Some Extremal Problems in Graphs

令v1,v2,…vm,是Rn中的向量。对于λi∈R,λi≥0,Σλi=1,v=λ1v1+λ2v2+…λmvm被称为是{v1,v2,…vm}的凸组合。{v1,v2,…vm}所有凸组合的集合就是{v1,v2,…vm}的一个凸包。在Rn中的有限个向量的集合X的凸包被称为是一个多面体，并且用P来表示。一个多面体P的顶点是多面体的零维面，并且P中的两个顶点u和v相邻当且仅当它们是多面体的同一个一维面中的点。一个多面体P的图（或者骨架）G(P)是一个图，它的顶点是多面体的顶点，并且它有一条边连接一对顶点，如果对应于多面体的这对顶点是相邻的，也就是这对顶点被多面体的一条边所连。这类图的研究已经有较长的历史。...In 1984, Naddef and Pulleyblank introduced the (0,1)-polytope and showed that if the graph G(P) of a (0,1)-polytope is bipartite graph, then G(P) is a hypercube; if G(P) is non-bipartite graph, then G(P) is hamilton connected. The class of (0,1)-polytope includes many well-known classes of polytope, such as: matching polytope, matorid basis polytope, stable set polytope and permutation polytope. N...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：1702005140301

Enumerating tree-like polyphenyl isomers

NSFC [10831001]Enumeration of molecules is one of the fundamental problems in bioinformatics and plays an important role in drug discovery, experimental structure elucidation (e.g., by using NMR or mass spectrometry), molecular design and virtual library construction. We consider the enumeration of tree-like polyphenyls (C(6)nH(4n+2)). For this purpose, we de fine two generating functions T (x) and R (x) involving the numbers t(n) and r(n) of tree-like polyphenyls (TL-polyphenyls) and monosubstituted tree-like polyphenyls (MTL-polyphenyls), respectively. By characterizing the symmetry groups with respect to TL-polyphenyls and MTL-polyphenyls, we establish two functional equations for these two generating functions. This yields for the first time an efficient recursion formula for calculating the numbers t(n) and r(n). The two functional equations are also the fundamentals for analyzing their asymptotic behaviors, from which we derive the precise asymptotic values for both r(n) and t(n). The resulting asymptotic values are shown to fit well to the numerical results obtained by using our recursion formula. Finally, we give an explicit enumerating expression for TL-polyphenyls of a particular type: the linear polyphenyls

Extremal polyphenyl chains concerning k-matchings and k-independent sets

Higher Education Program of Xinjiang [XJEDU2009S65]; Xinjiang Normal University [xjnubs0806]; NSFC [10831001]Denote by A(n) the set of the polyphenyl chains with n hexagons. For any A(n) is an element of A(n), let m(k)(A(n)) and m(k)(A(n)) and i(k)(A(n)) be the numbers of k-matchings and k-independent sets of A(n), respectively. In the paper, we show that for any A(n) is an element of A(n) and for any k >= 0, m(k)(M(n)) = i(k)(A(n)) >= i(k)(O(n)), with the equalities hold if A(n) = M(n) or A(n) = O(n), where M(n) and O(n) are the meta-chain and the ortho-chain, respectively. These generalize some related results in [1]

Extremal polyphenyl chains concerning k-matchings and k-independent sets

Higher Education Program of Xinjiang [XJEDU2009S65]; Xinjiang Normal University [xjnubs0806]; NSFC [10831001]Denote by A(n) the set of the polyphenyl chains with n hexagons. For any A(n) is an element of A(n), let m(k)(A(n)) and m(k)(A(n)) and i(k)(A(n)) be the numbers of k-matchings and k-independent sets of A(n), respectively. In the paper, we show that for any A(n) is an element of A(n) and for any k >= 0, m(k)(M(n)) = i(k)(A(n)) >= i(k)(O(n)), with the equalities hold if A(n) = M(n) or A(n) = O(n), where M(n) and O(n) are the meta-chain and the ortho-chain, respectively. These generalize some related results in [1]