On the inertia set of a signed graph with loops.

On the inertia set of a signed graph with loops.

Marina Arav, Hein van der Holst111Corresponding author, E-mail: hvanderholst@gsu.edu, John Sinkovic
Department of Mathematics and Statistics
Georgia State University
Atlanta, GA 30303, USA
Abstract

A signed graph is a pair , where is a graph (in which parallel edges and loops are permitted) with and . The edges in are called odd edges and the other edges of even. By we denote the set of all symmetric real matrices such that if , then there must be an even edge connecting and ; if , then there must be an odd edge connecting and ; and if , then either there must be an odd edge and an even edge connecting and , or there are no edges connecting and . (Here we allow .) For a symmetric real matrix , the partial inertia of is the pair , where and are the number of positive and negative eigenvalues of , respectively. If is a signed graph, we define the inertia set of as the set of the partial inertias of all matrices .

In this paper, we present a formula that allows us to obtain the minimal elements of the inertia set of in case has a -separation using the inertia sets of certain signed graphs associated to the -separation.

Introduction

A signed graph is a pair , where is a graph (in which parallel edges and loops are permitted) with and . The edges in are called odd edges and the other edges of even edges. By we denote the set of all symmetric real matrices such that

  • if , then there must be an even edge connecting and ,

  • if , then there must be an odd edge connecting and , and

  • if , then either there must be an odd edge and an even edge connecting and , or there are no edges connecting and .

Here we allow , in which case loops might occur at vertex . For example the matrix

belongs to , where is the signed graph shown in Figure 1.

Figure 1: Thick edges denote odd edges, while thin edges denote even edges.

For a symmetric real matrix , the partial inertia of , denoted by , is the pair , where and are the number of positive and negative eigenvalues of , respectively. If is a signed graph, we define the inertia set of as the set ; we denote the inertia set of by .

A separation of a graph is a pair of subgraphs of such that and ; its order is the cardinality of . If the order of a separation is , we also say that is a -separation. The notions of separations transfer without change to signed graphs.

Before presenting the formula, we need introduce some other notation. Let ; in this paper we include in the set . If for each , there exists an such that and , then we write . If and , then we write . Call a pair minimal if there is no pair with , and . Then if and only if every minimal pair in also belongs to and every minimal pair in also belongs to . By we denote the set .

Let be a -separation of a signed graph and let be the vertex in . For , let and be the signed graphs obtained from by adding at an even loop and an odd loop, respectively. For , let be the signed graph obtained from by deleting vertex . In this paper, we prove that the following formula holds:

(1)

If is a -separation of a graph , then Formula (1) is analogous to the formula for the inertia set of -sums of graphs. This formula was found by Barrett, Hall, and Loewy [1]. For a graph with (in which no parallel edges and loops are permitted), demotes the set of all symmetric real matrices such that , if and only if and are adjacent. The inertia set of is and is denoted by . If is a -separation of the graph and , then is also called a -sum of and at . If and , then denotes the subset of consisting of all pairs with . Barrett, Hall, and Loewy proved that .

An important lemma about the inertia set in the case of graphs is the Northeast Lemma. This lemma says that if has order and satisfies , then also for any satisfying and . We note that, contrary to the graphical case, in the case of signed graphs in which parallel edges and loops are permitted, the Northeast lemma does not always hold. For example, if is the signed graph with exactly one vertex and no edges, then , while .

The minimum rank of a graph is and is denoted by . The inertia set of a graph includes the minimum rank of , since . Mikkelson [5] gives a formula for the minimum rank of -sums of graphs that allow loops. In this case, a diagonal entry of a matrix corresponding to the graphs is zero or nonzero as to whether there is no loop or a loop at the corresponding vertex. See Fallat and Hogben [3] for a survey on the minimum ranks of graphs.

In this paper, we allow matrices to have zero rows or zero columns. For example, if is and is , then is . If is , then each vector belongs to . If , then denotes the matrix, while if , then denotes the identity matrix.

1 Arrows on symmetric matrices

If and are symmetric real matrices, we write

if there exists a real matrix such that . It is clear that if and , then . If for two symmetric real matrices and , and , then we write

The following lemma is a variant of Sylvester’s Law of Inertia (see, for example, Theorem 20.3 in [2]).

Lemma 1.

Let and be symmetric real matrices. If and has positive and negative eigenvalues, then has at most positive and at most negative eigenvalues. If , then and have the same number of positive and the same number of negative eigenvalues.

If and are sets of symmetric real matrices, we write

if for every matrix , there exists a matrix such that . If consists of a single matrix , then for we also write . We write

if and .

For a set of symmetric real matrices , we denote by the set . So .

Lemma 2.

Let and be sets of symmetric real matrices. If , then . Consequently, if , then .

Proof.

Let . Then there exists a matrix with positive and negative eigenvalues. Since , there exists a matrix such that . By Lemma 1, has at most positive and at most negative eigenvalues. Hence there exists a pair with . Thus, . ∎

2 Inequalities

Let and let

be symmetric and real matrices, respectively, where and are . Then the -subdirect sum of and (see [4]), which is denoted by , is the matrix

Let . If and are sets of symmetric and of symmetric real matrices, respectively, then the -subdirect sum of and , denoted by is the set of matrices

If , we write for , and if the set consists of a single symmetric real matrix , then we also write for .

Throughout the paper we denote by the matrix .

To prove Formula 1, it therefore suffices to prove that

(2)

We now prove some lemmas and show one direction of the arrows. In the next section, we will finish the proof.

Lemma 3.

Let . Let and be a symmetric and a symmetric real matrices, respectively. Then .

Proof.

We may write

and

where and are matrices. Let

Then

From the previous lemma, we immediately obtain the following lemma.

Lemma 4.

Let and be sets of symmetric real matrices and symmetric real matrices, respectively. Then

Lemma 5.

Let and let be a symmetric real matrix, where is a scalar. Then .

Proof.

To see that , let

Then

To see that , let

Then . ∎

If is a symmetric matrix and , then denotes the principal submatrix of obtained by removing the th row and column in . If is a set of symmetric matrices and , then denotes the set .

From Lemma 3 and the previous lemma, we obtain the following lemma.

Lemma 6.

Let and be positive integers, and let and be sets of symmetric real matrices and symmetric real matrices, respectively. Then

Lemma 7.

Let be a signed graph which has a -separation . Let be the vertex in . Then each of the following holds:

  1. ,

  2. ,

  3. , and

  4. .

Proof.

By Lemma 4,

and by Lemma 6,

We now show that

Let and . If , then we can find a positive scalar such that . (To see this, notice that if is sufficiently small, then has the same sign as , while if is sufficiently large, then has the same sign as .) If and , then has an odd loop at , and so has an odd loop at . Hence . The case where and is similar. If and , then has an odd loop at , and so has an odd loop at . Hence . If and , then has an even loop at , and so has an even loop. Hence . The cases where and either or are similar. If , then has an odd loop at and has an even loop at . Hence has an even and an odd loop at , and so . The proof that

is similar. ∎

3 The inertia set of a signed graph with a 1-separation

In this section we finish the proof that Formula (2) is correct. This finishes the proof that Formula (1) is correct. First we prove some lemmas.

Lemma 8.

Let and let be a symmetric real matrix, where is a scalar. Let . Then

Proof.

Let

Then is invertible and

hence the lemma follows. ∎

Lemma 9.

Let be a symmetric real matrix, where is a symmetric real matrix. If , then

Proof.

Let

Then

so

The other direction follows from Lemma 5. ∎

Lemma 10.

Let be a symmetric matrix, where is and is . Then at least one of the following holds:

  1. there exists a matrix such that

    or

  2. there exists a nonzero vector such that

Proof.

If for each , then there exists a matrix matrix such that

and so there exists an matrix such that . Let

Then

From this and Lemma 3, it follows that

Thus we may assume that there exists a vector with

nonzero. By Lemma 9,

which finishes the proof. ∎

Theorem 11.

Let be a symmetric real matrix, where is , is , and is a scalar. Then at least one of the following holds:

  1. there exists an such that

    or

  2. .

Proof.

By Lemma 10 there exists a vector such that

or there exists a nonzero scalar such that

By Lemma 8,

Lemma 12.

Let be a -separation of the signed graph , and let . If , then there exist matrices and such that .

Proof.

Except for the entries and , all other entries of and are determined by .

Suppose . Then has an odd edge at vertex . Hence or has an odd edge at vertex ; by symmetry, we may assume that has an odd edge at . If has an odd edge at , let . Otherwise, always. Then, we let . The case where is similar.

Suppose now that . If has no loops at , then both and have no loops at . Then . We now assume that has loops at . Then there is at least one even and at least one odd loop at . If has no even loops at , then has an even loop at . If has an odd loop at , then we let and . If has no odd loops at , then has an odd loop at . Then we let . The case where has no odd loops at is similar. So we may assume that and, by symmetry also , have an even and an odd loop. Then we let . ∎

Theorem 13.

Let be a signed graph which has a -separation . Let be the vertex in . Then

(3)
Proof.

By the previous section,

We now show that the converse direction also holds.

Let

Then, by Theorem 11, at least one of the following holds:

  1. .

  2. There exists a vector such that

Suppose first that holds. Then

Suppose now that holds. By Lemma 12, there exist matrices

such that . (So .) If , then

and

Hence . The cases where and are similar.

4 All terms are needed

We now exhibit several examples of signed graphs illustrating that each term in Formula (1) is needed.

To see that the term is needed in Formula (1), let be the signed graph where is a -path, all edges are odd, and none of the vertices has a loop. Let be the signed subgraph of consisting of one odd edge , and let be the signed subgraph of consisting of the other odd edge. Then Formula (1) shows that , while

To see that the term is needed in Formula (1), let be the signed graph consisting of three isolated vertices and no edges. Let and be distinct signed subgraphs, each consisting of two vertices. Then , while