Nonsingular (Vertex-Weighted) Block Graphs

05/06/2019 ∙ by Ranveer Singh, et al. ∙ Technion IIT Rajasthan 0

A graph G is nonsingular (singular) if its adjacency matrix A(G) is nonsingular (singular). In this article, we consider the nonsingularity of block graphs, i.e., graphs in which every block is a clique. Extending the problem, we characterize nonsingular vertex-weighted block graphs in terms of reduced vertex-weighted graphs resulting after successive deletion and contraction of pendant blocks. Special cases where nonsingularity of block graphs may be directly determined are discussed.

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1 Introduction

In 1957 Collatz and Sinogowitz proposed the problem of characterizing nonsingular graphs, i.e, graphs whose adjacency matrix is nonsingular [17]. This problem is of much interest in various branches of science, in particular quantum chemistry, Hückel molecular orbital theory [7, 10] and social networks theory [11]. Significant work was done towards a solution to this problem for special classes of undirected graphs, such as trees, unicyclic and bicyclic graphs [4, 6, 15, 8, 9, 16, 12, 1, 3, 19, 14, 13]. In particular, a tree is nonsingular if and only if it has a perfect matching [5]. Block graphs are a natural generalization of trees. A block in a graph is a maximal connected subgraph with no cut-vertex. A block graph is a graph in which each block is a clique (i.e., a complete subgraph), see [18, p. 15], [2]. In this article we study nonsingularity of block graphs.

It turns out that in order to characterize nonsingular block graphs, it is useful to consider vertex-weighted graph. A vertex-weighted graph is a pair where is a simple graph with vertex set , edge set , and

is a vector of vertex weights,

is the weight of vertex . A graph is the vertex-weighted block graph , where is the zero vector. The adjacency matrix of is given by

where is a diagonal matrix whose -th diagonal entry is . If is a vertex-weighted graph, and is a subgraph of , we denote by the restriction of the vector to the vertices of . We refer to as a subgraph of , and if is a component of we refer to as a component of .

A vertex-weighted block graph is nonsingular (singular) if is nonsingular (singular). In Section 2, we give a necessary and sufficient condition for a vertex-weighted block graph to be singular in terms of its reduced graphs resulting after successive contraction and deletion of pendant blocks. We then, in Section 3, present several families of nonsingular block graphs. In Section 4, we show that replacing edge blocks by paths of even order preserve nonsingularity/singularity.

The following terms and notations are used in the paper. A graph is a coalescence (at the vertex ) of two disjoint graphs and if it is attained by identifying a vertex and a vertex , merging the two vertices into a single vertex . We use

to denote an all-ones matrix, an all-ones column vector, a zero matrix, a zero column vector and a

-vector of suitable order, respectively. The standard basis vectors in are denoted by . A clique on vertices is denoted by . If is a subgraph of , then denotes the induced subgraph of on the vertex subset . If consists of a single vertex we will write for . The determinant of a graph is . For a nonzero , we use in this paper the following notation:

For a diagonal matrix with nonzero real diagonal entries, and are interpreted accordingly.

2 Characterizing nonsingular vertex-weighted block graphs

We start with a complete characterization of nonsingular vertex-weighted complete graphs, and some implications for vertex-weighted graphs that have a pendant block which is a clique. Note that elementary row and column operations do not change the rank of a matrix, and we use this fact in checking the singularity of . In particular, simultaneous permutations of rows and columns of do not change the rank, thus in checking whether a vertex-weighted block graph is singular or not we may relabel the vertices of , and reorder accordingly, as convenient.

Theorem 2.1.

Let .

  1. If exactly one of is equal to 1, then is nonsingular.

  2. If at least two of are equal to 1, then is singular.

  3. If , , let

    (1)

    then

    1. is nonsingular if and only if .

    2. if is singular, then for any vector such that , , and , the graph is nonsingular.

  4. If for and is nonsingular, then any matrix of the form

    can be transformed, using elementary row and column operations, to the following matrix

    where

    (2)
Proof.

Let . Then

  1. Without loss of generality, let . By subtracting the first row from the next rows we get that is row-equivalent to the matrix

    whose determinant is nonzero.

  2. In this case two rows (or columns) are equal.

    1. Denote

      (3)

      Then

      Thus is nonsingular if and only if

      is nonsingular. Since the eigenvalues of

      are and , the eigenvalues of the matrix are and . Thus , and , are nonsingular if and only if . As ,

      and is nonsingular if and only if .

    2. If , then if ,

      and is nonsingular by part 3(a); and if , is nonsingular by part 1 of the theorem.

    1. Let , . For every ,

      Therefore if ,

      Thus if is invertible, where , then by (3a) above,

      Hence

      Let

      Then

      where

    2. When exactly one of is equal to 1, we may assume without loss of generality that . If in

      we subtract the first column from column , and then the first row from row , we get the following matrix:

      where

Remark 1.

In part 4 of Theorem 2.1, note the following special cases for :

  1. If for every , and for at least one , then Hence in this case .

  2. If is a zero vector, is a matrix with all diagonal elements equal to and all off diagonal elements equal to In this case we get that

Remark 2.

Part 2 of Theorem 2.1 may be generalized: If a vertex-weighted block graph has a block such that for two non-cut-vertices , then is singular.

For a block of a vertex-weighted block graph , we denote by the sub-vector of consisting of the entries corresponding to the non-cut-vertices in . If for every non-cut-vertex in , we define

(5)

We simplify the notation to when no confusion may arise.

We now define two operations on using its pendant blocks.

Definition 1.
  1. PB-deletion. Let be a pendant block such that for every , and . A PB-deletion of is the operation of deleting all the vertices of and the corresponding entries of the weights vector , yielding a subgraph , where .

  2. PB-contraction. Let be a pendant block of with a cut-vertex , such that either exactly one entry in is 1, or for every and . A PB-contraction of is the operation of merging all the vertices of to the cut vertex , deleting the entries of from , and adding the weight to , where

Note that when is a vertex-weighted block graph, both PB-deletion and PB-contraction generate a vertex-weighted block graph. Also, PB-deletions may disconnect a connected vertex-weighted block graph, but PB-contractions preserve connectivity.

Lemma 2.2.

Let be a pendant block of such that for every non-cut-vertex in , and Let be obtained from by PB-deletion of . Then is singular if and only if is singular.

Proof.

Without loss of generality we may assume that the vertices of are , and is the cut-vertex. Then

where and . Any nonzero minor on the first rows and some columns, cannot have a zero column, cannot have more than one column of the form , and cannot consist of the first columns and a column of the form , since is singular. Thus every such nonzero minor includes the -th column, and any nonzero minor that does not include all the first columns has a zero complementary minor. Hence the Laplace expansion of along the first rows yields

(see also [16, Lemma 2.3].)

By part 3(b) of Theorem 2.1, is nonsingular. Thus is nonsingular if and only if is nonsingular. ∎

Lemma 2.3.

Let be a pendant block of such that either for of and , or exactly one entry in is 1. Let be obtained from by a PB-contraction of . Then is singular if and only if is singular.

Proof.

Without loss of generality we may assume that the vertices of are , and is the cut-vertex. Then

where . If either for every non-cut-vertex of and , or exactly one entry in is 1, the matrix is nonsingular by part 3(a) and part 1 of Theorem 2.1. By part 4 of that theorem, is similar to the matrix

where

Hence is nonsingular if and only if

is nonsingular. ∎

Remark 3.

Note that PB-deletion and PB-contraction may be used for any vertex-weighted graph which has a pendant block , where is a clique, and the proper conditions on are satisfied. Lemmas 2.2 and 2.3 hold in this case too.

Definition 2.

Reduced vertex-weighted block graph. A vertex-weighted block graph is a reduced vertex-weighted block graph of the vertex-weighted block if it is obtained from by a finite number of PB-deletions and PB-contractions.

Lemmas 2.2 and 2.3 imply that if is a reduced vertex-weighted block graph of , then is nonsingular if and only if is nonsingular. We can now prove the main theorem.

Theorem 2.4.

A vertex-weighted block graph is singular if and only if there exists a reduced vertex-weighted block graph that has one of the following:

  1. A component , where is a clique and for every vertex and

  2. A block for which at least two entries of are equal to 1.

Proof.

If is a reduced vertex-weighted block graph of , and satisfies 1 or 2, then is singular by part 3(a) of Theorem 2.1 or by Remark 2, respectively. By Lemmas 2.2 and 2.3 this implies that is singular.

Now suppose no reduced vertex-weighted block graph of satisfies 1 or 2. Perform PB-deletions and PB-contractions on until a reduced graph is obtained, for which no further PB-deletion or PB-contraction is possible. As cannot be further reduced, and does not satisfy 2, it does not have any pendant blocks. That is, each of its components is of the form , where is a clique. Since 1 and 2 are not satisfied, either for exactly one vertex of , or for every vertex of and . Hence by Theorem 2.1, each component of is nonsingular, and so is . ∎

We conclude the section with two of examples of families of vertex-weighted block graphs, where nonsingularity may be easily checked (without actually reducing the vertex-weighted block graph).

Theorem 2.5.

Let be a vertex-weighted block graph that satisfies the following two properties:

  1. for every vertex .

  2. for every cut-vertex .

  3. For every block of , .

Then is nonsingular.

Proof.

We show that such may be reduced by PB-contractions to a vertex-weighted clique satisfying (a) and (b). Since such a reduced graph is nonsingular by Theorem 2.1, this will complete the proof.

It suffices to show that if is a pendant block of satisfying (a)–(c), then may be PB-contracted and the resulting vertex-weighted block graph will also satisfy (a)–(c).

Let be the cut vertex of a pendant block of . By (a)–(c), this pendant block may be PB-contracted. The resulting vertex-weighted block graph satisfies for every vertex of other than , and . As , . Also, for every block of , if is not a vertex in , or is a cut-vertex in , then clearly . If is a non-cut-vertex of in , , since . ∎

Theorem 2.6.

Let be a vertex-weighted block graph, that satisfies the following three properties:

  1. for every vertex .

  2. Each block of has at least vertices.

  3. For every block of , there exists such that .

Then is nonsingular.

Proof.

Note that if consists of a single block satisfying (a)–(d), then is nonsingular: If , where and, without loss of generality, ,

by (a), and thus is nonsingular by Theorem 2.1.

If has a pendant block , this block may be PB-contracted since by the first part of Remark 1. As in the previous theorem, the resulting also satisfies (a)–(c). Such may be reduced by successive PB-contractions to a single vertex-weighted block satisfying (a)–(c), and is therefore nonsingular. ∎

Remark 4.

The two families of vertex-weighted block graphs in Theorems 2.5 and 2.6 are not mutually exclusive, but none of these families fully contains the other.

However, a block graph satisfies the conditions of Theorem 2.5 if and only if each block of has two non-cut-vertices. A block graph satisfies the conditions of Theorem 2.6 if and only if each block of has at least three vertices, at least one of which is a non-cut-vertex. That is, the family of block graphs satisfying Theorem 2.6 contains all the block graphs satisfying Theorem 2.5.

There are nonsingular block graphs that do not satisfy the requirements in Theorem 2.5. An example of one such graph is given in Figure 0(d) (see Theorem 3.3).

3 Some classes of nonsingular block graphs

In this section we use Theorem 2.4 to identify some families of nonsingular block graphs. First we name the graphs discussed at the end of the previous section.

Definition 3.

block graph. A block graph is a block graph if each block has at least three vertices, at least one of which is a non-cut-vertex.

From Theorem 4 and Remark 4 we deduce the following.

Theorem 3.1.

Every block graph is nonsingular

We observe that using Theorem 2.4 one obtains a new proof the following known result.

Theorem 3.2.

Let a graph be a forest on vertices. Then is nonsingular if and only if it has a perfect matching.

Proof.

Let be a forest, and let be any pendant edge in . Then and may be PB-deleted, yielding a forest . Note that has a perfect matching if and only if has a perfect matching: if the deleted pendant edge is , with the cut-vertex, then in the PB-deletion all edges incident with are deleted. Thus if has a perfect matching, adding to this matching yields a perfect matching of . And if has a perfect matching, has to be one of the edges in the matching, and removing it yields a perfect matching of .

Given a forest , reduce as much as possible by PB-deletions, until you get a forest that has no pendant edges. Each component of is either an edge, or a singleton. Then is nonsingular if and only if no component is a singleton, but also has a perfect matching if and only if no component of is a singleton. By the above, has a perfect matching if and only if has a perfect matching, and by Theorem 2.4 is nonsingular if and only if is. ∎

Next we consider block graphs of a special construction.

Theorem 3.3.

Let be a block graph consisting of a block , , to which at each vertex , blocks of orders , each greater than 2 are attached. Then is nonsingular if and only if

Proof.

Successively perform PB-contraction of each pendant block of . Then is reduced to a vertex-weighted block graph . By Remark 1, . As , we get that

The result follows by Theorem 2.4. ∎

A special case of Corollary 3.3, where the result is simplified is the following. Let be three integers. We define a family of block graph using these three integers. Let us coalesce pendant blocks at each vertex of . We call the resulting graph an -block graph. As an example the -block graph is shown in Figure 1(a). In the case of -block graphs the necessary and sufficient condition for nonsingularity in Corollary 3.3 becomes simple:

Corollary 3.4.

For , an -block graph is singular if and only if

Another special case of Theorem 3.3 is the case that .

Corollary 3.5.

Let be a block graph consisting of a block , to which at each of the two vertices some blocks of order greater than 2 each are attached. Then is nonsingular.

Proof.

This follows from Theorem 3.3 for , as

(a) A singular -block graph
(b) A nonsingular block graph, Theorem 3.6
(c) A -block graph
(d) A nonsingular block graph, Theorem 3.3
Figure 1: Examples of block graphs.
(a) A singular block graph
(b) A singular block graph
Figure 2: Examples in support of the necessity of conditions (a) and (b), respectively, in Theorem 3.7.

Next we consider the following construction.

Definition 4.

A tree of block graphs. Let be a tree on vertices, and let be block graphs. For every edge of , choose a vertex of and of , and connect and by an edge. The resulting graph is a block graph, and we call such graph a tree of . We refer to each of the edges in as a skeleton edge, and to the vertices and as skeleton vertices. The graph is considered pendant in the tree of if the vertex is pendant in .

The first result on a tree of block graphs generalizes Corollary 3.5.

Theorem 3.6.

Let be a tree with vertices , and let be the degree of vertex in . Let be the graph obtained by coalescing cliques , each of order at least , at each vertex of . If

for every , then is nonsingular.

Proof.

By PB-contractions of all pendant blocks in we obtain the reduced vertex-weighted tree , where

If for every , then is a strictly diagonal dominant matrix, and therefore nonsingular. The result now follows from Theorem 2.4. ∎

Next consider trees of block graphs.

Theorem 3.7.

Let be a tree of block graphs , in which

  1. no two skeleton edges share a vertex,

  2. there is at least one non-cut vertex in any block that has at vertices or more.

Then is nonsingular.

Proof.

For such , consider weight vectors with the following three properties:

  1. for every .

  2. for any skeleton vertex.

  3. For any block of with at least three vertices for at least one vertex .

We show, by induction on , that if is as in the theorem, and a weight vector for satisfies 1–3, then is nonsingular. (As the weight vector satisfies 1–3, this will prove the theorem.)

For , this holds by Theorem 2.6. Suppose the result holds for any such vertex-weighted tree of block graphs, and let be a tree of block graphs that satisfies (a) and (b), and is a weight vector for , satisfying 1–3. Without loss of generality, is pendant in . Let and be skeleton vertices. Then

where and are (0,1)-column vectors, and is , and is the adjacency matrix of . As each block graph has at leas two pendant blocks, we may perform subsequent PB-contractions of blocks in , leaving the block containing the skeleton vertex in to last. After these contractions, the remaining block satisfies (a)–(c) of Theorem 2.6. Moreover, at least one non-cut-vertex of . Thus , and we may contract it also. The adjacency matrix of the resulting vertex-weighted graph is

where by part 1 of Remark 1. The pendant edge of this graph has and may be PB-contracted, resulting in a weight of to the vertex . The resulting vertex-weighted graph is , where is a tree of , and for every vertex except , whose weight is . Note that is not a skeleton vertex in (due to the assumption that in no two skeleton edges share a vertex). Thus satisfies 1–3, and by the induction hypothesis is nonsingular. By Theorem 2.4 so is

None of the two conditions (a) and (b) in Theorem 3.7 may be dropped — see examples in Figure 2.

Theorem 3.8.

Let be a block graph, in which each block has at least two non-cut-vertices. Then any graph obtained by coalescing edges at some of the cut vertices of is nonsingular.

Proof.

By PB-deletion of the coalesced pendant edges, the cut vertices at which they were coalesced are also deleted. The resulting graph is a subgraph of , whose components are block graphs, and is thus nonsingular, implying nonsingularity of . ∎

Starting with a graph like of Theorem 3.8, and some nonsingular graphs, we can construct another nonsingular tree of block graphs.

Theorem 3.9.

Let be a block graph, in which each block has at least two non-cut-vertices. Let is obtained as in Theorem 3.8 by coalescing edges at different cut vertices , and let be nonsingular block graphs, . Let be a star graph . The tree of block graphs of obtained by choosing , , and letting the skeleton edges be , , is nonsingular.

Proof.

PB-delete each of the pendant edges. In the resulting graph each component is either a block graph, or a graph like the one in Theorem 3.8, or one of . Thus each component is nonsingular, and so is . ∎

(a) A tree of -block graphs

(b) A nonsingular block graph
Figure 3:

4 Replacing edge blocks by even order paths

We prove here some results on the determinant of a graph obtained by coalescing two graphs, or combining them by a bridge. These results will imply ways to construct more nonsingular block graphs from known block graphs.

Most of the results in this section are based on [16, Lemma 2.3], restated here for simple graphs with no vertex weights. In this lemma, denotes the characteristic polynomial of the graph .

Lemma 4.1.

[16] Let be a coalescence of and at a vertex . Then

Using this lemma, we deduce the following.

Lemma 4.2.

If is a coalescence of and at a vertex , then

Proof.

Obtain by substituting in in Lemma 4.1. This yields

Corollary 4.3.

If a graph has a pendant edge with the cut vertex, then .

Proof.

In this case, is the coalescence of and consisting of the edge . It is easy to see that and , the result follows. ∎

Corollary 4.4.

A coalescence of any two singular graphs is singular.

Proof.

Let be coalescence of singular graphs and As ,

Note that a coalescence of nonsingular graphs may be singular: e.g., the coalescence of two edges results in a singular tree. More generally, we have the following corollary of Lemma 4.2.

Corollary 4.5.

If is any graph, and two pendant edges are coalesced with it at the same vertex , then the resulting graph is singular.

Proof.

In Lemma 4.2 let be the coalescence of one of the pendant edges with , and the second pendant edge. Then is a singleton, and has a singleton component, thus