Mathematical Computation September 2015, Volume 4, Issue 3, PP.56-59
The Triangle Size of Zero-nonzero Patterns Gufang Mou College of Mathematics and Information Science, Leshan Normal University, Leshan 614000, China Email: mougufang1010@163.com
Abstract For an asymmetric matrix, the zero-nonzero pattern P of entries can be described by a digraph Γ( P ) which has an arc if an entry is nonzero. The minimum rank of a zero-nonzero pattern is defined to be the smallest possible rank over all real matrices having the given zero-nonzero pattern. Definitions of various graph parameters that have been used to bound minimum rank of a zerononzero pattern, including path cover number and edit distance, and the triangle size tri(P). In this paper, by converting a digraph into an undirected bipartite graph G(U, V), we present an algorithm for constructing a sub-bipartite graph with the unique maximum perfect matching M', and obtain that tri(P)=|M'| for P. Keywords: Zero-Nonzero Pattern; Triangle Size; Bipartite Graph
1 INTRODUCTION The minimum rank problem, which asks us to determine the minimum rank among all real matrices whose zerononzero pattern of entries is described by a given graph. The minimum rank problem for a graph is to determine the minimum among the ranks of all real matrices (symmetric or asymmetric) whose zero-nonzero pattern is descried by a given graph (undirected or directed). The solution to the minimum rank problem is equivalent to the determination of maximum nullity, and the relationship between maximum nullity and the related graph parameters has received considerable attention (see [1], [2], [3]). These problems have been studied extensively over the last ten years (see [7]). An asymmetric zero-nonzero pattern P can be described by a digraph Γ( P) . The minimum possible rank of a zero-nonzero pattern is at least the size of a sub-pattern that is permutation equivalent to a triangular pattern with nonzero diagonal (see [4], [5]). In [5], the presence of triangles of sizes less than 7 were given, and the relationship between triangle size and minimum rank for m × n (m=5, 6) zero-nonzero patterns were discussed. In this paper, we obtain the triangle size for an n × n zero-nonzero pattern P by converting a digraph Γ( P) of a zero-nonzero pattern into an undirected bipartite graph G(U, V).
2 THE TRIANGLE SIZE OF A ZERO-NONZERO PATTERN P In this section, by converting a digraph of a zero-nonzero pattern into an undirected bipartite graph G(U, V), we present an algorithm for constructing a sub-bipartite graph with the unique maximum perfect matching M' in G(U, V), and obtain that tri(P)=|M'| for a zero-nonzero pattern P. A graph G(U, V) is called bipartite if its vertices may be partitioned into two disjoint subsets U = {1, 2, , m} and V = {1′, 2′, , n′} so that each edge {i, j ′} of G has exactly one endpoint i ∈ U and the other endpoint j ′ ∈ V . We describe the pattern of nonzero entries in an m × n matrix A = aij with the bipartite graph G(A)=G(U,V) whose vertices are {1, 2, , m} (corresponding to the rows of A) and {1′, 2′, , n′} (corresponding to the columns of A). When aij is nonzero, there is an undirected edge {i, j ′} in G(U, V). An k × k square sub-matrix S of A corresponds to k column vertices C(A) and k row vertices R(A), there is an ssociate sub-bipartite graph G(S)=G(U',V') of G(A). A matching M in a bipartite graph G(U,V) is a set of edges without common vertices. Equivalently it is an injective mapping σ from one of the vertex sets to the other respecting the graph. A matching M' in a bipartite graph G(U,V) is called perfect if it covers all vertices in of the two vertex sets. A maximum matching M0 in a bipartite graph G(U,V) is a matching with the maximum number of edges among all matching in G(U,V). The matching number of M - 56 www.ivypub.org/mc
denoted by |M|, is the number of edges in a maximum matching. For a vertex w ∈ U ( w′ ∈ V ), let n(w)(n(w')) denote the set of all vertices adjacent to w(w'). n(w)(n(w')) is called the neighborhood of w(w'). A vertex i ∈ U ( j ′ ∈ V ) is a duplicate of w ∈ U ( w′ ∈ V ) of bipartite graph G(U,V) if their neighborhoods are the same. The degree of a vertex v in a bipartite graph G(U,V) is the number of edges incident with v and is denoted by deg v. The minimum degree of G(U,V) is the minimum degree among the vertices of G(U,V) and is denoted by δ (G (U ,V )) ; the maximum degree of G(U,V) is denoted by ∆(G (U ,V )) . An edge coloring of a nonempty bipartite graph G(U,V) is an assignment of colors to the edges of G(U,V), one color to each edge, such that adjacent edges are assigned different colors. The minimum number of colors that can be used to color the edges of G(U,V) is called the edge chromatic number and is denoted by χ (G (U ,V )). Lemma 2.1. Given a bipartite graph G(U,V) with bipartition U and V, if U ≠ V then there doesn't exist a perfect matching for G(U,V). Lemma 2.2 [6]. Let G(U,V) with bipartition U and V be a bipartite graph with the unique perfect matching. Then there exist two vertices of degree 1 with one in U and another in V. Remark 2.3. The condition of two vertices of degree 1 with one in U and another in V is not sufficient to ensure the unique perfect matching in G(U,V). Example 2.4. Let * P(Γ) =0 0 0
0 * * * be an 4 × 4 not necessarily symmetric zero-nonzero pattern. The digraph of P(Γ) is Γ( P) . It is converted into an undirected bipartite graph G(U,V) with one set vertices U=R({1,2,3,4}) based on rows of P(Γ) and the other set vertices V=C({1',2',3',4'}) based on columns of P(Γ) . There is a perfect matching M in G(U,V), and there are two vertices of degree 1 with one in U and another in V, while the perfect matching is not unique. 0 * 0 *
0 * * 0
We will present Algorithm 2.5 for searching for a sub-bipartite graph G( U',V') with the unique maximum perfect matching M' in G(U,V) by Lemma 2.3 and Algorithm ABMP [8] (construction of a maximum matching in G(U,V)). Algorithm 2.5. Construction of a sub-bipartite graph G( U',V') with the unique maximum perfect matching M' in G(U,V). Let M0 be a maximum matching in G(U,V) and G(U0, V0) with bipartition U 0 ⊆ U and V0 ⊆ V be a sub-bipartite graph relative to M0. (1) Search for a maximum matching M0 in G(U,V) by Algorithm ABMP, and construct a sub-bipartite graph G(U0, V0) relative to M0. From G(U0, V0) construct a sub-bipartite graph G( U',V') with the unique maximum perfect matching M' in G(U,V). (2) If M'= M0 then G (U ′,V ′) = G (U 0 ,V0 ) .
(3) while M ′ ≠ M 0 If there is a vertex i (i') with degree 1 then delete i (i') from U0 (V0) and remove all edges adjacent to i (i') from G(U0, V0). G(U0, V0)=G( U0-{i}, V0-{i'}). Elseif choose a vertex r (r') of maximum degree then delete r (r') from U0 (V0 ) and remove all edges adjacent to r (r') from G(U0, V0). G(U0, V0)=G( U0-{r}, V0-{r'}) Elseif there is a duplicate vertex u (u') then delete u(u') from U0(V0 ). G(U0, V0)=G( U0-{u}, V0-{u'}). - 57 www.ivypub.org/mc
Let S be removed set from U0 and S' be removed set from V0. G(U0, V0)=G( U0-S, V0- S'). If M'= M0 then G (U ′,V ′) = G (U 0 {i},V0 {i ′}).
Else goto (3). End while Example 2.6. Assume an 7 × 7 zero-nonzero pattern * 0 0 P = * 0 * *
0 0 * 0 * * * Let Γ( P) be the digraph of P. Γ( P) is converted into an undirected bipartite graph G(U0, V0) (see Fig. 1) with one set vertices U 0 = {1, 2, , 7} based on rows of P and the other set vertices V0 = {1′, 2′, , 7′} based on columns of P. * * 0 0 * 0 *
* 0 * * * 0 0
0 * 0 * * * 0
* * * 0 0 * 0
0 * * * 0 0 *
In G(U0, V0), there is a maximum perfect matching M0 but not unique, and there does not exist any vertex of degree 1 in U0 or V0. Thus we choose a vertex 1 ∈ U 0 ( 7′ ∈ V0 ) with the maximum degree and delete 1 (7') from U0 (V0 ) and all edges adjacent to 1 (7') from G(U0, V0), and obtain G(U1, V1) (see Fig. 2). In G(U1, V1), there does not exist any vertex of degree 1 in U1 or V1, then we delete {1,6} ({5', 6'}) from U1 ( V1 ) and all edges adjacent to {1,6} ({5', 6'}) from G(U1, V1), and obtain G(U2, V2). In G(U2, V2), there is a vertex of degree 1 in U2 or V2 , then we delete {4} ({1'}) from U1 ( V1 ) and all edges adjacent to {4} ({1'}) from G(U2, V2), and obtain G(U3, V3). In G(U3, V3), there does not exist any vertex of degree 1 in U3 or V3, then we choose a vertex 2 ∈ U 0 ( 2′ ∈ V0 ) with the maximum degree and delete 2 (2') from U3 (V3 ) and all edges adjacent to 2 (2') from G(U3, V3). We obtain the sub-bipartite graph G( U',V') with the unique maximum perfect matching M' (see Fig. 3), where U'={3, 4, 5}, V'={1', 3', 4'}.
FIG. 1
FIG. 2
FIG. 3
Definition 2.7 [6]. Let M be a perfect matching in bipartite graph G(U, V). An alternating cycle in G(U,V) with respect to M is a cycle of even length that has alternating one edge in M and one edge out of M . Theorem 2.8 (K o ning's Theorem). If G(U, V) is a nonempty bipartite graph,
χ (G (U ,V )) = ∆(G (U ,V )) . Theorem 2.10. Let G(U, V) be a undirected bipartite graph. The sub-bipartite graph G( U', V') of G(U, V) has the unique maximum perfect matching M' by Algorithm 2.5. Proof. If M'= M0, this is certainly true by Algorithm 2.5. Otherwise, we suppose M ′ ≠ M 0 (|M'|<| M0|). Let G(Ui, Vi) be a sub-bipartite graph of G(U0, V0) deleted r (r') of maximum degree from G(U0, V0) and all edges adjacent to r (r') from G(U0, V0). Let Ri be a minimum partition edge set of G(Ui, Vi). Let Mi be a maximum matching relative to G(Ui, Vi) (i = 1, 2,) . - 58 www.ivypub.org/mc
|Ri| ( χ (G (U i ,Vi ))) is less than |R0| ( χ (G (U 0 ,V0 ))) , then ∆(G (U i ,Vi )) is less than ∆(G (U 0 ,V0 )) according to Theorem 2.9. Thus, we decide to delete the vertex r (r') of maximum degree from Ui-1(V0) in G(U0, V0) and delete all edges adjacent to r (r') from G(U0, V0), and obtain G(Ui, Vi). If M'=M1, then G(U', V')=G(U1-{k}, V1-{k'}) by Algorithm 2.5. Otherwise, we continue the step (3) of Algorithm 2.5 until we arrive at Mi=M' (i = 1, 2,) , and obtain a sub-bipartite graph G(U', V')= G(Ui, Vi) with the unique perfect matching M'. If m (m') from U0(V0) is deleted vertex with degree 1 in G(U0, V0), then G (U ′,V ′) = G (U i {m},Vi {m′})
with the unique maximum perfect matching M' in G(U, V). According to Algorithm 2.5 and Theorem 2.10, we will easily obtain Theorem 2.11. Theorem 2.11. Let P be a zero-nonzero pattern having entries in {*, 0}. The triangle size of P is equivalent to |M'| of the sub-bipartite graph G(U', V') by Algorithm 2.5. In example 2.6, the triangle size of P is 3.
3 CONCLUSION By converting a digraph of a zero-nonzero pattern into an undirected bipartite graph G(U, V), we have provided an algorithm for constructing a sub-bipartite graph G(U', V') with the unique maximum perfect matching M' in G(U, V), and obtained that the triangle size of P tri(P) is equivalent to |M'| for any zero-nonzero pattern P.
REFERENCES [1]
A. Berman, S. Friedland, L. Hogben, U. G. Rothblum, B. Shader, An upper bound for the minimum rank of a graph, Linear
[2]
L. Hogben, Minimum rank problems, \ Linear Algebra and its Applications, 432 (2010) 1961-1974
[3]
L. Hogben, B. Shader, Generic maximum nullity of a graph, Linear Algebra and its Applications, 432 (2010) 857-866
[4]
R. Canto, C. R. Johnson, The relationship between maximum triangle size and mimnimum rank for zero-nonzero patterns, Textos
[5]
C. R. Johnson, J. A. Link, The extent to which triangular sub-patterns explain minimum rank, Discrete Applied Mathematics, 156
[6]
L. LovCasz, M. D. Plummer, Matching Theory, in: Annals of Discrete Mathematics, Vol. 29, North-Holland, Amsterdam, 1986.
[7]
S. M. Fallat, L. Hogben, The Minimum Rank of Symmetric Matrices Described by a Graph: A Survey, Linear Algebra and its
Algebra and its Applications, 429 (2008) 1629-1638
de Matematica, 39 (2006) 39-48 (2008) 1637-1651
Applications, 426 (2007) 558-582 [8]
ggkvist, Bipartite graphs and their applications, Published in the United States of America by A. S. Asratian, T. M. Denley, R. H a Cambridge University Press, New York, 1998
AUTHOR Gufang Mou Was born in HuBei province in 1981. PhD in School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu. The author’s major field is properties of special matrices by applying theories and methods of graphs.
- 59 www.ivypub.org/mc