On the Hosoya index of graphs

Let G be a (molecular) graph. The Hosoya index Z(G) of G is defined as the number of subsets of the edge set E(G) in which no two edges are adjacent in G, i.e., Z(G) is the total number of matchings of G. In this paper, we determine all the connected graphs G with n + 1 ≤ Z(G) ≤ 5n ? 17 for n ≥ 19. As a byproduct, the graphs of n vertices with Hosoya index from the second smallest value to the twenty first smallest value are obtained for n ≥ 19.     

心形图的匹配能序及Hosoya指标排序

以G(a, 4, b)(a 1, b 1)表示a+b+4个点的心形图.从匹配多项式的定义出发,通过递推关系式刻画出了这类心形图的匹配能级和它们的Hosoya指标的全排序.     

On the matching polynomial of subdivision graphs

Let G be a simple graph and let S(G) be the subdivision graph of G, which is obtained from G by replacing each edge of G by a path of length two. In this paper, by the Principle of Inclusion and Exclusion we express the matching polynomial and Hosoya index of S(G) in terms of the matchings of G. Particularly, if G is a regular graph or a semi-regular bipartite graph, then the closed formulae of the matching polynomial and Hosoya index of S(G) are obtained. As an application, we prove a combinatorial identity.     

Enumeration of perfect matchings of a type of Cartesian products of graphs

Let G be a graph and let Pm(G) denote the number of perfect matchings of G.We denote the path with m vertices by P_m and the Cartesian product of graphs G and H by G×H. In this paper, as the continuance of our paper [W. Yan, F. Zhang, Enumeration of perfect matchings of graphs with reflective symmetry by Pfaffians, Adv. Appl. Math. 32 (2004) 175-188], we enumerate perfect matchings in a type of Cartesian products of graphs by the Pfaffian method, which was discovered by Kasteleyn. Here are some of our results:1. Let T be a tree and let C_n denote the cycle with n vertices. Then Pm(C₄×T)=∏(2+α²), where the product ranges over all eigenvalues α of T. Moreover, we prove that Pm(C₄×T) is always a square or double a square.2. Let T be a tree. Then Pm(P₄×T)=∏(1+3α²+α⁴), where the product ranges over all non-negative eigenvalues α of T.3. Let T be a tree with a perfect matching. Then Pm(P₃×T)=∏(2+α²), where the product ranges over all positive eigenvalues α of T. Moreover, we prove that Pm(C₄×T)=[Pm(P₃×T)]².     

Rainbow number of matchings in planar graphs

The rainbow number$rb(G,H)$ for the graph $H$ in $G$ is defined to be the minimum integer $c$ such that any $c$-edge-coloring of $G$ contains a rainbow $H$. As one of the most important structures in graphs, the rainbow number of matchings has drawn much attention and has been extensively studied. Jendrol et al. initiated the rainbow number of matchings in planar graphs and they obtained bounds for the rainbow number of the matching $k{K}_2$ in the plane triangulations, where the gap between the lower and upper bounds is $O\left(k^3\right)$. In this paper, we show that the rainbow number of the matching $k{K}_2$ in maximal outerplanar graphs of order $n$ is $n+O\left(k\right)$. Using this technique, we show that the rainbow number of the matching $k{K}_2$ in some subfamilies of plane triangulations of order $n$ is $2n+O\left(k\right)$. The gaps between our lower and upper bounds are only $O\left(k\right)$.     

图的特征多项式的若干性质

图的特征多项式有许多性质，本文给出了特征多项式的指数表达式，特征多项式的导数的几个不同表达式以及高阶导数的图论意义。     

On the extremal Merrifield-Simmons index and Hosoya index of quasi-tree graphs

It is well known that the two graph invariants, “the Merrifield-Simmons index” and “the Hosoya index” are important in structural chemistry. A graph G is called a quasi-tree graph, if there exists u₀ in V(G) such that G−u₀ is a tree. In this paper, at first we characterize the n-vertex quasi-tree graphs with the largest, the second-largest, the smallest and the second-smallest Merrifield-Simmons indices. Then we characterize the n-vertex quasi-tree graphs with the largest, the second-largest, the smallest and the second-smallest Hosoya indices, as well as those n-vertex quasi-tree graphs with k pendent vertices having the smallest Hosoya index.     

On a Poset of Trees II

In this article, we study problems where one has to prove that certain graph parameter attains its maximum at the star and its minimum at the path among the trees on a fixed number of vertices. We give many applications of the so‐called generalized tree shift which seems to be a powerful tool to attack the problems of the above‐mentioned kind. We show that the generalized tree shift increases the largest eigenvalue of the adjacency matrix and Laplacian matrix, decreases the coefficients of the characteristic polynomials of these matrices in absolute value. We will prove similar theorems for the independence polynomial and the edge cover polynomial. The generalized tree shift induces a partially ordered set on trees having fixed number of vertices. The smallest element of this poset is the path, largest element is the star. Hence, the above‐mentioned results imply the extremality of the path and the star for these parameters.     

Enumeration of perfect matchings of graphs with reflective symmetry by Pfaffians

The Pfaffian method enumerating perfect matchings of plane graphs was discovered by Kasteleyn. We use this method to enumerate perfect matchings in a type of graphs with reflective symmetry which is different from the symmetric graphs considered in [J. Combin. Theory Ser. A 77 (1997) 67, MATCH—Commun. Math. Comput. Chem. 48 (2003) 117]. Here are some of our results: (1) If G is a reflective symmetric plane graph without vertices on the symmetry axis, then the number of perfect matchings of G can be expressed by a determinant of order |G|/2, where |G| denotes the number of vertices of G. (2) If G contains no subgraph which is, after the contraction of at most one cycle of odd length, an even subdivision of K_2,3, then the number of perfect matchings of G×K₂ can be expressed by a determinant of order |G|. (3) Let G be a bipartite graph without cycles of length 4s, s{1,2,…}. Then the number of perfect matchings of G×K₂ equals ∏(1+θ²)^m_θ, where the product ranges over all non-negative eigenvalues θ of G and m_θ is the multiplicity of eigenvalue θ. Particularly, if T is a tree then the number of perfect matchings of T×K₂ equals ∏(1+θ²)^m_θ, where the product ranges over all non-negative eigenvalues θ of T and m_θ is the multiplicity of eigenvalue θ.     

饱和链烃的Hosoya指数与其分子结构的匹配多项式及根

一个图的Hosoya指数实际上是该图的匹配多项式的系数之和.我们计算了40种饱和链烃的Hosoya指数和其分子结构图的匹配多项式以及匹配多项式的最大根,最后经过通过SPSS10.0软件进行曲线拟合发现各饱和链烃的沸点与Hosoya指数具有对数函数的关系,如下：其沸点b.p.（℃）=-98.674＋64.5488＊ln（...     

非正则图的最大特征值

通过对图的最大特征分量与顶点度之间的关系的刻画,得到了图的谱半径与参数最大度和次大度之间的不等关系,进而获得了简单连通非正则图的谱半径的若干上界.     

On the minimal energy ordering of trees with perfect matchings

The energy of a graph is defined as the sum of the absolute values of all eigenvalues of the adjacency matrix of the graph. Zhang and Li [F. Zhang, H. Li, On acyclic conjugated molecules with minimal energies, Discrete Appl. Math. 92 (1999) 71–84] determined the first two smallest-energy trees of a fixed size with a perfect matching and showed that the third minimal energy is between two trees. This paper characterizes trees of a fixed size with a perfect matching with third minimal, fourth minimal and fifth minimal energies for n≥86 and third minimal, fourth minimal energies for 14≤n≤84.