Some New Results Of Tensors And Polycyclic Hexagonal Systems

The Perron-Frobenius Theorem is a fundamental result for nonnegative matrices. It has numerous applications, not only in many branches of mathematics, such as Markov chains, graph theory, game theory, and numerical analysis, but in various fields of science and technology, e.g. economics, operational research, and recently, page rank in the internet,as well. In particular, the Perron Frobenius Theorem for nonnegative tensors is related to measuring higher order connectivity in linked objects[1]and hypergraphs[2]. K. C. Chang et al.[3]and Y. Yang, Qingzhi Yang[4]give some new results on the Perron-Frobenius Theorem for nonnegative rectangular tensors. In this paper, we give some results on the PerronFrobenius Theorem for nonnegative generalized rectangular tensor.The idea of “forcing” has long been used in many research fields, such as colorings,orientations, geodetics and dominating sets in graph theory, as well as Latin squares, block designs and Steiner systems in combinatorics(see[5]and the references therein). Recently,the “forcing” on perfect matchings has been attracting more researchers’ attention. A forcing set of a perfect matching M of a graph G is a subset of M contained in no other perfect matchings of G. A global forcing set of G, introduced by Vukiˇcevi′c et al., is a subset of E(G) on which there are distinct restrictions of any two different perfect matchings of G.Combining the above “forcing” and “global” ideas, Xu et al.[6]introduced a complete forcing set of G defined as a subset of E(G) on which the restriction of any perfect matching M of G is a forcing set of M. The minimum cardinality of complete forcing sets is the complete forcing number of G. In this paper, we give the explicit expressions for the complete forcing number of several classes of polyphenyl systems and spiro hexagonal systems.The Kirchhoff index Kf(G) of a graph G is the sum of resistance distances between all pairs of vertices in G. The Hosoya index m(G) and the Merrifield-Simmons index i(G)of a graph G are the number of matchings and the number of independent sets in G. In this paper, we establish exact formulas for the expected values of the Kirchhoff indices of the random phenylene and hexagonal chains; and the Hosoya index and Merrifield-Simmons index of a random spiro chains.The text of paper main consists of four sections as Some new results of tensors and polycyclic hexagonal systems, and the paper consists of four sections.In the first section, the research background and some research results of the PerronFrobenius Theorem, forcing set, Kirchhoff index, Hosoya index and Merrifield-Simmons index are Introduced, the common concepts and symbols of the following chapters are given.Firstly, we give the main theorems of this paper.In the second section, we give some results on the Perron-Frobenius Theorem for nonnegative generalized rectangular tensor.In the third section, we give the explicit expressions for the complete forcing number of several classes of polyphenyl systems and spiro hexagonal systems.In the fourth section, we establish exact formulas for the expected values of the Kirchhoff indices of the random phenylene and hexagonal Chains; and the Hosoya index and Merrifield-Simmons index of a random spiro chains.