1. Introduction
Let G be a simple undirected graph, having no loops, no multiple edges and no weighted edges. The vertex set of G is . The monic characteristic polynomial of G in the variable x, denoted by , is the determinant of , where is the identity matrix and is the adjacency matrix of G. The roots of are the eigenvalues of G. Since is a real and symmetric matrix, the eigenvalues of G are real numbers. An eigenvector of an eigenvalue of G is a nonzero vector satisfying . The eigenspace associated with an eigenvalue of G, denoted by , is the vector space containing the zero vector together with every possible eigenvector of . For any vertex v, the graph is the graph obtained from G by removing v and the edges incident to v.
In 1973, the polynomial reconstruction problem was posed by Cvetković at the XVIII International Scientific Colloquium held in Ilmenau [
1,
2]. It asked the following question: Is it true that for
, the characteristic polynomial
of a simple graph
G on
n vertices is determined uniquely by the polynomial deck
of characteristic polynomials of vertex deleted subgraphs of
G? The problem, which was also posed by Schwenk [
3] independently of Cvetković, is still open, although it was answered in the affirmative for several classes of graphs. Examples of such graph classes include the class containing graphs whose eigenvalues are bounded below by
; for these graphs, the answer is ‘yes’ for both the connected [
4] and the disconnected cases [
5]. Other graph classes for which the problem has an affirmative answer are trees [
6], unicyclic graphs [
7] and regular graphs [
8]. Further results on the reconstruction of
from
were also put forward for graphs having terminal vertices [
9], for bipartite graphs [
8] and for disconnected graphs [
10].
In this paper, we show that has enough information for us to deduce whether or not shares more than half of its roots (where multiple roots are counted as many times as their respective multiplicities) with one of the polynomials in . If more than half of the eigenvalues of G are indeed shared by those of one of its vertex deleted subgraphs , say, then is shown to be reconstructible from , even though these common eigenvalues in G and are not known prior to the completion of this reconstruction.
To achieve our goals, we make use of graph walks. A walk in
G of length
k is a sequence
where
are vertices of
G,
are edges of
G, and each edge
in this sequence connects vertices
and
(neither the vertices nor the edges in this sequence are necessarily distinct). The walk is closed if
. We consider
n walk matrices of
G,
, each containing walk enumerations of
G that end at vertex
in
G, respectively. Even though as far as this paper is concerned, these matrices are not themselves reconstructible from
, if for some
, the greatest common divisor of
and
is of degree more than
, then we shall be able to deduce enough information about
for us to accomplish our task of reconstructing
.
The proof of the main result of this paper is in
Section 5, specifically in Theorem 13. Before we present this proof, the next three sections contain results for walk matrices, for companion matrices, and for eigenvalues of both the graph
G and for any of its vertex deleted subgraphs.
Section 6 illustrates our techniques by successfully reconstructing the characteristic polynomial of two example graphs from their respective polynomial decks. The final section contains a remarkable consequence of our main result, Theorem 14, stating that the polynomial reconstruction problem is settled in the affirmative for a large subclass of disconnected graphs. This result further strengthens a result first obtained by Sciriha and Formosa in [
10].
2. Walk Matrices
In the literature, a walk matrix
is a matrix of the form
where
is a 0–1 vector. Usually,
is taken to be
, the vector of all ones [
11,
12,
13,
14], but there are exceptions [
15,
16,
17,
18]. For every
i and
j, the entry in the
ith row and
jth column of
is equal to the number of walks of length
that start from vertex
i and end at any vertex in
S, where
S is the subset of
indicated by the entries in
that are equal to 1. It is known (see [
15,
19]) that, for any indicator vector
and any number of columns
k of
we choose the matrix to have, there is a number
r such that the rank of
is
k for all
and is
r for all
. For this reason,
is either assumed to have
r columns [
11,
15,
19], or
n columns [
12,
16,
18].
In this paper, we consider walk matrices , , where is the standard n dimensional vector basis for . To slightly simplify the notation, henceforth we denote the matrix by . Moreover, for each vertex v, the number of columns of is the minimum possible number of columns such that its rank is maximized; in other words, the number of columns of is the number r described in the previous paragraph. Of course, for distinct vertices u and v, the ranks of and may differ. Thus, the n matrices may not have the same number of columns, but for each of them, the rank will stay the same if the number of columns is increased by any number, and will decrease by s if the number of columns is decreased by s, for any feasible s.
As in the References [
15,
16], for each walk matrix
, we consider the Gram matrix of its columns, which is the matrix
. It is well-known that
is a positive semidefinite matrix having the same rank as
[
20]. In this case, since, by definition,
has full rank,
is invertible and, hence, positive definite. Moreover,
so that, for all
j and
k, the
th entry of
is the number of closed walks of length
in
G that start and end at vertex
v. Note also that
has constant skew diagonals, so it is a Hankel matrix. We denote these walk enumerations by
or by
if the vertex
v in question is inferable from the context.
Thus, for any
t, the matrix
has the same rank as the matrix
and, as we said earlier, there is a number
r such that the rank of (
3) is equal to
r for all
and is equal to
t for all
. Thus, the determinant of (
2) is 0 for all
and is nonzero for all
, leading to the following result.
Theorem 1. Let be the number of closed walks in G of length that start and end at vertex v. Then there exists a number r such that the determinant ofis zero for all and is nonzero for all . Moreover, the rank of is r. Note also that, referring to (
1), each of the upper
submatrices of
are invertible. This may be deduced from the fact that
is positive definite. Alternatively, any upper
submatrix of the
matrix
is equal to
, where
is
restricted to its first
p columns, and this matrix has a full rank by Theorem 1.
3. Companion Matrices and Eigenvalues
The rank of a walk matrix
may also be evaluated by finding the number of eigenvalues of
G having an associated eigenvector that is not orthogonal to
[
17,
18]. In our case, since the walk matrices under discussion have
, where
, we can equivalently say the following.
Theorem 2 ([
17]).
The rank of is the number of eigenvalues of G with an associated eigenvector having a nonzero vth entry. The significance of zero and nonzero entries in eigenvectors has found applications in control theory [
17,
18,
21] and in molecular conduction [
2,
22,
23].
For any walk matrix
, its companion matrix
is the matrix satisfying
For any vertex
, we denote the companion matrix of
by
. As we shall see soon,
may be different for each
. Indeed, if
is an
matrix, then
is the
matrix
for an appropriate column vector
. As can be seen if we evaluate the determinant of
using the Laplace determinant expansion along the last column, the characteristic polynomial of
is
where the vector
is
. It is known that the coefficients of
are integers [
16].
Moreover, and more importantly,
divides the characteristic polynomial of
G [
15,
19]. Here, we provide an alternative proof of this result by stating the roots of
.
Theorem 3. Let be the companion matrix of such that and let be the characteristic polynomial of . The roots of are the eigenvalues of G whose eigenspaces contain an eigenvector with a nonzero entry in its vth position.
Proof. Let
be any eigenvalue of
with associated eigenvector
. We take the transpose on both sides of the relation
to obtain
. By postmultiplying both sides of this equality by
, we get
But
Hence
if and only if
has a zero entry in its
vth position. Thus, referring to (
4) and (
5), whenever an eigenvector
associated with
in
has a nonzero entry in its
vth position, the (nonzero) vector
would be an eigenvector associated with
in
. Since the choice of the eigenvalue
in
was arbitrary, we have proved the result. □
By combining Theorems 2 and 3 together, the following corollary is determined immediately.
Corollary 1. The rank of is equal to the degree of .
The following theorem conveys the fact that if
r is the rank of
, then the
companion matrix
may be found solely from walk enumerations of closed walks of length up to
that start and end at vertex
v in
G. The argument of Theorem 4 below is based on the result in ([
24] p. 43) stating that the last column of
is equal to
.
Theorem 4. The companion matrix of is the matrix , whereand are the number of closed walks of length that start and end at vertex v. Proof. By comparing the last columns of the equality
, we obtain the relationship
. Thus,
, which may be rewritten as
or as
The proof is complete since
is invertible. □
The reference ([
15] Theorem 2.3) contains a result akin to Theorem 4 that instead extracts each entry of
one at a time from the walk enumerations
.
We emphasize what we have accomplished in Theorem 4 by presenting the following corollary.
Corollary 2. For any vertex v, let the characteristic polynomial of be . If is of degree r, then is deducible from .
4. Eigenvalues of Vertex Deleted Subgraphs
Recall that the vertex deleted subgraph of G is obtained from G by removing vertex v and all edges incident to it. The common eigenvalues of and G are also able to tell us the rank of . The reason for this is presented in the proof of Theorem 7 below.
Before proving Theorem 7, we first prove the following result that allows us to deduce the common eigenvalues of G and from the eigenspaces of G. The result of Theorem 5 is required for the proof of Theorem 7.
Theorem 5 ([
17]).
Let λ be an eigenvalue of G with eigenspace . Then λ is also an eigenvalue of if and only if contains an eigenvector whose vth entry is zero. Proof. Let the adjacency matrices of
G and
be
and
, respectively. After reordering the vertices of
G if necessary,
may be partitioned into the block matrix
for some indicator vector
of the adjacencies of vertex
v. We also have
for some eigenvector
in
.
Suppose
is an eigenvalue of both
G and of
. We prove that
contains an eigenvector with a zero in its
vth position. Let
for some eigenvector
pertaining to the eigenvalue
in the eigenspace of
. Let
be partitioned into
, where
is
-dimensional. Then
may be rewritten as
from which we obtain, in particular, the relation
By premultiplying both sides of (
6) by
, we obtain
which simplifies to
Hence, either
or
. If
, then we have proved the result, since then
. If
, then we claim that
; indeed
Thus, either way,
has an eigenvector in
with a zero entry in its
vth position, proving sufficiency.
Conversely, suppose
is an eigenvalue of
G with an associated eigenvector having a zero in its
vth position. We prove that
is also an eigenvalue of
. Let
be partitioned into
. Since
,
Thus, in particular,
, proving that
is also an eigenvalue of
. □
We arrive at yet another way of determining the rank of
, which is illustrated in Theorem 7. The proof of this result uses the so-called Interlacing theorem (see [
25], for instance), presented below for the case of eigenvalues of graphs.
Theorem 6 (Interlacing theorem [
25]).
Let G be a graph on n vertices having the n eigenvalues and let the vertex deleted subgraph have the eigenvalues . Thenthat is, the eigenvalues of G and interlace. As an immediate corollary of Theorem 6, the multiplicity of any eigenvalue of a graph G must be at most one more than the multiplicity of the same eigenvalue in the vertex deleted subgraph . The statement of Theorem 7 below concerns eigenvalues of G whose multiplicities are exactly one more than those for the same eigenvalues in . In its proof argument, an eigenvalue is assumed to have multiplicity zero if it is not an eigenvalue of its adjacency matrix.
Theorem 7. Let λ be an eigenvalue of G whose multiplicity is one more than the multiplicity of λ in the vertex deleted subgraph . The rank of is the number of all such distinct eigenvalues of G.
Proof. Let have multiplicity q in G and let be an eigenbasis for in G. We assume that all of the eigenvectors have their vth entry equal to zero. If this is not the case, so that, without loss of generality, both and have a nonzero ith entry, then replace by .
By the Interlacing theorem (Theorem 6), the multiplicity of in cannot be less than . Let be the vector without its vth entry. By the argument presented in the proof of Theorem 5, all of the eigenvectors will also be eigenvectors of in , and clearly these eigenvectors are linearly independent. If also has its vth entry equal to zero, then will also be an eigenvector of in that is linearly independent of , so that the multiplicity of in would be at least q. However, then that would mean that any linear combination of the vectors in the eigenbasis of in G would have a zero in its vth position, which, by Theorem 2, would lead to not contributing to the rank of . Thus, the eigenvalues that contribute to the rank of are precisely those whose multiplicity is one more than that of the same eigenvalue in , as required. □
A vertex
v whose removal from
G reduces the multiplicity of an eigenvalue
in
G by one is called a
-core vertex [
26] or a downer vertex [
27]. Thus, Theorem 7 may be restated as follows:
Theorem 8. The rank of is the number of distinct eigenvalues of G for which v is a core/downer vertex.
The eigenvalues of G described in the statement of Theorem 7 or Theorem 8 are precisely the roots of . The following theorem proves this, and more.
Theorem 9. Let G be a graph and let v be any of its vertices. The r distinct roots of are the eigenvalues of G for which v is a core/downer vertex. Moreover, the remaining eigenvalues of G (including multiplicities) are also eigenvalues of .
Proof. Let be any root of . By Theorem 3, is an eigenvalue of G whose eigenspace contains an eigenvector with a nonzero number in its vth entry. As described in the first paragraph of the proof of Theorem 7, a basis for can be chosen such that only has a nonzero entry in its vth position. By the argument in the second paragraph of the proof of Theorem 7, the multiplicity of in must be , proving the first part of the theorem statement.
Now suppose is an eigenvalue of G that is not a root of . Then by Theorem 3, contains only vectors whose vth entry is zero. Thus, any eigenbasis for must be made up of such vectors as well. By again applying the argument used in Theorem 7, the multiplicity of in must be at least equal to that of in G, which proves the second part of the theorem statement. □
We end this section by summarizing the various ways described in this paper to obtain the rank of .
Theorem 10. The following are all equal to the rank of :
One less than the order of the smallest singular matrix whose skew diagonal entries are constant and equal to from left to right (Theorem 1);
The degree of (Corollary 1);
The number of distinct eigenvalues of G with an associated eigenvector having a nonzero entry in its vth position (Theorem 2);
The number of distinct eigenvalues of G for which v is a core/downer vertex (Theorem 7).
5. Reconstruction
Let be the polynomial deck containing the (unordered) characteristic polynomials of all vertex deleted subgraphs of G. As is well-known, the derivative of the characteristic polynomial of G may be reconstructed from as in the following result.
Theorem 11 ([
28,
29]).
The derivative of the characteristic polynomial of G is equal to . Thus, all the coefficients of may be reconstructed from , except possibly the constant one. Furthermore, if one of the members of has a multiple root, then by Theorem 6, this root must also be a root of , and is reconstructed immediately.
The number of closed walks of length
starting and ending at any vertex of
G are also reconstructible from
. This was proved in [
1]. Below, we slightly elaborate on the proof that is provided there.
Theorem 12 ([
1] Theorem 2).
For all , the walks may be reconstructed from . Proof. From ([
29] p. 34), the formal power series
may be generated by the following generating function:
Note that the rational function (
7) may be rewritten as
where the notation
represents the reflected polynomial of
, that is, the expression
, the polynomial
with its coefficients in reverse order [
30,
31]. Hence
Since the only unknown coefficient of
is the leading one, the numbers
may be discovered by expanding the left hand side of (
8) and comparing coefficients. □
We have finally conveyed all the results we require in order to prove the main result of this paper, which is presented in Theorem 13 underneath. It is assumed that, when counting the roots of any polynomial mentioned in the statement of Theorem 13, multiple roots are counted as many times as their multiplicity.
Theorem 13. Let be the polynomial deck containing the characteristic polynomials of all vertex deleted subgraphs of G. Then determines whether or not has more than half of its n roots included among the roots of one of its characteristic polynomials. When this is the case, is reconstructible from .
Proof. For each vertex
v, we first obtain the walk enumerations
using Theorem 12. Moreover, for each
v, we calculate the determinant
where
is either
or
, depending on the parity of
n. If one of these determinants is zero for some particular vertex
v, then by Theorem 1, the rank of
,
r, is the order of the largest nonsingular principal upper submatrix of this determinant, which is less than
t. Indeed, if this happens, then when
n is odd,
, so
, and when
n is even,
, so
. Either way, we infer that
r is less than half
n whenever (
9) is zero for some vertex
v.
For this particular vertex
v, we now use Theorem 4 to obtain the companion characteristic polynomial
from
. By Theorem 3, the
r roots of
are
r of the
n roots of
. Moreover, by Theorem 9, the remaining
roots of
will also be eigenvalues of
. However, since
,
. We have determined, therefore, that when the determinant (
9) is zero for some vertex
v, then
G must share more than half of its eigenvalues (if we include repetitions) with
. Since (
9) may be found from
, the fact that
G shares more than half of its eigenvalues with
is recognizable from
.
To reconstruct , we use the fact that divides . By Theorem 11, all the coefficients of bar the constant one K are reconstructible from . Since is available, we reconstruct by, for instance, performing the polynomial division and solving for K. □
Remark 1. Focusing on the last sentence of the proof of Theorem 13 for a moment, the eigenvalues shared by G and would be the roots of the polynomial .
6. Examples
We now illustrate the techniques in the proof of Theorem 13 by successfully reconstructing the characteristic polynomials of the following two example polynomial decks.
6.1. First Example
The polynomial deck contains the seven polynomials
All of have six distinct roots. The sum of these seven polynomials is ; thus, by integration, is the polynomial , according to Theorem 11. Thus, as soon as we discover the value of K, the reconstruction of would be complete.
The roots of are . We show that more than half of the seven roots of (in this case, more than three) are among these six roots.
We find
, the number of closed walks of length
starting and ending at vertex 7 in
G. By using Theorem 12, particularly (
8):
whence we obtain
. In line with Theorem 1, we calculate the determinant
Since this determinant is zero,
G must share at least four eigenvalues with
, by Theorem 13. The upper principal minor
is nonzero, so the rank of
is three, by Theorem 1. Hence
G shares exactly
eigenvalues with
. However, we do not know which are these four eigenvalues yet.
In accordance with Theorem 4, we evaluate the matrix product
obtaining
. Thus, the companion polynomial of
,
, is equal to
. This polynomial divides
, by Theorem 3. By performing polynomial division of
by
, say, we deduce that
.
We have thus successfully reconstructed as being . Note that its roots are . The three roots are those of , while the four remaining roots are shared with those of , confirming that G does share four of its eigenvalues with .
6.2. Second Example
The polynomial deck contains the eight polynomials
All of have seven distinct roots. The sum of these eight polynomials is ; thus, by Theorem 11, , after integrating this sum with respect to x. As before, we now endeavour to determine the value of K.
The roots of are . We show that more than half—that is, more than four—of the eight roots of are among these seven roots.
The number of closed walks of length
starting and ending at vertex 8 in
G may again be obtained by comparing the coefficients of both sides of two appropriate formal power series in agreement with (
8):
We obtain the values
. By showing that the determinant
is zero (Theorem 1) we deduce that
G must share at least five eigenvalues with
(Theorem 13). Note that
is not used here, because the number of vertices of
G is even. The upper principal minor
is nonzero—thus, the rank of
is three. This means that
G shares exactly
eigenvalues with
; however, these five eigenvalues are unknown, for now.
We now discover
, the companion polynomial of
. To this end, we work out the product
obtaining
. Hence, in line with Theorem 4, the companion polynomial of
is
. Since
divides
(Theorem 3), we deduce that
, and hence
.
The roots of are . The three roots are those of , while the five remaining roots are shared with those of .
7. Disconnected Graphs
Our main result, Theorem 13, reconstructs from by obtaining a companion polynomial after deducing that the degree of is smaller than half of . This companion polynomial, having integer coefficients, is a factor of .
It is known that if
G is a connected graph, then the largest eigenvalue of
is always a root of
for any
[
16]. Hence, Theorem 13 cannot be used for connected graphs whose largest eigenvalue has a minimal polynomial whose degree is at least half of
. This includes connected graphs whose characteristic polynomial is irreducible over
.
Note, however, that if
G is a disconnected graph having two components
and
, then
. If
, then Theorem 13 is always able to reconstruct
from
. If
G has more than two components or
is odd, then it will always be the case that one of these components has less than
vertices, allowing the use of Theorem 13 to reconstruct
from
. The details are in the proof of the following corollary, which was proved in [
10] using a different approach.
Corollary 3 ([
10]).
Let G be a disconnected graph on n vertices. If not all components of G has exactly vertices, then is reconstructible from . In particular, if G has more than two components or has an odd number of vertices, then is reconstructible from . Proof. Let v be a vertex of a component K of G having less than vertices. Then and G have more than common eigenvalues. By Theorem 13, this fact is inferable from , which leads to the reconstruction of from .
Clearly if a disconnected graph G has more than two components or has an odd number of vertices, then G must have such a component K. □
Note that if
G is known to be disconnected, then its characteristic polynomial can be reconstructed from
[
32]. However, we need to stress here that the result of Corollary 3 is true irrespective of whether
G is known to be disconnected. Only disconnected graphs having two components, each having
vertices, are left out by Corollary 3.
The results of this paper allow us to include more disconnected graphs than those described in Corollary 3, however. This corollary states that only disconnected graphs having two components with an equal number of vertices may possibly not have their characteristic polynomials reconstructed from their polynomial deck. Let G be one such disconnected graph on vertices, whose two components and have k vertices each. Clearly, every vertex deleted subgraph of G will have at least k eigenvalues in common with those of G. Thus, our main result, Theorem 13, will only be inapplicable to G if it so happens that every vertex deleted subgraph of G has exactly k eigenvalues in common with those of G.
However, by Theorem 5, if either of the components of G has an eigenvector associated with one of its eigenvalues with a zero entry in its vth position, then this eigenvalue will also be present in , and hence G will have more than half of its eigenvalues shared by those of . This will allow the reconstruction of from by applying Theorem 13.
Consequently, among all disconnected graphs, only disconnected graphs having two components
and
of equal order, where both
and
have no zeros in their eigenvectors, may possibly be counterexamples to the polynomial reconstruction conjecture. Such graph components with no zeros in their eigenvectors are called omnicontrollable in [
17].
We thus conclude our paper with the following remarkable result that strengthens Corollary 3.
Theorem 14. Let G be a disconnected graph with polynomial deck . Then is reconstructible from , except possibly if G has two components of equal order, both of which are omnicontrollable graphs.