of degree at most with exactly terms such that the condition given in (4.2) holds.
Consider the 4-cycle, . In the notation of Theorem 4.7, , so that . The polynomial has while and are nonzero. Hence, the condition given in (4.2) is met, so that Theorem 4.7 applies. Thus, as and has terms, there must exist a cyclically symmetric orthogonal representation for the -cycle of rank , so that . In particular, as the normalized coefficient vector of is , the vectors
form a cyclically symmetric orthogonal representation for by Proposition 4.6. Since has support of size , by Lemma 3.11 the rank of this representation should be , which in fact is visibly the case. Finally, it is easy to verify that , so that in fact
Hence, the -cycle is an example of a graph for which there is a positive semidefinite circulant matrix achieving the minimum rank, and hence the minimum semidefinite rank as well.
As Example 4.8 illustrates, Proposition 4.6 allows the construction of a cyclically symmetric orthogonal representation for a given circulant graph on vertices in terms of a polynomial with nonnegative coefficients that vanishes on a corresponding selection of the complex th roots of unity. Such a polynomial can be taken with degree at most , and then the rank of the resulting representation is simply the number of terms appearing in .
This naturally leads to an interest in the following question. Given precisely which complex th roots of unity are zeros of a certain polynomial with nonnegative coefficients, how few terms may appear in that polynomial? In addressing this question, a useful upper bound is provided by the following lemma, the proof of which rests on a fundamental result of convex geometry.
Let be a self-conjugate set of complex th roots of unity with . Then there exists a polynomial with and at most terms such that for all .
Assume without loss of generality that and take such that
For each , let
|if , and otherwise let|
Every element of is a root of , and it follows that . In particular, is in the convex hull of the vectors. It follows by Carathéodory’s Theorem (see, e.g., [5, Theorem 2.3]) that there exist nonnegative such that with . Hence, taking
gives a polynomial as desired. ∎
Note that standard proofs of Carathéodory’s Theorem are both elementary and constructive. Hence, under the hypotheses of Lemma 4.9, the polynomial that is asserted to exist can in fact be calculated in a finite number of steps.
When the goal is to apply Theorem 4.7, the serious limitation of Lemma 4.9 is that nothing about the result or its proof provides any guarantee as to which complex th roots of unity are not zeros of the promised polynomial. Under some circumstances, however, this limitation can be overcome, as we will see in the next section.
5. Minimum circulant rank for particular classes of circulants
In general, determination of the minimum semidefinite rank for a particular graph is a difficult problem; the necessary upper and lower bounds may both be difficult to obtain.
A remarkable upper bound that holds in general was proved by probabilistic methods in [16, 17] and gives the following connection between and the vertex connectivity of , namely the smallest number of vertices whose deletion from leaves a disconnected graph, which we denote by .
Theorem 5.1 ([16, Corollary 1.4]).
For every graph on vertices, .
In particular, let . Since deleting all neighbors of a single vertex is sufficient to disconnect the graph, certainly . When equality holds, Theorem 5.1 gives . Interestingly, results of  show that in fact does hold under certain conditions, including both when is a consecutive circulant (see Definition 2.3) and when is prime. Therefore, we have the following.
Let . If is a consecutive circulant or is prime, then .
Also appearing in  is the so-called Delta Conjecture, attributed to Maehara, which asserts that in fact for every graph on vertices with minimum degree . Since is regular of degree , the truth of this conjecture would imply that the bound of Theorem 5.2 in fact holds for all circulant graphs.
In what follows, we strengthen the result of Theorem 5.2 by showing first that can be replaced with , and then that, moreover, this modification actually gives equality. In particular, we prove this under the hypothesis that is a consecutive circulant in Subsection 5.1, and under the hypothesis that is prime in Subsection 5.2. Moreover, our methods of proof are constructive, so that, when the hypothesis of Theorem 5.2 holds, a matrix achieving the minimum of can be found in a finite number of steps.
5.1. Computing for consecutive circulants
Before we proceed, a remark about lower bounds is also in order. In particular, a combinatorial graph parameter called the zero forcing number of , introduced in  and denoted by , provides a lower bound for the minimum rank over any field.
Theorem 5.3 ([12, Proposition 2.4]).
Let be a graph on vertices and let be a field. Then .
For consecutive circulant graphs, the zero forcing number behaves in a predictable way. In particular, the following is straightforward to show.
If is a consecutive circulant, then .
for any consecutive circulant graph , so that, in particular, equality holds throughout, i.e., . Theorem 5.7, which follows, strengthens this result by showing that as well. Hence, for a consecutive circulant graph, the minimum rank over can always be achieved by a positive semidefinite circulant matrix. (Example 6.1 will show that this does not hold over , however.)
Since our goal is to show the existence of an appropriate circulant matrix, we turn our attention to constructing polynomials of the kind required to invoke Theorem 4.7. Since these must be polynomials with nonnegative coefficients, we will find the following result, which we quote verbatim from , to be very useful.
Theorem 5.5 ([2, Theorem 1.1]).
Let be a polynomial of degree , , with nonnegative coefficients and zeros . For write
Then if , all of the coefficients of are positive.
Taking as the initial polynomial of Theorem 5.5, we obtain that any monic polynomial whose zeros are precisely for some with will have only positive coefficients. That observation is the essential content of the following lemma.
Whenever and are integers with , the polynomial has degree and positive real coefficients.
If is a consecutive circulant graph, then
If is a complete graph, then , and the result follows by Observation 2.18. Hence, assume is not a complete graph. Then