On an Application of Bipartite Graphs (original) (raw)
Bipartite Graphs Related to Mutually Disjoint S-Permutation Matrices
2012
Some numerical characteristics of bipartite graphs in relation to the problem of finding all disjoint pairs of S-permutation matrices in the general n 2 × n 2 case are discussed in this paper. All bipartite graphs of the type g = Rg ∪ Cg, Eg , where |Rg| = |Cg| = 2 or |Rg| = |Cg| = 3 are provided. The cardinality of the sets of mutually disjoint S-permutation matrices in both the 4 × 4 and 9 × 9 cases are calculated.
Calculation of the number of all pairs of disjoint S-permutation matrices
Applied Mathematics and Computation, 2015
The concept of S-permutation matrix is considered. A general formula for counting all disjoint pairs of n 2 × n 2 S-permutation matrices as a function of the positive integer n is formulated and proven in this paper. To do that, the graph theory techniques have been used. It has been shown that to count the number of disjoint pairs of n 2 × n 2 S-permutation matrices, it is sufficient to obtain some numerical characteristics of all n × n bipartite graphs.
On the number of disjoint pairs of S-permutation matrices
Discrete Applied Mathematics, 2013
In [Journal of Statistical Planning and Inference (141) (2011) 3697-3704], Roberto Fontana offers an algorithm for obtaining Sudoku matrices. Introduced by Geir Dahl concept disjoint pairs of S-permutation matrices [Linear Algebra and its Applications (430) (2009) 2457-2463] is used in this algorithm. Analyzing the works of G. Dahl and R. Fontana, the question of finding a general formula for counting disjoint pairs of n 2 × n 2 S-permutation matrices as a function of the integer n naturally arises. This is an interesting combinatorial problem that deserves its consideration. The present work solves this problem. To do that, the graph theory techniques have been used. It has been shown that to count the number of disjoint pairs of n 2 × n 2 S-permutation matrices, it is sufficient to obtain some numerical characteristics of the set of all bipartite graphs of the type g = R g ∪ C g , E g , where V = R g ∪ C g is the set of vertices, and E g is the set of edges of the graph g, R g ∩ C g = ∅, |R g | = |C g | = n.
On the number of mutually disjoint pairs of S-permutation matrices
2016
This work examines the concept of S-permutation matrices, namely n^2 × n^2 permutation matrices containing a single 1 in each canonical n × n subsquare (block). The article suggests a formula for counting mutually disjoint pairs of n^2 × n^2 S-permutation matrices in the general case by restricting this task to the problem of finding some numerical characteristics of the elements of specially defined for this purpose factor-set of the set of n × n binary matrices. The paper describe an algorithm that solves the main problem. To do that, every n× n binary matrix is represented uniquely as a n-tuple of integers.
2016
In [Journal of Statistical Planning and Inference (141) (2011) 3697-3704], Roberto Fontana offers an algorithm for obtaining Sudoku matrices. Introduced by Geir Dahl concept disjoint pairs of S-permutation matrices [Linear Algebra and its Applications (430) (2009) 2457-2463] is used in this algorithm. Analyzing the works of G. Dahl and R. Fontana, the question of finding a general formula for counting disjoint pairs of n 2 × n 2 S-permutation matrices as a function of the integer n naturally arises. This is an interesting combinatorial problem that deserves its consideration. The present work solves this problem. To do that, the graph theory techniques have been used. It has been shown that to count the number of disjoint pairs of n 2 × n 2 S-permutation matrices, it is sufficient to obtain some numerical characteristics of the set of all bipartite graphs of the type g = R g ∪ C g , E g , where V = R g ∪ C g is the set of vertices, and E g is the set of edges of the graph g, R g ∩ C g = ∅, |R g | = |C g | = n.
Disjoint S-permutation Matrices
ArXiv, 2012
The concept of S-permutation matrix is considered in this paper. When two binary matrices are disjoint is defined. For an arbitrary n2timesn2n^2 \times n^2n2timesn2 S-permutation matrix, the number of all disjoint whit it S-permutation matrices is found. A formula for counting the number of all disjoint pairs of n2timesn2n^2 \times n^2n2timesn2 S-permutation matrices is formulated and proven. In particular, a new shorter proof of a known equality is obtained in the work.
A pr 2 01 6 Canonical binary matrices related to bipartite graphs
2016
The current paper is dedicated to the problem of finding the number of mutually non isomorphic bipartite graphs of the type g = 〈Rg, Cg, Eg〉 at given n = |Rg | and m = |Cg |, where Rg and Cg are the two disjoint parts of the vertices of the graphs g, and Eg is the set of edges, Eg ⊆ Rg ×Cg. For this purpose, the concept of canonical binary matrix is introduced. The different canonical matrices unambiguously describe the different with exactness up to isomorphism bipartite graphs. We have found a necessary and sufficient condition an arbitrary matrix to be canonical. This condition could be the base for realizing recursive algorithm for finding all n×m canonical binary matrices and consequently for finding all with exactness up to isomorphism binary matrices with cardinality of each part equal to n and m.
On the probability of two randomly generated S-permutation matrices to be disjoint
2012
The concept of S-permutation matrix is considered in this paper. It defines when two binary matrices are disjoint. For an arbitrary n^2 × n^2 S-permutation matrix, a lower band of the number of all disjoint with it S-permutation matrices is found. A formula for counting a lower band of the number of all disjoint pairs of n^2 × n^2 S-permutation matrices is formulated and proven. As a consequence, a lower band of the probability of two randomly generated S-permutation matrices to be disjoint is found. In particular, a different proof of a known assertion is obtained in the work. The cases when n=2 and n=3 are discussed in detail.
Canonical binary matrices related to bipartite graphs
2016
The current paper is dedicated to the problem of finding the number of mutually non isomorphic bipartite graphs of the type g=langleRg,Cg,Egrangleg=\langle R_g ,C_g ,E_g \rangleg=langleRg,Cg,Egrangle at given n=∣Rg∣n=|R_g |n=∣Rg∣ and m=∣Cg∣m=|C_g |m=∣Cg∣, where RgR_gRg and CgC_gCg are the two disjoint parts of the vertices of the graphs ggg, and EgE_gEg is the set of edges, EgsubseteqRgtimesCgEg \subseteq R_g \times C_gEgsubseteqRgtimesCg. For this purpose, the concept of canonical binary matrix is introduced. The different canonical matrices unambiguously describe the different with exactness up to isomorphism bipartite graphs. We have found a necessary and sufficient condition an arbitrary matrix to be canonical. This condition could be the base for realizing recursive algorithm for finding all ntimesmn \times mntimesm canonical binary matrices and consequently for finding all with exactness up to isomorphism binary matrices with cardinality of each part equal to nnn and mmm.
Combinatorially orthogonal matrices and related graphs
Linear Algebra and its Applications, 1998
Let G be a graph and let c(x,y) denote the number of vertices in G adjacent to both of the vertices x and y. We call G quadrangular if c(x,y) ~ 1 whenever x and y are distinct vertices in G. Reid and Thomassen proved that IE(G)I >t 21V(G)I-4 for each connected quadrangular graph (7, and characterized those graphs for which the lower bound is attained. Their result implies lower bounds on the number of l's in m x n combinatorially orthogonal (0,1)-matrices, where a (0. I)-matrix A is said to be combinatorially orthogonal if the inner product of each pair of rows and each pair of columns of A is never one. Thus the result of Reid and Thomassen is related to a paper of Beasley, Brualdi and Shader in which they show that a fully indecomposable, combinatorially orthogonal (0,1)-matrix of order n ~ 2 has at least 4n-4 l's and characterize those matrices for which equality holds. One of the results obtained here is equivalent to the result of Beasley, Brualdi and Shader. We also prove that IE(G)I >I 2IV(G)I-I for each connected quadrangular nonbipartite graph G with at least 5 vertices, and characterize the graphs for which the lower bound is attained. In addition, we obtain optimal lower bounds on the number of l's in m x n combinatorially row-orthogonal (0,1)-matrices.