Let be a simple graph with vertex set and edge set . Let the cardinality (also called the order of ) be equal to . The adjacency matrix of is the square matrix of order defined by
where A graph is called nonsingular (singular) if is nonsingular (singular). The rank of , denoted by is the rank of the adjacency matrix . If has full rank, that is, , then is nonsingular, otherwise, it is singular. The nullity of
is equal to the number of zero eigenvalues of. Thus, a zero nullity of a graph implies that it is nonsingular while a positive nullity implies that it is singular. A cut-vertex of is a vertex whose removal results in an increase in the number of connected components. A block in a graph is a maximal connected subgraph that has no cut-vertex ([18, p. 15]). Note that if a connected graph has no cut-vertex, then it itself is a block. A block in a graph is called a pendant block if it has one cut-vertex of . A complete graph on vertices is denoted by . If every block of a graph is a complete graph then the graph is called a block graph. Except for the graph in Figure 1(b), all the graphs in the figures of this paper are block graphs. By a -forbidden block graph we mean a block graph in which no block is an edge. In other words, -forbidden block graphs are those having blocks of order greater or equal to 3. By a -allowed block graph we mean a block graph in which blocks may also be edges. For more graph theoretic preliminaries see [18, 1]. For more details on block graphs see (, Chapter 7).
A well-known problem, proposed in 1957 by Collatz and Sinogowitz, is to characterize graphs with positive nullity . Nullity of graphs is applicable in various branches of science, in particular quantum chemistry, Hückel molecular orbital theory [7, 10] and social networks theory . There has been significant work on the nullity of undirected graphs like trees, unicyclic graphs, and bicyclic graphs [5, 6, 8, 9, 12, 9, 2, 4, 19, 14, 13]. It is well-known that a tree is nonsingular if and only if it has a perfect matching. Since a tree is a block graph, it is natural to investigate which block graphs are nonsingular in general. Combinatorial formulae for the determinant of block graphs are known in terms of size and arrangements of its blocks [3, 16, 15]
. But due to a myriad of possibilities of sizes and arrangements of the blocks in general, it is difficult to classify nonsingular or singular block graphs. Hence, it is an open problem. In this paper, some classes of singular and nonsingular blocks graphs and some related conjectures are presented.
1.1 Some additional notations
We frequently use mathematical induction and elementary row or column operations. The elementary operations transform the adjacency matrix to another matrix which may have nonzero diagonal entries. The graph corresponding to can be obtained by a change of weights on the edges and addition of loops on the vertices in according to the elementary operations on . Thus, during the proofs, we often encounter graphs with some possible loops on the vertices. Reordering the vertices of or of the graph resulting after elementary operations keeps the rank unchanged. If is a subgraph of , then denotes the induced subgraph of on the vertex subset . Here, is the standard set-theoretic subtraction of vertex sets. If consists of a single vertex we will write for . If is an edge of on vertices , then denotes the graph with vertex set and edge set . Let and be graphs on disjoint sets of vertices. Their disjoint union is the graph A coalescence of graphs and is any graph obtained from the disjoint union by identifying a vertex of with a vertex of , that is, merging two vertices, one from each graph, into a single vertex. By we denote a path graph on vertices. Consider a graph Let be obtained by coalescing with at some vertex then is called a pendant path of .
Consider any vertices, say of a complete graph If at each vertex a loop of weight is added, we denote the resulting graph by . The matrix is the matrix resulting by replacing the corresponding diagonal entries in by weights .denotes a -column vector of suitable order, and denotes its transpose.
The rest of the paper is organized as the follows. In subsection 1.2 we give some preliminary results on general graphs, which are later used for block graphs. In section 2 we consider -forbidden block graphs, and -allowed block graphs are discussed in section 3. Finally, in section 4 we consider combining nonsingular block graphs in a tree structure.
1.2 Some preliminaries
If is a coalescence of and at a vertex , then
By [16, Lemma 2.3],
where for every graph , is the characteristic polynomial of , and is the weight of the loop at . Since is obtained in substituting in , and in our case , we get that
A coalescence of any two singular graphs is singular.
Let be coalescence of singular graphs and Let be the common vertex of and in the coalescence. By Lemma 1.1
As are singular, ∎
Note that in general it is not necessarily true that if we add an edge between the vertices of two nonsingular graphs then the resulting graph is also nonsingular. The following corollary makes it clear.
Let and be two graphs. If we add an edge between vertices and , , then the resulting graph is singular if and only if
Let be the resulting graph. Let denote the edge between the vertices and and the graph, which is the coalescence of and the graph . Note that , and .
Then using Corollary 1.2 twice, first for with the cut vertex , and then for and the cut vertex , we get that
If or is a null graph then the determinant by convention is equal to 1. Thus is nonsingular if and only if
Let be any graph with a pendant path where is even. Then,
We prove the result using induction on . For By a suitable reordering of the vertices, the adjacency matrix can be written as the follows.
On subtracting multiples of the first row from subsequent rows and then subtracting multiples of the first column from subsequent columns we get
As row and column operations do not change rank, it is clear that . Assume that the result holds for any even . Now consider a with a pendant path . Assume that is the pendant vertex (other than the coalesced vertex) of and is its adjacent vertex. Let be the edge between Then the adjacency matrix
is converted to following matrix by elementary row and column operations.
which has rank equals to , where is a graph with a pendant path Using the induction hypothesis . Thus , which proves the result. ∎
A related observation is:
Let have a pendant edge . Then
Without loss of generality let be the pendant edge with the cut-vertex. Then,
Compute the determinant by the first row, and then the first column of the resulting minor. ∎
Let and be two graphs. Let be the graph obtained by adding a path between a vertex of and a vertex of . Then
If the order of
is odd,is nonsingular if the coalescence of and by identifying and is nonsingular.
If the order of is even, is nonsingular if the graph obtained by connecting and by a single edge is nonsingular.
For given graphs and , let us denote by the graph obtained by adding a path between the vertex of and the vertex of . It suffices to show that for every , is nonsingular if and only if is nonsingular. Let . Choose a vertex on the path of order between and , whose distance from each of the two end vertices is at least 1. Let be the part of the path connecting to (including), the part of connecting and . Let be the neighbor of in , and the neighbor of in . Finally, let and , and . Note that the coalescence of and by identifying and results in a . By Lemma 1.1,
2 -forbidden block graphs
First, consider a graph In general is always nonsingular for We prove it in two ways. The first proof is by induction. The second proof (alternative proof) generalizes the result by giving a necessary and sufficient condition for to be nonsingular.
If then any is nonsingular.
We prove the result by induction on . Note that any matrix corresponding to a can be obtained on replacing diagonal entries of by weights less than 1. Consider . Without loss of generality let us replace the first diagonal entry of by . Then the matrix
has determinant equal to . Hence is always nonsingular. Assume that every is nonsingular for We need to prove that is nonsingular for The corresponding matrix can be written as
with weights , , as the first diagonal entries. As is nonsingular by assumption, is linearly dependent on the columns of Let us write as the linear combination of the columns of with the coefficient respectively. Then,
From the top equations we have
As , at least one of is positive. Let us say that is positive. Using Equation (2),
This completes the proof. ∎
We now provide an alternative proof of Lemma 2.1.
Alternative proof of Lemma 2.1: Consider the complete graph . Let us add a loop of weight on the -th vertex of The matrix corresponding to the resulting graph can be written as where is a diagonal matrix, . So
Let Then is nonsingular if and only if is nonsingular. Since the eigenvalues of are and 0, the matrix is nonsingular if and only if That is, is nonsingular if and only if
Now consider the graph As clearly is nonsingular using (3).
and therefore if
Thus if is invertible, where then by the above,
Further, consider a matrix
where matrix is nonsingular and is some arbitrary number. Then,
where . So is nonsingular if and only if . In particular,
if is a matrix with all diagonal elements equal to and all off diagonal elements equal to and
In these cases, since , is nonsingular.
Let be a graph with possible loops on the vertices. Let be a pendant block of with at least one noncut-vertex without loop. Let be the only cut-vertex of and suppose has a loop of weight . Then there exists such that
where except for the weight of the loop at which is .
We now define a class of -forbidden block graphs.
-block graph: a block graph , in which each block has at least three vertices, and at least one of them a noncut-vertex, is called a -block graph.
An example of -block graph is given in Figure 0(a).
Any -block graph is nonsingular. Moreover, every graph obtained from a block graph by adding loops of weights less than 1 to some vertices, except for at least one noncut-vertex from each block, is also nonsingular.
We prove that every obtained from -block graph as described in the theorem is nonsingular by induction on the number of blocks in , . When the result is true by Lemma 2.1. Suppose that the result holds for any with blocks. Now consider a with blocks. Let , be a pendant block in with a loop of weight at the cut-vertex . Then we can write
After elementary operations on the first rows and columns can be converted to the following matrix.
is nonsingular by the induction hypothesis Thus is nonsingular. Any -block graph is a special case of such , and hence it is nonsingular. ∎
We give an alternative proof that any -block graph is nonsingular. Let us first recall a determinant formula for the block graphs.
 Let be a block graph with vertices. Let be the blocks of . Then
where the summation is over all -tuples of nonnegative integers satisfying the following conditions:
for any nonempty
where denotes the subgraph of induced by the blocks
Consider the block graph in Theorem 2.4. Let If is a pendant block, then in -tuples
either or [3, Lemma 1]. Also, in [15, Lemma 3.3]
it is shown that each -tuple satisfying 1 and 2 corresponds to
partition of into induced subgraphs and vice versa, where . Thus if has
cut-vertices, then in any -tuple satisfying 1 and 2,
Alternative proof: As in a -block graph each block has order at least 3 and has at least one noncut-vertex, for any -tuple that satisfies 1 and 2. Thus using Equation (6), the product terms inside the summation are either zero or positive. This implies that is nonsingular if there exists a -tuple , whose product term is positive, that is, . We claim the following.
For any -block graph with blocks there exists a -tuple , such that, .
We prove the claim by induction on . For the result is true, as in this case the -block graph is a complete graph of order at least 3. Suppose that the result is true for any -block graph on blocks. Now consider a -block graph on vertices with blocks. Let , , be a pendant block in with the cut-vertex . Since is a -block graph with blocks, there exists by the induction hypothesis a -tuple , such that , and . Then for the -tuple satisfies conditions 1 and 2 of Theorem 2.4. Also, as is -block graph, any other -tuple has . Hence is nonsingular. Thus any -block graph is nonsingular. ∎
We present the following conjecture on the nullity of -forbidden block graphs.
Let be a connected block graph where each block is of order at least 3. Then the nullity of is at most 1.
3 -allowed block graphs
In this section, we consider -allowed block graphs, that is, now some blocks may be edges. We also present some -forbidden block graphs, which are nonsingular and are not -block graphs.
3.1 -block graphs
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 0(b). Next, we give a sufficient and necessary condition for an -block graph to be singular.
For , an -block graph is singular if and only if
where . The eigenvalues of are with multiplicities , respectively.
the eigenvalues of are with multiplicities , respectively, and is singular. Then , whereas the total number of vertices in is , hence is singular.
Now suppose is singular. Since the order of is , by Equation (7) must be have a zero eigenvalue. That is, either or . But as , the only possibility is
Let us slightly generalize -block graphs to give another family of nonsingular block graphs.
Let be block graph consisting of a block to which at each vertex , blocks of order are attached. If
then is nonsingular.
More generally, by the alternative proof of Lemma 2.1, we get the following.
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