Extremal polygonal chains with respect to the Kirchhoff index

The Kirchhoff index is defined as the sum of resistance distances between all pairs of vertices in a graph. This index is a critical parameter for measuring graph structures. In this paper, we characterize polygonal chains with the minimum Kirchhoff index, and characterize even (odd) polygonal chains with the maximum Kirchhoff index, which extends the results of \cite{45}, \cite{14} and \cite{2,13,44} to a more general case.Comment: 13 pages. arXiv admin note: substantial text overlap with arXiv:2209.1026

Approximately Sampling Elements with Fixed Rank in Graded Posets

Graded posets frequently arise throughout combinatorics, where it is natural to try to count the number of elements of a fixed rank. These counting problems are often $\#\textbf{P}$-complete, so we consider approximation algorithms for counting and uniform sampling. We show that for certain classes of posets, biased Markov chains that walk along edges of their Hasse diagrams allow us to approximately generate samples with any fixed rank in expected polynomial time. Our arguments do not rely on the typical proofs of log-concavity, which are used to construct a stationary distribution with a specific mode in order to give a lower bound on the probability of outputting an element of the desired rank. Instead, we infer this directly from bounds on the mixing time of the chains through a method we call $\textit{balanced bias}$. A noteworthy application of our method is sampling restricted classes of integer partitions of $n$. We give the first provably efficient Markov chain algorithm to uniformly sample integer partitions of $n$ from general restricted classes. Several observations allow us to improve the efficiency of this chain to require $O(n^{1/2}\log(n))$ space, and for unrestricted integer partitions, expected $O(n^{9/4})$ time. Related applications include sampling permutations with a fixed number of inversions and lozenge tilings on the triangular lattice with a fixed average height.Comment: 23 pages, 12 figure

Some Extremal Problems on the Topological Indices of Polygonal Chains

如果一个简单无向图G=(V,E)的每个顶点代表分子中的一个原子，每条边代表原子之 间形成的化学键，这种图就叫分子图。众所周知,图论学科的产生与发展与化学分子图的研究非常密切. 分子拓扑指数以及分子图的不变量的研究是现代化学图论中最活跃的研究领域之一. 对于化学分子图的某些拓扑性质,人们已经得到了很多结果, 其中有关数学方面的研究主要集中在覆盖问题、非同构计数问题、匹配计数、独立点集计数与相关的排序问题等方面. 在图论中,匹配数(在化学上称为Hosoya指标)、独立集数 (在化学上称为Merrifield-Simmons指标)和Wiener 指标是三个具有重要意义的图参数.它们有着...Let G=(V, E) be a simple, undirected graph. If each vertex of G represents an atom of molecule and each edge represents the chemical bond between the atoms respectively, then G is called an molecular graph. It is well known that the appearance and the development of graph theory are closely connected with the research of chemical molecular graph. The study of molecular topological indices an...学位：理学博士院系专业：数学科学学院数学与应用数学系_应用数学学号：B20042301

The Structure of 4-Clusters in Fullerenes

Fullerenes can be considered to be either molecules of pure carbon or the trivalent plane graphs with all hexagonal and (exactly 12) pentagonal faces that models these molecules. Since carbon atoms have valence 4 and our models have valence 3, the edges of a perfect matching are doubled to bring the valence up to 4 at each vertex. The edges in this perfect matching are called a Kekule structure and the hexagonal faces bounded by three Kekule edges are called benzene rings. A maximal independent (disjoint) set of benzene rings for a given Kekule structure is called a Clar set, and the maximum possible size of a Clar set over all Kekule structures is the Clar number of the fullerene. For any patch of hexagonal faces in the fullerene away from all pentagonal faces, there is a perfect Kekule structure: a Kekule structure for which the faces of an independent set of benzene rings are packed together as tightly as possible. Starting with such a patch and extending it as far as possible results in a perfect Kekule structure except for isolated regions, called clusters, containing the pentagonal faces. It has been shown that clusters must contain even numbers of pentagonal faces. It has also been shown that the Kekule structure of the patch can be extended into each of these clusters to give a full Kekule structure. However, these Kekule extensions will not admit as tightly packed benzene rings as in the patch external to the clusters. A basic problem in computing the Clar number of a fullerene is to make these extensions in a way that maximizes the number of benzene rings in each cluster. The simplest case, that of 2-clusters, has been completely solved. This thesis is devoted to developing a complete understanding of the Clar structures of 4-clusters