1 Introduction
A wellknown 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 cutvertices 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 cutvertex 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 cutvertex. A vertex in a digraph is a cutvertex if it is a cutvertex 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 cutvertex is called nonseparable. A block of a digraph is pendant block if it contains at most one cutvertex of . In Figure 0(a) a digraph with seven blocks is presented. These blocks are the induced subdigraphs on the vertexsets 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 cutvertex . 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:

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

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,

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

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,

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
if and only if .

if and only if .

if and only if one of the following hold

and .

and

and but and .

Proof.

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 .

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 ,

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 .
2 Case I
Theorem 2.1.
Let be a digraph with a cutvertex . 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 cutvertex , 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 cutvertex and be a noncutvertex of the digraph . An edge between and is an edge, where ( are nonzero), and there is no loop on the noncutvertex . 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 biarc tree. If loops are allowed only at cutvertices then we call as a cutloop biarc tree. That is, in a cutloop biarc tree, any noncut vertex does not have a loop while any cutvertex 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 biarc tree having the maximum matching of size . Then .
Proof.
We denote by any biarc 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 cutvertex. Let be the components (biarc 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 cutvertices and blocks having cutvertices of respectively. Then denotes the induced subgraph of on the noncutvertices, . And denotes a digraph obtained after removing of cutvertices 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 cutvertices from decrease its rank by . Hence in any pendant subdigraph with the cutvertex 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 cutvertex such that,
Definition 3.
digraph: A digraph is called digraph if at each cutvertex of the digraph there is an block.
Notice that any digraph can be extended to an digraph by adding blocks at all the cutvertices. 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 cutvertices.
Theorem 2.5.
Let be an digraph having blocks, and cutvertices. 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 cutvertices does not change the rank of an digraph.
Proof.
Theorem 2.7.
Let be an digraph with cutvertices. Consider digraphs . Let an edge be added between one arbitrary vertex of and one arbitrary cutvertex 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 cutvertices, the rank increases by the number of edges or arcs. Thus the rank is unchanged by the addition of simple edges at its cutvertices.
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 biarc tree. If at each cutvertex 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
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 cutvertices. 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 cutvertices. 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 cutvertices and blocks. Choose pendant edges, one at each cutvertex. If each of the remaining blocks has at least two noncutvertices, then is nonsingular.
Corollary 2.11.
Let be an biblock graph with cutvertices and blocks. Choose pendant edges, one at each cutvertex. If each of the remaining blocks has at least two noncutvertices in different partition, then
Proof.
Let be the blocks which are pendant edges selected one from each cutvertex 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 cutvertex 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 cutvertex . Let be a nonempty induced subdigraph which includes such that there is no arc or , where and . If , and any of the following happens,


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 cutvertex 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 cutvertices. 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 cutvertex . 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 noncutvertices 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 cutvertex 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 cutvertex . Let be a nonempty induced subdigraph which includes such that there is no arc or , where and . If , then

If ,

If ,
Proof.
With suitable relabelling of the vertices, we can write

, 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.

, 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 cutvertex . Let be a nonempty induced subdigraph which includes such that there is no arc or , where and . If , and if and but and Then
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