Inverses of Bipartite Graphs

# Inverses of Bipartite Graphs

Yujun Yang  and Dong Ye School of Mathematics and Information Science, Yantai University, Yantai, Shandong 264005, China. Partially supported by a grant from National Natural Sciences Foundation of China (No. 11671347).Corresponding author. Department of Mathematical Sciences and Center for Computational Sciences, Middle Tennessee State University, Murfreesboro, TN 37132; Email: dong.ye@mtsu.edu. Partially supported by a grant from Simons Foundation (No. 359516).
###### Abstract

Let be a bipartite graph and its adjacency matrix . If has a unique perfect matching, then has an inverse which is a symmetric integral matrix, and hence the adjacency matrix of a multigraph. The inverses of bipartite graphs with unique perfect matchings have a strong connection to Möbius functions of posets. In this note, we characterize all bipartite graphs with a unique perfect matching whose adjacency matrices have inverses diagonally similar to non-negative matrices, which settles an open problem of Godsil on inverses of bipartite graphs in [Godsil, Inverses of Trees, Combinatorica 5 (1985) 33-39].

## 1 Introduction

Throughout the paper, a graph means a simple graph (no loops and parallel edges). If parallel edges and loops are allowed, we use multigraph instead. Let be a bipartite graph with bipartition . The adjacency matrix of is defined such that the -entry if , and 0 otherwise. The bipartite adjacency matrix of is defined as the -entry for and . So is an -matrix and

 A=[0BB⊺0].

A perfect matching of is a set of disjoint edges covering all vertices of . If a bipartite graph has a perfect matching, then its bipartite adjacency matrix is a square matrix. Godsil proved that if a bipartite graph has a unique perfect matching, then is similar to a lower triangular matrix with all diagonal entries equal to 1 by permuting rows and columns (, see also ). So in the following, we always assume that the bipartite adjacency matrix of a bipartite graph with a unique perfect matching is a lower triangular matrix. Clearly, is invertible and its inverse is an integral matrix (cf. [5, 17]). If is non-negative (i.e. all entries are non-negative), then it is the bipartite adjacency matrix of another bipartite multigraph: the -entry is the number of edges joining the vertices and . However, the adjacency matrix of a graph has a non-negative inverse if and only if the graph is the disjoint union of ’s and ’s (cf. Lemma 1.1 in , and ).

The inverse of is diagonally similar to a non-negative integral matrix if there exists a diagonal matrix with -1 and 1 on its diagonal such that . So is a bipartite adjacency matrix of a bipartite multigraph that is called the inverse of the bipartite graph in  (a broad definition of graph inverse is given in the next section). The following is a problem raised by Godsil in  which is still open .

###### Problem 1.1 (Godsil, ).

Characterize the bipartite graphs with unique perfect matchings such that is diagonally similar to a non-negative matrix.

The bipartite graphs with unique perfect matchings are of particular interest because of the combinatorial interest of their inverses (cf. [5, 12]). Let be a bipartite graph with a unique perfect matching , and be the bipartition of . Let be the digraph obtained from by orienting all edges from to and then contracting all edges in . Simion and Cao proved that the digraph is acyclic (). For example, see Figure 1. The acyclic digraph corresponds to a poset such that for , there is a directed path from to in if and only if in . The Zeta matrix of is defined as follows (cf. Chapter 4 in )

 (Z)ij:={1if ai≤aj;0otherwise,

The modified Zeta matrix of is obtained by replacing the entry 1 by a variable for a comparable pair of which is not an arc of the digraph . Then the Zeta matrix of , and the bipartite adjacency matrix of . Note that is the adjacency matrix of and is the adjacency matrix of the transitive closure of . The Möbius function on the interval of in is (see Ex. 22 in Chapter 2 of Lovász on page 216 in ), and is the Möbius matrix of . On the other hand, the Zeta matrix of a poset is a lower triangular matrix, corresponding to a bipartite adjacency matrix of a bipartite graph with a unique perfect matching. This sets up a connection between inverses of bipartite graphs with unique perfect matchings and Möbius functions of posets.

As observed in , if is a geometric lattice (a finite matroid lattice ) or the face-lattice of a convex polytope ), then the Möbius matrix of is diagonally similar to a non-negative matrix (cf. Corollary 4.34 in ). Godsil  proved that if is a tree with a perfect matching, then the inverse of its adjacency matrix is diagonally similar to a non-negative matrix. Further, it has been observed that if and are two bipartite graphs with the property stated in Problem 1.1, then the Kronecker product is again a bipartite graph with the property . The following is a partial solution to Problem 1.1.

###### Theorem 1.2 (Godsil, ).

Let be a bipartite graph with a unique perfect matching such that is bipartite. Then is diagonally similar to a non-negative matrix.

Godsil’s result was generalized to weighted bipartite graphs with unique perfect matchings by Panda and Pati in . In this paper, we provide a solution to Problem 1.1 as follows.

###### Theorem 1.3.

Let be a bipartite graph with a unique perfect matching . Then is diagonally similar to a non-negative matrix if and only if does not contain an odd flower as a subgraph.

To define odd flower, we need more notation. Let be a bipartite multigraph with a perfect matching . A path of is -alternating if is a perfect matching of . For two vertices and of , let be the number of -alternating paths of joining and . Further, let be the number of -alternating paths of joining and such that is odd, and be the number of -alternating paths joining and such that is even. For a subset of , the -span of is defined as a subgraph of consisting of all -alternating paths joining and for any , denoted by . An -span is called a flower if the vertices of can be ordered such that if and only if (mod ). A flower is odd if there is an odd number of vertex pairs with . For example, see Figure 1. In Section 4, it will be shown that the existence of an odd flower means simply that the vertices in induce a cycle with an odd number of negative edges in the inverse of .

## 2 Inverses of weighted graphs

A weighted multigraph is a multigraph with a weight-function where is a field. We always assume that a weighted multigraph has no parallel edges since all parallel edges joining a pair of vertices and can be replaced by one edge with weight . The adjacency matrix of a weighted multigraph , denoted by , is defined as

 (Aw)ij:={w(ij)if ij∈E(G);0otherwise

where loops, with , are allowed. A weighted multigraph is invertible over if its adjacency matrix is invertible over . Note that is a symmetric matrix. Its inverse is also symmetric and therefore is the adjacency matrix of some weighted graph, which is called the inverse of . The inverse of is defined as a weighted graph whose vertex set is and whose edge set is , and whose weight function is . Note that this definition of graph inverse is different from the definitions given in  and .

Let be a graph. A Sachs subgraph of is a spanning subgraph with only copies of and cycles (including loops) as components. For example, a perfect matching of is a Sachs subgraph. For convenience, a Sachs subgraph is denoted by where consists of the cycles of (including loops), and consists of all components of isomorphic to . The following result shows how to compute the determinant of the adjacency matrix of a graph.

###### Theorem 2.1 (Harary, ).

Let be a graph and be the adjacency matrix of . Then

 det(A)=∑S2|C|(−1)|C|+|E(S)|,

where is a Sachs subgraph.

If is a bipartite graph with a Sachs subgraph , then every cycle in is of even size and hence its edge set can be decomposed into two disjoint perfect matchings of . Therefore, has at least perfect matchings. So if is a bipartite graph with a unique perfect matching , then is the unique Sachs subgraph of . Hence we have the following corollary of the above result, which can also be derived easily from a result of Godsil (Lemma 2.1 in ).

###### Corollary 2.2.

Let be a bipartite graph with a unique perfect matching . Then

 det(A)=(−1)|M|,

where is the adjacency matrix of .

By Corollary 2.2, the determinant of the adjacency matrix of a bipartite graph with a unique perfect matching is either 1 or . So a bipartite graph with a unique perfect matching is always invertible. The inverse of a graph can be characterized in terms of its Sachs subgraphs as shown in the following theorem, which was originally proved in . However, to make the paper self-contained, we include the proof here as well.

###### Theorem 2.3 ().

Let be a graph with adjacency matrix , and

 Pij={P|P is a path joining i and j≠i % such that G∖V(P) has a Sachs subgraph S}.

If has an inverse , then

 w(ij)=⎧⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪⎩1det(A)∑P∈Pij∑S2|C|(−1)|C|+|E(S)∪E(P)|if i≠j;1det(A)det(Ai,i)otherwise

where is a Sachs subgraph of and is the matrix obtained by deleting -th row and -th column from .

###### Proof.

Let be an invertible graph and be its inverse. Assume has vertices and . By the definition of the inverse of a graph, .

Note that is symmetric and hence is also symmetric. By Cramer’s rule,

 (A−1)ij=(A−1)ji=cijdet(A)

where where is the matrix obtained from by deleting -th row and -th column. Let be the matrix obtained from by replacing the -entry by 1 and all other entries in the -th row and -th column by 0. Then by the Laplace expansion,

If , then . So . So the theorem holds for . In the following, assume that .

Let be the -entry of . Recall that the Leibniz formula for the determinant of is

 det(Mi,j)=∑π∈Snsgn(π)∏mkπ(k),

where the sum is computed over all permutations of the set . Since all -entries () of are equal to 0 but the -entry is 1, only permutations such that contribute to the the determinant of . Let be the family of all permutations on such that . Denote the cycle of permuting to by . For convenience, is also used to denote the set of vertices which corresponds to the elements in the permutation cycle , for example, denotes the set of vertices in but not in . Denote the permutation of restricted on by . Then

 det(Mi,j) =∑π∈Πi→jsgn(π)∏k∈V(G)∖{i}mkπ(k) =∑π∈Πi→j(sgn(πij)∏k∈πij∖{i}mkπ(k)) (sgn(π∖πij)∏k∈V(G)∖πijmkπ(k)).

By the definition of , if or , then , the -entry of .

If the permutation cycle does not correspond to a cycle of , then for some , is not an edge of and hence . So . If the permutation cycle does correspond to a cycle in the graph , let be the path from to following the permutation order in . Then . Note that is the determinant of the adjacency matrix of the graph . By Theorem 2.1, it follows that

 sgn(π∖πij)∏k∈V(G)∖πmkπ(k)=∑S2|C|(−1)|C|+|E(S)|,

where is a Sachs subgraph of . For the case that has no Sachs subgraphs, then . Hence,

 det(Mi,j)=∑P∈Pij(−1)|E(P)|(∑S2|C|(−1)|C|+|E(S)|)=∑P∈Pij∑S2|C|(−1)|C|+|E(S)∪E(P)|,

where is a Sachs subgraph of . The theorem follows immediately from . This completes the proof. ∎

For a bipartite graph with a unique perfect matching, the weight function of its inverse can be simplified as shown below.

###### Theorem 2.4.

Let be a bipartite graph with a unique perfect matching , and let

 Pij={P|P is an M-alternating path joining i and j}.

Then has an inverse such that

 w(ij)=⎧⎪⎨⎪⎩∑P∈Pij(−1)|E(P)∖M|if i≠j;0otherwise.
###### Proof.

Let be a bipartite graph with a unique perfect matching . By Corollary 2.2, has an inverse which is a weighted graph .

For any two vertices and , let be a path joining and .

Claim: has a Sachs subgraph if and only if is an -alternating path.

Proof of Claim: If is an -alternating path, then has a perfect matching. So has a Sachs subgraph.

Now assume that has a Sachs subgraph. Note that is a bipartite graph. Every cycle of a Sachs subgraph of is of even size. So has a perfect matching . Therefore, is a path with even number of vertices and has a perfect matching . Hence is a perfect matching of . Since has a unique perfect matching, it follows that . So is an -alternating path. This completes the proof of Claim.

Let be a path in . Then has a unique perfect matching , which is also its unique Sachs subgraph. By Claim and Theorem 2.3, for , we have

 w(ij)=(−1)|M|∑P∈Pij(−1)|(M∖E(P))∪E(P)|=∑P∈Pij(−1)|E(P)∖M|.

If , then has no perfect matching and hence no Sachs subgraph. By Theorem 2.1, . By Theorem 2.3, it follows that . This completes the proof. ∎

## 3 Balanced weighted graphs

Let be a weighted graph. An edge of a weighted graph is positive if and negative if . A cycle of is negative if . A signed graph is a special weighted graph with a weight function , where is called the signature of (see ). Signed graphs are well-studied combinatorial structures due to their applications in combinatorics, geometry and matroid theory (cf. [18, 20]).

A switching function of a weighted graph is a function , and the switched weight-function of defined by is . Two weight-functions and of a graph are equivalent to each other if there exists a switching function such that . A weighted graph is balanced if there exists a switching function such that for any edge . The following is a characterization of balanced signed graphs obtained by Harary .

###### Proposition 3.1 ().

Let be a signed graph. Then is balanced if and only if has a bipartition and such that .

For a weighted graph , define a signed graph such that for any edge . Then is balanced if and only if is balanced. Therefore, the above result can be easily extended to weighted graphs as follows.

###### Proposition 3.2.

Let be a weighted graph. Then is balanced if and only if has a bipartition and such that .

Remark. Let be a weighted graph such that is connected, and let . Let be the graph obtained from by contracting all edges in and deleting all loops. Then by Theorem 3.2, is balanced if and only if is a bipartite multigraph. Therefore, it takes steps to determine whether a weighted graph is balanced or not, where is the total number of edges of .

A direct corollary of the above theorem is the following result.

###### Corollary 3.3.

Let be a weighted graph. Then is balanced if and only if it does not contain a negative cycle.

Let be a weighted graph and be its adjacency matrix. For a switching function , define to be a diagonal matrix with . Then is equivalent to if and only if for some switching function . So the adjacency matrices of two equivalent weighted graphs are diagonally similar to each other.

###### Lemma 3.4.

Let be a bipartite graph with a unique perfect matching . Then is diagonally similar to a non-negative matrix if and only if the inverse of is a balanced weighted graph.

###### Proof.

Since is invertible, let be the inverse of by Theorem 2.4. Let be the adjacency matrix of such that

 A=[0BB⊺0],

where is the bipartite adjacency matrix of , which we assume without loss of generality to be a lower triangular matrix with 1 on the diagonal. Then the inverse of is the adjacency matrix of as follows,

 A−1=[0(B⊺)−1B−10].

Note that is diagonally similar to a non-negative matrix if and only if is diagonally similar to a non-negative matrix. In other words, if and only if there exists a diagonal matrix with such that is non-negative. Define a switching function such that . Note that

 wζ(ij)=ζ(i)w(ij)ζ(j)=ζ(i)(A−1)ijζ(j)=(DA−1D)ij.

Hence is diagonally similar to a non-negative matrix if and only if there exists a switching function such that . Let and . So the existence of the switching function is equivalent to the existence of a bipartition and of such that . By Proposition 3.2, it follows that is diagonally similar to a non-negative matrix if and only if is balanced. ∎

By Lemma 3.4, Godsil’s problem is equivalent to ask which bipartite graphs with unique perfect matchings have a balanced weighted graph as its inverse.

## 4 Proof of Theorem 1.3

Now, we are ready to prove our main result.

Theorem 1.3. Let be a bipartite graph with a unique perfect matching . Then is diagonally similar to a non-negative matrix if and only if does not contain an odd flower as a subgraph.

###### Proof.

Let be a bipartite graph with a unique perfect matching and the bipartite adjacency matrix of . For any two vertices and of , let

 Pij={P |P is an M-alternating path joining i and% j}.

: Assume that is diagonally similar to a non-negative matrix. We need to show that does not contain an odd flower. Suppose on the contrary that does contain a vertex subset such that is an odd flower. Then all paths in belong to . By Theorem 2.4, has an inverse where,

 w(xixi+1)=∑P∈Pxixi+1(−1)|E(P)∖M|.

So and if and only if . Note that is an odd flower. So is a negative cycle in . By Corollary 3.3, is not balanced. Hence is not diagonally similar to a non-negative matrix by Lemma 3.4, a contradiction.

: Assume that does not contain an odd flower as a subgraph. We need to show that is diagonally similar to a non-negative matrix. Suppose on the contrary that is not diagonally similar to a non-negative matrix. Then by Lemma 3.4, its inverse is not balanced, and hence contains a negative cycle by Corollary 3.3. Choose a shortest negative cycle (i.e., is as small as possible). Then as is an edge of (subscripts modulo ). Hence (subscripts modulo ). Let . In the following, we are going to prove is an odd flower.

Since is a smallest negative cycle of , it follows that has no chord, which implies that if and are not consecutive on . In other words, if and only if (mod ). Note that is a negative cycle. So contains an odd number of negative edges. Hence, there is an odd number of vertex pairs such that . Hence is an odd flower, a contradiction. This completes the proof. ∎

Remark. For a matrix , its inverse can be found in steps. Note that it takes steps to determine whether the inverse of is balanced or not. Hence, it can be determined in whether has a balanced weighted graph as inverse or not.

## Acknowledgement

The authors would like to thank the anonymous referees for their valuable comments to improve the final version of the paper.

## References

•  M. Aigner, Combinatorial Theory, Springer, Berlin, 1979.
•  R.B. Bapat and E. Ghorbani, Inverses of triangular matrices and bipartite graphs, Linear Algebra Appl. 447 (2014) 68-73.
•  D. Cvetković, I. Gutman and S. Simić, On self-pseudo-inverse graphs, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. fiz. 602-633 (1978) 111–117.
•  R. Donaghey and L.W. Shapiro, Motzkin numbers, J. Combin. Theory Ser. A 23 (1977) 291–301.
•  C.D. Godsil, Inverses of trees, Combinatorica 5 (1985) 33–39.
•  C.D. Godsil, Personal communication with D. Ye, 2015.
•  F. Harary, On the notion of balance of a signed graph, Michigan Math. J. 2 (2) (1953) 143–146.
•  F. Harary, The determinant of the adjacency matrix of a graph, SIAM Rev. 4 (1962) 202–210.
•  F. Harary and H. Minc, Which nonnegative matrices are self-inverse? Math. Mag. 49 (2) (1976) 91–92.
•  D.J. Klein, Treediagonal matrices and their inverses, Linear Algebra Appl. 42 (1982) 109–117.
•  L. Lovász, Combinatorial Problems and Exercises, North-Holland, Amsterdam, 1979.
•  C. McLeman and E. McNicholas, Graph invertibility, Graphs Combin. 30 (2014) 977–1002.
•  H. Minc, Nonnegative Matrices, Wiley, New York, 1988.
•  S.K. Panda and S. Pati, On the inverese of a class of graphs with unique perfect matchings, Elect. J. Linear Algebra 29 (2015) 89–101.
•  R. Simion and D. Cao, Solution to a problem of C.D. Godsil regarding bipartite graphs with unique perfect matching, Combinatorica 9 (1989) 85–89.
•  D.J.A. Welsh, Matroid Theory, Dover Publications, 2010.
•  D. Ye, Y. Yang, B. Mandal and D.J. Klein, Graph invertibility and median eigenvalues, Linear Algebra Appl. 513 (2017) 304–323.
•  T. Zaslavsky, Signed graphs, Discrete Appl. Math. 4 (1982) 47–74.
•  T. Zaslavsky, A mathematical bibliography of signed and gain graphs and allied areas, Electron. J. Combin. 8 (1998) #DS8: 1–124.
•  T. Zaslavsky, Signed graphs and geometry, arXiv:1303.2770 [math.CO].
You are adding the first comment!
How to quickly get a good reply:
• Give credit where it’s due by listing out the positive aspects of a paper before getting into which changes should be made.
• Be specific in your critique, and provide supporting evidence with appropriate references to substantiate general statements.
• Your comment should inspire ideas to flow and help the author improves the paper.

The better we are at sharing our knowledge with each other, the faster we move forward.
The feedback must be of minimum 40 characters and the title a minimum of 5 characters   