A Matrix Approach to Hypergraph Stable Set and Coloring Problems with Its Application to Storing Problem

Abstract

This paper considers the stable set and coloring problems of hypergraphs and presents several new results and algorithms using the semitensor product of matrices. By the definitions of an incidence matrix of a hypergraph and characteristic logical vector of a vertex subset, an equivalent algebraic condition is established for hypergraph stable sets, as well as a new algorithm, which can be used to search all the stable sets of any hypergraph. Then, the vertex coloring problem is investigated, and a necessary and sufficient condition in the form of algebraic inequalities is derived. Furthermore, with an algorithm, all the coloring schemes and minimum coloring partitions with the given q colors can be calculated for any hypergraph. Finally, one illustrative example and its application to storing problem are provided to show the effectiveness and applicability of the theoretical results.

Solving Quaternion Linear System Based on Semi-Tensor Product of Quaternion Matrices

“…Semi-tensor product of matrices is designed in such a way that the product rule can automatically search the proper position for each factor of multiplier. At present, semi-tensor product of matrices is widely used in biological system and life science [2,3], game theory [4,5], graph theory and formation control [6,7], fuzzy control [8,9], coding theory, and algorithm implementation [10,11]. In addition, some scholars proposed a new quaternion real vector representation method [12,13] based on semi-tensor product of matrices, and applied this method to the solution of quaternion linear system.…”

Semi‐tensor product of quaternion matrices and its application

“…Semi‐tensor product of matrices proposed by Professor Cheng 1 opens up a new field for us to explore matrix theory, and the various forms of semi‐tensor product were defined as Cheng et al 2,3 : matrix‐vector semi‐tensor product and semi‐tensor product of matrices. Semi‐tensor product of matrices is widely used in many fields such as biological system, 4,5 game theory, 6,7 graph theory, 8,9 fuzzy control, 10,11 and coding theory 12,13 …”

“…Semi-tensor product of matrices proposed by Professor Cheng 1 opens up a new field for us to explore matrix theory, and the various forms of semi-tensor product were defined as Cheng et al 2,3 : matrix-vector semi-tensor product and semi-tensor product of matrices. Semi-tensor product of matrices is widely used in many fields such as biological system, 4,5 game theory, 6,7 graph theory, 8,9 fuzzy control, 10,11 and coding theory. 12,13 As a noncommutative four-dimensional associative algebra, quaternion is a linear combination of real number and three imaginary units i, j, k, expressed as where i, j, k satisfy i 2 = j 2 = k 2 = −1, ij = −ji = k, jk = −kj = i, ki = −ik = j.…”

Algebraic Relations among Four Types of Right Semi-Tensor Product

“…anks to the left MM STP, a Boolean network can be converted into a linear discrete-time form, which stimulates the development of Boolean networks [2][3][4]. In addition, the left MM STP also plays an important role in finite game [5][6][7], fuzzy systems [8,9], and graph theory [10,11].…”