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Journal of Applied Mathematics
Volume 2013 (2013), Article ID 823769, 5 pages
http://dx.doi.org/10.1155/2013/823769
Research Article

Invariants for Weighted Digraphs under One-Sided State Splittings

Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China

Received 7 February 2013; Accepted 9 April 2013

Academic Editor: Md Sazzad Chowdhury

Copyright © 2013 Sheng Chen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Using Matrix-Forest theorem and Matrix-Tree theorem, we present some invariants for weighted digraphs under state in-splittings or out-splittings.

1. Introduction

State in-splittings and out-splittings are very important operations in the theory of one-sided, or two-sided Markov shifts ([1, 2]). Lind and Tuncel introduced a spanning tree invariant for Markov shifts in [3]. Spanning tree invariants are further studied in [46]. Motivated by these works, we consider some other graph structures like cycles and forests and present some invariants for weighted digraphs under state in-splittings or out-splittings.

Firstly we give some basic definitions in graph theory and a brief introduction of Matrix-Forest theorem for digraphs. Readers can refer to [7, 8] for more details.

In this paper, a digraph is an ordered pair of finite sets, where is called the vertex set and is called the edge set. For an edge , and are called the initial and terminal ends of the edge, respectively. The number of edges having as the initial end is defined to be the outdegree of and denoted by . The number of edges having as the terminal end is defined to be the indegree of . A walk of length is a sequence of edges and can be denoted by ; moreover, if is the same as , we call the walk a closed one. A directed forest is a digraph without closed walks such that the indegree of each vertex is no more than one. The vertices with indegree zero of a forest are called roots. We say that is a spanning subgraph of if and .

Suppose that is a digraph with vertex set . Let be a weight function on the edge set. We then say that is a weighted digraph and is the weight matrix of . The Kirchhoff matrix of is defined as , where is a diagonal matrix and . The product of the weights of all edges that belong to a subgraph of is defined to be the weight of and denoted by .

Let be the set of all spanning rooted forests of and the set of those spanning rooted forests of such that and belong to the same tree rooted at . For a matrix , denotes the cofactor of the -entry of . The Matrix-Forest theorem then states as follows.

Lemma 1 (cf. [8]). Let be a weighted digraph. Let be the Kirchhoff matrix of . Then one has (1); (2)for any , .

2. Invariants for Weighted Digraphs under State In-Splitting

Before giving the main result, we recall the definition of state in-splitting.

Definition 2. Let be a weighted digraph. For a vertex of , denotes the set of edges of with terminal end . The state in-splitting of at induces a new weighted digraph in the following way: let be a partition of . The vertex set of the new digraph is . The edge set and weight of are defined as follows.(i)For , if and only if and in this case . (ii)For , if and only if and in this case . (iii)For , if and only if and in this case . (iv)If , then , for , and in this case .
For more details about state splittings, readers can refer to [2, 3, 9]. Now we give the definition of our new invariant.

Definition 3. Let be a weighted digraph. We define as where runs over and denotes the set of closed walks of with length at vertex . Furthermore, we define the generating function as
Let be a square matrix. The trace of is defined to be the sum of the elements on the main diagonal and denoted by . For a digraph , the diagonal matrix denotes the outdegree matrix of that is, . Then we have the following result.

Theorem 4. Let be a weighted digraph with weight matrix . Then is an invariant under state in-splitting and can be computed in the following way:

Proof. We firstly prove the invariance of for . Without loss of generality, if there is a loop at vertex , we assume that it belongs to , where denotes the partition of as in the definition of state in-splitting.
We define the mapping in the following way: for a closed walk of with length , if , then ; otherwise, we replace each maximum path of of the form with if . it is not difficult to see that where , and are both weight-preserving bijections.
Since is the same for and and , we know that for , and the invariance of follows.
Finally, we notice that . Thus

Example 5. Let be a weighted digraph as in the left of Figure 1. is the opposite of (see the right of Figure 1), that is, the digraph obtained from by reversing the direction of all its edges. It is easy to see that and have the same outdegree sequence . The weight of any edge or is defined to be . Since and , we know that cannot be obtained from by a sequence of in-splittings or reverse operations.

823769.fig.001
Figure 1

Let be a nonnegative matrix. is called row stochastic if the summation of each row equals 1 and column stochastic if the summation of each column equals 1 is called double stochastic if it is row and column stochastic.

Definition 6. Let be a row-stochastic matrix and a real positive number. Let be the weighted digraph with weight matrix . We define as where runs over all vertices of , runs over all spanning directed forests of , and runs over all spanning directed forests including as a root.

In general, is not an invariant under state in-splitting, but the following result shows that it indeed reflects some invariance.

Corollary 7. Let be a row-stochastic matrix and a real positive number. Let be a weighted digraph with weight matrix . Then is an integer independent of .

Proof. Let be the outdegree matrix of . Then we get by Lemma 1 that Since is stochastic and , we have where .
Therefore
By Theorem 4, we know that is an invariant under in-splitting; thus The result follows.

Lind and Tuncel defined a spanning tree invariant for Markov shifts in [3] as follows: Here the weight matrix of is an irreducible row-stochastic matrix, and runs over all spanning trees of .

By considering the outdegree matrix as in Definitions 3 and 6, we can define a new spanning tree invariant as where is as above, and denotes the outdegree of the root of .

Corollary 8. is an invariant under in-splitting.

Proof. Let be the weight matrix of and thus row stochastic as in [3]. By the Matrix-Tree theorem (Theorem 2 in [8]), we have
By Theorem 4, we know that is an invariant under state in-splitting. it is also well known that is an invariant under state splitting. Therefore Since is a constant and , we have The result follows.

Let be a weighted digraph. The out-weighted line digraph of is a weighted digraph defined in the following way: the vertex set of is ; if and only if , and in this case, . Similarly, if we let in the above definition, then we get the in-weighted line digraph . Galeana-Sánchez and Gómez show that can be obtained by sequences of state in-splittings from (see Proposition 2.2 in [9], which has a small typo there by stating can be obtained by sequences of state in-splittings). Now the following conclusion is an immediate result of Corollary 8.

Corollary 9. is an invariant under out-weighted line digraph operation.

3. The State Out-Splitting Case

Let be a row-stochastic matrix. Let be the weighted digraph with weight matrix . We first give the definition of state out-splitting, which is a little more complicated than the case of state in-splitting. Readers can refer to [3] for more details.

Definition 10. For a vertex of , let denote the set of edges of with initial end . The state out-splitting of at induces a new weighted digraph in the following way: let be a partition of . Let denote the sum of the weights of edges in . The vertex set of the new digraph is . The edge set and weight of are defined as follows.(i)For , if and only if and in this case . (ii)For , if and only if and in this case . (iii)For , if and only if and in this case . (iv)If , then , for , and in this case .
In the definition of and , by replacing outdegrees with indegrees, we get and ; that is, where is the indegree of .

Theorem 11. Let be a row-stochastic matrix. Let be the weighted digraph with weight matrix . Then is an invariant under state out-splitting, and can be computed as where is the indegree matrix of .

Proof. We just need to prove the invariance of for . Without loss of generality, if there is a loop at vertex , we assume that it belongs to , where denotes the partition of as in the definition of state out-splitting.
We define the mapping in the following way: for a closed walk of with length , if , then ; otherwise, we replace each maximum path of of the form with if . By the definition of state out-splitting, it is not difficult to prove that where , and are both bijections.
We now prove that they are also weight-preserving. In fact, if , then , since . On the other hand, for any walk of of the form , we have . (1)If , we have (2)If and , we have (3)If and , we have
Thus the maps above are weight preserving. Since is the same for and , and , we know that , for , and the invariance of follows.
The proof of the equality is similar to that of Theorem 4.

Similarly, we can define and prove that it is also an invariant under state out-splitting on the basis of the above result.

Now, we consider some weighted digraphs from [10] in the following two examples.

Example 12. The weight matrices of two weighted digraphs are as follows: By some computation, we get that , , and . Thus cannot be archived by a sequence of in-splittings or reverse operations, but may be archived by a sequence of out-splittings or reverse operations.

Example 13. The weight matrices of three weighted digraphs are as follows: By some computation, we get that , , . Thus for any pair of them, we cannot get one from the other and by a sequence of in-splittings or reverse operations either nor by a sequence of out-splittings or reverse operations.

4. Invariants for Weighted Digraphs with Double-Stochastic Matrices

Let be a weighted digraph. If the weight matrix is column stochastic, the weight distribution after state out-splitting can be defined in an easier way, that is, without multiplying by the coefficients about in Definition 10. Under this definition, we can get that is still an invariant under state out-splitting, the proof of which is similar to that of Corollary 8. We also know from [9] that the in-weighted line digraph can be obtained by a sequence of such state out-splittings, so the following result is immediate.

Corollary 14. Let be a weighted digraph. If the weight matrix is column stochastic, then is an invariant under in-weighted line digraph operation.

Especially, if the weight matrix is doubly stochastic, we have the following result.

Corollary 15. Let be a weighted digraph. If the weight matrix is doubly stochastic, then .

Proof. Since is doubly stochastic, we have by Corollary 8 that and by Corollary 14 that By Matrix-Tree theorem (Theorm 2 in [8]), we know that both and are row-constant matrices, where is transposed. Thus is a constant matrix. Since the sum of indegrees is equal to that of outdegrees, the result follows.

Acknowledgments

The research was supported by National Natural Science Foundation of China (Grant nos. 11001064 and 11101105) and by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry. Xiaomei Chen was supported by China Scholarship Council.

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