给定悬挂点数的双圈图中具有最小Hosoya指数的图

Let G be a graph. The Hosoya index Z（G） of a graph G is defined to be the total number of its matchings. In this paper, we characterize the graph with the smallest Hosoya index of bicyclic graphs with given pendent vertices. Finally, we present a new proof about the smallest Hosoya index of bicyclic graphs.     

线团-收敛图

一个图的线团图就是这个图的线图的团图。对于自然数n，一个图被称为n-线团－收敛的，如果它的n次线团图同构于一个固定的图。否则称之为发散的。本刻画了线团－收敛图与发散图，给出一个线团－收敛图的构造方法，并且，讨论了线团－收敛图的线团－收敛指数。     

On the spectral radius of unicyclic graphs with prescribed degree sequence

We consider the set of unicyclic graphs with prescribed degree sequence. In this set we determine the (unique) graph with the largest spectral radius (or index) with respect to the adjacency matrix. In addition, we give a conjecture about the (unique) graph with the largest index in the set of connected graphs with prescribed degree sequence.     

The spectral characterization of graphs of index less than 2 with no path as a component

A graph is said to be determined by the adjacency and Laplacian spectrum (or to be a DS graph, for short) if there is no other non-isomorphic graph with the same adjacency and Laplacian spectrum, respectively. It is known that connected graphs of index less than 2 are determined by their adjacency spectrum. In this paper, we focus on the problem of characterization of DS graphs of index less than 2. First, we give various infinite families of cospectral graphs with respect to the adjacency matrix. Subsequently, the results will be used to characterize all DS graphs (with respect to the adjacency matrix) of index less than 2 with no path as a component. Moreover, we show that most of these graphs are DS with respect to the Laplacian matrix.     

Almost self-centered graphs

The center of a graph is the set of vertices with minimum eccentricity. Graphs in which all vertices are central are called self-centered graphs. In this paper almost self-centered (ASC) graphs are introduced as the graphs with exactly two non-central vertices. The block structure of these graphs is described and constructions for generating such graphs are proposed. Embeddings of arbitrary graphs into ASC graphs are studied. In particular it is shown that any graph can be embedded into an ASC graph of prescribed radius. Embeddings into ASC graphs of radius two are studied in more detail. ASC index of a graph G is introduced as the smallest number of vertices needed to add to G such that G is an induced subgraph of an ASC graph.     

Tricyclic graphs with maximum Merrifield-Simmons index

It is well known that the graph invariant, ‘the Merrifield-Simmons index’ is important one in structural chemistry. The connected acyclic graphs with maximal and minimal Merrifield-Simmons indices are determined by Prodinger and Tichy [H. Prodinger, R.F. Tichy, Fibonacci numbers of graphs, Fibonacci Quart. 20 (1982) 16-21]. The sharp upper and lower bounds for the Merrifield-Simmons indices of unicyclic graphs are characterized by Pedersen and Vestergaard [A.S. Pedersen, P.D. Vestergaard, The number of independent sets in unicyclic graphs, Discrete Appl. Math. 152 (2005) 246-256]. The sharp upper bound for the Merrifield-Simmons index of bicyclic graphs is obtained by Deng, Chen and Zhang [H. Deng, S. Chen, J. Zhang, The Merrifield-Simmons index in (n,n+1)-graphs, J. Math. Chem. 43 (1) (2008) 75-91]. The sharp lower bound for the Merrifield-Simmons index of bicyclic graphs is determined by Jing and Li [W. Jing, S. Li, The number of independent sets in bicyclic graphs, Ars Combin, 2008 (in press)]. In this paper, we will consider the tricyclic graph, i.e., a connected graph with cyclomatic number 3. The tricyclic graph with n vertices having maximum Merrifield-Simmons index is determined.     

Connected graphs of fixed order and size with maximal index: Some spectral bounds

The index (or spectral radius) of a simple graph is the largest eigenvalue of its adjacency matrix. For connected graphs of fixed order and size the graphs with maximal index are not yet identified (in the general case). It is known (for a long time) that these graphs are nested split graphs (or threshold graphs). In this paper we use the eigenvector techniques for getting some new (lower and upper) bounds on the index of nested split graphs. Besides we give some computational results in order to compare these bounds.     

若干图类的Wiener指数的极值

一个图的Wiener指数是指这个图中所有点对的距离和.Wiener指数在理论化学中有广泛应用. 本文刻画了给定顶点数及特定参数如色数或团数的图中Wiener指数达最小值的图, 同时也刻画了给定顶点数及团数的图中Wiener指数达最大值的图.     

On bounds for the index of double nested graphs

The index of a simple graph is the largest eigenvalue of its adjacency matrix. It is well-known that in the set of all connected graphs with fixed order and size the graphs with maximal index are nested split graphs. It was recently observed that double nested graphs assume the same role if we restrict ourselves to bipartite graphs. In this paper we provide some bounds (lower and upper) for the index of double nested graphs. Some computational results are also included.     

The largest Wiener index of unicyclic graphs given girth or maximum degree

The Wiener index of a connected graph G is defined to be the sum of all distances of pairs of distinct vertices of G. Yu and Feng (Ars Comb 94:361–369, 2010) determined the unique graph having the largest Wiener index among all unicyclic graphs given girth. In the article we first give two new graphic transformations. Then with them we determine the graphs having the second largest Wiener index among all unicyclic graphs given girth and we also determine the graphs having the largest Wiener index among all unicyclic graphs given maximum degree.     

On bags and bugs

Usual graph classes, such as complete graphs, paths, cycles and stars, frequently appear as extremal graphs in graph theory problems. Here we want to turn the reader's attention to two novel, simply defined, graph classes that appear as extremal graphs in several graph theory problems. We call them bags and bugs. As examples of problems where bags and bugs appear, we show that balanced bugs maximize the index of graphs with fixed number of vertices and diameter ?2, while odd bags maximize the index of graphs with fixed number of vertices and radius ?3.     

The Index of a Periodic Graph

For periodic graphs, a special class of infinite, but locally finite graphs, an index theory is developed that can serve in classifying these graphs and that enables connections with various graph invariants as in the case of finite graphs. The index is defined with the aid of certain finite matrices that result rather canonically from reductions of the infinite adjacency operator due to the periodicity. As a central result we derive a sharp global lower bound for the index of any periodic graph.     

Vertex and edge PI indices of Cartesian product graphs

The Padmakar-Ivan (PI) index of a graph G is the sum over all edges uv of G of the number of edges which are not equidistant from u and v. In this paper, the notion of vertex PI index of a graph is introduced. We apply this notion to compute an exact expression for the PI index of Cartesian product of graphs. This extends a result by Klavzar [On the PI index: PI-partitions and Cartesian product graphs, MATCH Commun. Math. Comput. Chem. 57 (2007) 573-586] for bipartite graphs. Some important properties of vertex PI index are also investigated.     

Independent sets and 2‐factors in edge‐chromatic‐critical graphs

In 1968, Vizing made the following two conjectures for graphs which are critical with respect to the chromatic index: (1) every critical graph has a 2‐factor, and (2) every independent vertex set in a critical graph contains at most half of the vertices. We prove both conjectures for critical graphs with many edges, and determine upper bounds for the size of independent vertex sets in those graphs. © 2003 Wiley Periodicals, Inc. J Graph Theory 45: 113–118, 2004     

Variable neighborhood search for extremal graphs. 16. Some conjectures related to the largest eigenvalue of a graph

We consider four conjectures related to the largest eigenvalue of (the adjacency matrix of) a graph (i.e., to the index of the graph). Three of them have been formulated after some experiments with the programming system AutoGraphiX, designed for finding extremal graphs with respect to given properties by the use of variable neighborhood search. The conjectures are related to the maximal value of the irregularity and spectral spread in n-vertex graphs, to a Nordhaus–Gaddum type upper bound for the index, and to the maximal value of the index for graphs with given numbers of vertices and edges. None of the conjectures has been resolved so far. We present partial results and provide some indications that the conjectures are very hard.     

一类三圈图的Wiener指数

图G的Wiener指数是指图G中所有顶点对间的距离之和,即W（G）=∑dc（u,u）,{u,u}CG其中de（u,u）表示G中顶点u,u之间的距离.三圈图是指边数与顶点数之差等于2的连通图,任意两个圈至多只有一个公共点的三圈图记为T_n~3.研究了三圈图T_n~3的Wiener指数,给出了其具有最小、次小Wiener指数的图结构.     

Combinatorial properties of boundary NLC graph languages

Node label controlled (NLC) grammars are graph grammars (operating on node labeled undirected graphs)which rewrite single nodes only and establish connections between the embedded graph and the neighbors of the rewritten node on the basis of the labels of the involved nodes only. They define (possibly infinite) languages of undirected node labeled graphs (or, if we just omit the labels, languages of unlabeled graphs). Boundary NLC (BNLC) grammars are NLC grammars with the property that whenever - in a graph already generated - two nodes may be rewritten, then these nodes are not adjacent. The graph languages generated by this type of grammars are called BNLC languages.In this paper we investigate the behaviour of graph invariants within BNLC languages. First we demonstrate that there is a dependency between the chromatic number and the clique number of graphs in BNLC languages (while this is wellknown not to be true for arbitrary graph languages). Secondly, we introduce a new graph invariant, the so-called index of a graph which seems to be very suitable for describing the adjacency structure of a graph. Then we prove that every BNLC language is of bounded index (which is shown not to be true for arbitrary graph languages). Thus we exhibit properties (concerning graph invariants) of BNLC languages which are intrinsic to this class. We use them to demonstrate that certain graph languages are not BNLC languages.     

三圈图的Harary指数

图G的Harary指数是指图G中所有顶点对间的距离倒数之和. 三圈图是指边数等于顶点数加2的连通图. 研究了三圈图的Harary数, 给出了所有三圈图中具有极大Harary指数的图的结构以及含有三个圈的三圈图中具有次大Harary指数的图的结构.     

Further results on the expected hitting time,the cover cost and the related invariants of graphs

A close relation between hitting times of the simple random walk on a graph, the Kirchhoff index, the resistance-centrality, and related invariants of unicyclic graphs is displayed. Combining graph transformations and some other techniques, sharp upper and lower bounds on the cover cost (resp. reverse cover cost) of a vertex in an $n$-vertex unicyclic graph are determined. All the corresponding extremal graphs are identified.     

Estimating the Szeged index

Lower and upper bounds on Szeged index of connected (molecular) graphs are established as well as Nordhaus–Gaddum-type results, relating the Szeged index of a graph and of its complement.