Rank of weighted digraphs with blocks

07/11/2018 ∙ by Ranveer Singh, et al. ∙ Technion 0

Let G be a digraph and r(G) be its rank. Many interesting results on the rank of an undirected graph appear in the literature, but not much information about the rank of a digraph is available. In this article, we study the rank of a digraph using the ranks of its blocks. In particular, we define classes of digraphs, namely r_2-digraph, and r_0-digraph, for which the rank can be exactly determined in terms of the ranks of subdigraphs of the blocks. Furthermore, the rank of directed trees, simple biblock graphs, and some simple block graphs are studied.

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1 Introduction

A well-known problem, proposed in 1957 by Collatz and Sinogowitz, is to characterize graphs with positive nullity [19]. Nullity of graphs is applicable in various branches of science, in particular, quantum chemistry, Hückel molecular orbital theory [10], [13] and social network theory [14]. For more detail on the applications see [10]. Many significant results on the nullity of undirected graphs are available in the literature, see [4, 7, 11, 12, 15, 12, 2, 3, 20].

Recently in [9, 8, 6] nullity of undirected graphs was studied using cut-vertices and connected components. The results are used to calculate the nullity of line graphs of undirected graphs. Apart from the connected components, it turns out that the blocks are also interesting subgraphs of a graph, which can be utilized to know its determinant [17, 16]. This motivates us to utilize blocks to determine the nullity or rank (for a square matrix of order nullity is equal to ) of digraphs.

We construct some interesting classes of digraphs, in particular -digraph and -digraph, where the rank can be determined by the ranks of subdigraphs of the blocks. Furthermore, the ranks of directed trees, simple biblock graphs, and block graphs are determined.

1.1 Notations and Preliminaries

A digraph consists of a vertex set and an arc set . An arc is called a loop at the vertex . A weighted digraph is a digraph equipped with a weight function . If then, the digraph is called a null graph. A subdigraph of is a digraph such that and . A subdigraph is an induced subdigraph of if and implies .

A simple graph consists of a vertex set and edge set where each edge is an unordered pair of vertices, With each directed graph we associate an underlying simple graph, with the same vertex set, and an edge (undirected) between two distinct vertices and if only if or is an arc of . A cut-vertex in a simple graph is a vertex whose removal increases the number of connected components. A block in a simple graph is a maximal connected induced subgraph with no cut-vertex. A vertex in a digraph is a cut-vertex if it is a cut-vertex of its underlying simple graph. Similarly a subdigraph is a block of the digraph if it corresponds to a block in the underlying simple graph. A graph or digraph with no cut-vertex is called nonseparable. A block of a digraph is pendant block if it contains at most one cut-vertex of . In Figure 0(a) a digraph with seven blocks is presented. These blocks are the induced subdigraphs on the vertex-sets respectively.

For every weighted digraph there corresponds a matrix with for every (in particular, is the weight of the loop at if it exists), and otherwise.

Let denote the column space of a matrix , that is, the space of all linear combinations of the columns of . Similarly, denotes the row space of .

Appending a row (column) to a matrix keeps its rank unchanged if the row (column) is in the row space (column space) of , otherwise, the rank is increased by 1. Thus, increasing the size of a square matrix by one increases its rank by at most 2. It leads to the following observation.

Observation 1.

Let be a digraph with a cut-vertex . Let be a nonempty induced subdigraph which includes such that there is no arc or , where and . Then one of the following three cases can occur.
CASE I: .
CASE II: .
CASE III: .

Some typical examples of simple graphs and weighted digraphs for the above three cases are the following:

Example 1.

CASE I:

  1. Simpe graph: Consider a nonsingular simple bipartite graph of order . For example, a simple bipartite graph with a unique perfect matching is nonsingular. Note that

    has to be even as the eigenvalues of a bipartite graph are symmetric with respect to 0, thus if

    is odd then the bipartite graph is singular. By Cauchy’s interlacing eiegnvalues property, if we remove any vertex

    from then the resulting bipartite graph will be singular having exactly one zero eigenvalue. Hence, That is,

  2. Digraph: Consider a digraph on two vertices without loops. Let the arcs have nonzero weights respectively. Then .

CASE II:

  1. Simple graph: Consider a simple complete bipartite graph . The nullity of is . For , if then If we remove a vertex from , the rank will still be 2. That is,

  2. Digraph: Consider a digraph on two vertices , with loop on but not on . Let has only one arc . Then

CASE III:

  1. Simple graph: A simple complete graph is nonsingular. Thus if then removal of any vertex results in a decrease of the rank by 1. That is,

  2. Digraph: Consider a digraph on two vertices , without loops, and exactly one arc . Then

In Section 2 we discuss CASE I and we define a new family of digraphs, -digraphs, to obtain results on ranks of -digraphs. In Section 3 we discuss CASE II, we define a new family of digraphs, -digraphs, to obtain results on their ranks. In Section 4 we discuss CASE III, and give some partial results, which includes results on ranks of block graphs discussed in [18].

Theorem 1.1.

Consider the following square matrix

where is a scalar, and

are column vectors of order

, and is a square matrix of order . Then

  1. if and only if .

  2. if and only if .

  3. if and only if one of the following hold

    1. and .

    2. and

    3. and but and .

Proof.
  1. Equality holds in the right inequality if and only if and in the left inequality if and only if . Similarly, if and only if and Thus if and . Conversely, if and then which implies .

  2. if and only if And if and only if . That is, if and only if and . Similarly, if and only if and . Hence, if and only if ,

  3. If then . If then . That is, if and then . Similarly, if and If and , then . Which implies that if , , then

Remark 1.2.

By part 1 of Lemma 1.1, for some if and only if for every .

(a)

(b)

(c)
Figure 1: The digraph in (a) is extented to an -digraph in (b) by adding -edges at the cut-vertices. (c) An -tree digraph. In this figure the undirected edges are simple edges, which are equivalent to two opposite arcs of same weight.

2 Case I

Theorem 2.1.

Let be a digraph with a cut-vertex . Let be a nonempty induced subdigraph which includes such that there is no arc or , where and . If , then

Proof.

With suitable reordering of the vertices in , we can write,

where the first row and the first column correspond to the cut-vertex , is the weight of the loop at , and

denotes the zero matrix of suitable order. Clearly the rank of the matrix

is . Consider the submatrix

The rank of can be at most . By the hypothesis and using Theorem 1.1(1), . Thus the first row of is not in the row space of the rest of the matrix. Hence the rank of is . Now, consider . Its rank can be at most . Again using Theorem 1.1(1) and by the hypothesis, the column vector , which implies that the first column of is not in the column space of . Hence

Which proves the result. ∎

2.1 Tree digraph

A directed edge or arc from a vertex to a vertex of weight is denoted by . A simple weighted edge of weight between and is an edge, where , and no loops on and . In general an edge between the vertices in a weighted directed graph is a set of two arcs, . We now define a few more types of edges for the purpose of our study. Let be a cut-vertex and be a noncut-vertex of the digraph . An -edge between and is an edge, where ( are nonzero), and there is no loop on the noncut-vertex . Similarly, an -arc between and is an arc either from to or to and no loop on . Analogously, an -edge between and is an edge, where ( are nonzero), and there is a loop on . Finally, an -arc between and is an arc either from to or to and a loop on . Matching in a digraph is a set of vertex disjoint pairs of arcs on the same vertices.

For an undirected tree having maximum matching of size , the rank is equal to [5]. We show that this result is also true for some categories of tree digraphs. Let denote a tree digraph obtained from an undirected tree by replacing its each simple edge by arcs of arbitrary nonzero weights We will call such a tree digraph a loopless bi-arc tree. If loops are allowed only at cut-vertices then we call as a cut-loop bi-arc tree. That is, in a cut-loop bi-arc tree, any noncut vertex does not have a loop while any cut-vertex may or may not have a loop. We consider a class of tree digraphs having loops and single arcs on the vertices in the next subsection.

Theorem 2.2.

Let be a loopless bi-arc tree having the maximum matching of size . Then .

Proof.

We denote by any bi-arc tree with vertices. We will prove the result by using induction on . For , clearly the result is true. Assume that the result is true for every , . Consider a tree . Let be a pendant edge of the cut-vertex. Let be the components (bi-arc trees of order less than ) of having maximum matching of size , respectively. Adding to the maximal matchings of the edge yields a matching in of size This is a maximal matching in since any matching in consists of at most one edge incident with and matchings in and thus has size of at most By the induction hypothesis . Using Theorem 2.1, It proves the result. ∎

Lemma 2.3.

Consider vertices of a digraph . Then if and only if for every subset of ,

Proof.

Without loss of generality let us consider a subset of . Assume that but . Note that as deletion of vertex from results in deletion of a row and a column in corresponding matrix hence the rank can be decrease at most by 2. Thus implies . If we increase the size of by adding vertices , that is . Then further there can be at most reduction in the rank. That is now Which is a contradiction, hence the result follows. ∎

Let be a digraph with cut-vertices and blocks having cut-vertices of respectively. Then denotes the induced subgraph of on the noncut-vertices, . And denotes a digraph obtained after removing of cut-vertices from .

Definition 1.

A block induced pendant subdigraph of is a maximal subdigraph of any block of such that it has exactly one cut vertex of .

Theorem 2.4.

If then

Proof.

Let be subdigraph of such that is obtained from by deleting exactly one cut vertex of for . In for . Using Lemma 2.3, for removal of cut-vertices from decrease its rank by . Hence in any pendant subdigraph with the cut-vertex of satisfies Applying Theorem 2.1 for a block induced pendant subdigraph of each , the result follows. ∎

Definition 2.

-block: An -block of a digraph is a pendant block , with the cut-vertex such that,

Definition 3.

-digraph: A digraph is called -digraph if at each cut-vertex of the digraph there is an -block.

Notice that any digraph can be extended to an -digraph by adding -blocks at all the cut-vertices. Moreover, an -digraph can be extended to a higher order -digraph by coalescing -blocks at arbitrary vertices. Thus a nonseparable digraph can also be converted to an -digraph. The smallest -block in a digraph could be an -edge. A digraph in Figure 0(a) is extended to an -digraph in Figure 0(b) by attaching -edges at the cut-vertices.

Theorem 2.5.

Let be an -digraph having blocks, and cut-vertices. Then

Proof.

The proof follows by using Theorem 2.1 for each -block of . ∎

Corollary 2.6.

Adding loops or weights on the loops at the cut-vertices does not change the rank of an -digraph.

Proof.

By Remark 1.2, the weights of the loops at the cut-vertices did not play any role in Theorem 2.1 and Theorem 2.5, and hence the result follows. ∎

Theorem 2.7.

Let be an -digraph with cut-vertices. Consider digraphs . Let an -edge be added between one arbitrary vertex of and one arbitrary cut-vertex of for . Let be the resulting digraph. Then

Proof.

The proof follows by Theorem 2.1 for each -block of . ∎

Corollary 2.8.

Let be an -digraph. On adding edges or arcs to its cut-vertices, the rank increases by the number of -edges or -arcs. Thus the rank is unchanged by the addition of simple edges at its cut-vertices.

Proof.

The proof directly follows by Theorem 2.7. ∎

We will now see some more categories of tree digraphs.

Definition 4.

-Tree Digraph: Let be a cutloop bi-arc tree. If at each cut-vertex of there exists an -edge then it is called an -tree digraph.

Corollary 2.9.

Let denote an -tree digraph with -edges or -arcs. If is the maximum matching in , then

Proof.

It follows by Theorem 2.2 and Corollary 2.8. ∎

(a)

(b)
Figure 2: (a) -block graph. (b) -biblock graph.

A simple graph is called a block graph if each of its blocks is a simple complete graph. When each block of a simple graph is simple complete bipartite graph then it is called a biblock graph. A block graph is called an -block graph if there exists a simple pendant edge at each of its cut-vertices. An example of an -block graph is given in Figure 1(a). Similarly, A biblock graph is called an -biblock graph if there exists a simple pendant edge at each of its cut-vertices. An example of an -biblock graph is given in Figure 1(b).

Corollary 2.10.

[18](Theorem 3.5) Let be an -block graph with cut-vertices and blocks. Choose pendant edges, one at each cut-vertex. If each of the remaining blocks has at least two noncut-vertices, then is nonsingular.

Corollary 2.11.

Let be an -biblock graph with cut-vertices and blocks. Choose pendant edges, one at each cut-vertex. If each of the remaining blocks has at least two noncut-vertices in different partition, then

Proof.

Let be the blocks which are pendant edges selected one from each cut-vertex of . Let be the rest of the blocks in . Using Theorem 2.5

Note that any is complete bipartite graph having at least 2 vertices, thus it has rank 2. Hence,

This completes the proof. ∎

3 Case Ii

Let be any subdigraph of a digraph . Let be a cut-vertex of with a loop of weight . By , we denote the column vector consisting of the weights of all the incoming edges from to . Similarly, by , we denote the row vector consisting of weights of all the outgoing edges from to .

Theorem 3.1.

Let be a digraph with a cut-vertex . Let be a nonempty induced subdigraph which includes such that there is no arc or , where and . If , and any of the following happens,

  1. or is independent of .

Then

Proof.

With suitable relabelling of the vertices, we can write

Notice that by the hypothesis the row vector . Similarly, the column vector .

Using elementary operations, can be transformed to the matrix

Clearly the result follows for . When or then for any value of ,

Hence the result follows. ∎

Definition 5.

Let be a block in a digraph. If for every cut-vertex in then is called an block.

Definition 6.

Let be a digraph having blocks. If at least blocks are blocks, then is called -digraphs.

Theorem 3.2.

Let be a graph having no loops at the cut-vertices. If is -digraph, then .

Proof.

We prove the result by induction on the number of blocks. For the result is trivial. Suppose the result holds for . Consider a digraph having blocks. Since has at least two pendant blocks, there exists a pendant block, without loss of generality which is an -block with the cut-vertex . Then using Theorem 3.1

Since, has blocks, all of them, except for at most one, -blocks, by the induction hypothesis

Hence which proves the result. ∎

Corollary 3.3.

Let be a biblock graph having blocks. If each block has at least two noncut-vertices in different partition sets, then

Proof.

Note that such a biblock graph is an -digraph. Hence the result follows from the Theorem 3.2. ∎

4 Case Iii

Let be any subdigraph of a digraph having a cut-vertex with a loop of weight . By we denote the column vector consisting of the weight of the loop at and weights of all the incoming edges from to . Similarly, by we denote the column vector consisting of the weight of the loop at and the weights of all the outgoing edges from to .

Theorem 4.1.

Let be a digraph with a cut-vertex . Let be a nonempty induced subdigraph which includes such that there is no arc or , where and . If , then

  1. If ,

  2. If ,

Proof.

With suitable relabelling of the vertices, we can write

  1. , but .

    In this case, the column vector , while . Using elementary operations can be transformed to the matrix

    Notice that the matrix

    has rank , and its rows are not in the row space of the matrix

    If then the rank of above matrix is otherwise . Hence the result follows.

  2. , but .

    In this case, the row , while . Using elementary operations can be transformed to the matrix

    Similar to the Case I, if then else

This proves the result. ∎

Theorem 4.2.

Let be a digraph with a cut-vertex . Let be a nonempty induced subdigraph which includes such that there is no arc or , where and . If , and if and but and Then