双圈图的最大Estrada指数

令Bn^＋表示顶点个数为礼的双圈二部图的集合．考虑了召吉中图依Estrada指数从大到小的排序问题．利用二部图的Estrada指数和最大特征值之间的关系,当n≥8时,得到了Bn^＋中具有最大和次大Estrada指数的图．     

图的广义Randi\'{c} Estrada 指标的界

受图的Randi\'{c} Estrada 指标和广义 Randi\'{c} 能量的启发, 定义了图的广义 Randi\'{c} Estrada 指标. 利用代数方法和初等分析方法给出了n 阶简 单连通图和 r-正则图的广义Randi\'{c} Estrada指标的上下界, 推广了Bozkurt等人有关Randi\'{c} Estrada指标的结论.     

Quadrature rule-based bounds for functions of adjacency matrices

Bounds for entries of matrix functions based on Gauss-type quadrature rules are applied to adjacency matrices associated with graphs. This technique allows to develop inexpensive and accurate upper and lower bounds for certain quantities (Estrada index, subgraph centrality, communicability) that describe properties of networks.     

New upper bounds for Estrada index of bipartite graphs

Suppose G is a graph and λ₁,λ₂,…,λ_n are the eigenvalues of G. The Estrada index EE(G) of G is defined as the sum of e^λ_i, 1≤i≤n. In this paper some new upper bounds for the Estrada index of bipartite graphs are presented. We apply our result on a (4,6)-fullerene to improve our bound given in an earlier paper.     

The Estrada index of trees

The Estrada index of a graph G is defined as , where λ₁,λ₂,…,λ_n are the eigenvalues of its adjacency matrix. We determine the unique tree with maximum Estrada index among the set of trees with given number of pendant vertices. As applications, we determine trees with maximum Estrada index among the set of trees with given matching number, independence number, and domination number, respectively. Finally, we give a proof of a conjecture in [J. Li, X. Li, L. Wang, The minimal Estrada index of trees with two maximum degree vertices, MATCH Commun. Math. Comput. Chem. 64 (2010) 799-810] on trees with minimum Estrada index among the set of trees with two adjacent vertices of maximum degree.     

图的预解Estrada指标的界的估计（英文）

n阶图G的子图中心度,即后来著名的Estrada指标定义为EE（G）=∑_（i=1）~N e~（λ2）.其中λ_1,λ_2……λ_n为图G的特征值.作为复杂网络的一种中心性测度和一种分子结构描述符,Estrada指标在许多研究领域有着广泛的应用.最近,Estrada和High-ama引进了一种新的复杂网络中心度,即∑_（i=1）~n n-1n-1λ_i：他们称之为预解中心度,后来又被称为预解Estrada指标.本文主要利用图G的顶点数和边数给出了图G的预解Estrada指标的若干界.     

On the sum of powers of Laplacian eigenvalues of bipartite graphs

For a bipartite graph G and a non-zero real α, we give bounds for the sum of the αth powers of the Laplacian eigenvalues of G using the sum of the squares of degrees, from which lower and upper bounds for the incidence energy, and lower bounds for the Kirchhoff index and the Laplacian Estrada index are deduced.     

On the Estrada and Laplacian Estrada indices of graphs

The Estrada index of a graph G is defined as , where λ₁,λ₂,…,λ_n are the eigenvalues of G. The Laplacian Estrada index of a graph G is defined as , where μ₁,μ₂,…,μ_n are the Laplacian eigenvalues of G. An edge grafting operation on a graph moves a pendent edge between two pendent paths. We study the change of Estrada index of graph under edge grafting operation between two pendent paths at two adjacent vertices. As the application, we give the result on the change of Laplacian Estrada index of bipartite graph under edge grafting operation between two pendent paths at the same vertex. We also determine the unique tree with minimum Laplacian Estrada index among the set of trees with given maximum degree, and the unique trees with maximum Laplacian Estrada indices among the set of trees with given diameter, number of pendent vertices, matching number, independence number and domination number, respectively.     

On Estrada index of trees

Let G be a graph on n vertices, and let λ₁,λ₂,…,λ_n be its eigenvalues. The Estrada index is defined as . We determine the unique tree with maximum Estrada index among the trees on n vertices with given matching number, and the unique tree with maximum Estrada index among the trees on n vertices with fixed diameter. For , we also determine the tree with maximum Estrada index among the trees on n vertices with maximum degree Δ. It gives a partial solution to the conjecture proposed by Ili? and Stevanovi? in Ref. [14].     

The energy of random graphs

In 1970s, Gutman introduced the concept of the energy E(G) for a simple graph G, which is defined as the sum of the absolute values of the eigenvalues of G. This graph invariant has attracted much attention, and many lower and upper bounds have been established for some classes of graphs among which bipartite graphs are of particular interest. But there are only a few graphs attaining the equalities of those bounds. We however obtain an exact estimate of the energy for almost all graphs by Wigner’s semi-circle law, which generalizes a result of Nikiforov. We further investigate the energy of random multipartite graphs by considering a generalization of Wigner matrix, and obtain some estimates of the energy for random multipartite graphs.     

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.     

Bounds of the Estrada index of graphs

Let G be a graph of order n and let λ1,λ2,...,λn be its eigenvalues. The Estrada index[2] of G is defined as EE = EE(G) =∑n i=1 eλi. In this paper, new bounds for EE are established, as well as some relations between EE and graph energy E.     

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.     

Estimating the Estrada index

Let G be a graph on n vertices, and let λ₁,λ₂,…,λ_n be its eigenvalues. The Estrada index of G is a recently introduced graph invariant, defined as . We establish lower and upper bounds for EE in terms of the number of vertices and number of edges. Also some inequalities between EE and the energy of G are obtained.     

Energy of generalized line graphs

The energy of a graph is equal to the sum of the absolute values of its eigenvalues. Line graphs play an important role in the study of graph theory. Generalized line graphs extend the ideas of both line graphs and cocktail party graphs. In this paper, we establish relations between the energy of the generalized line graph of a graph G and the Laplacian and signless Laplacian energies of G. We give upper and lower bounds for the energy of generalized line graphs. Finally, we present upper and lower bounds for some special graphs.     

Edge-forwarding index of star graphs and other Cayley graphs

We present a technique for building, in some Cayley graphs, a routing for which the load of every edge is almost the same. This technique enables us to find the edge-forwarding index of star graphs and complete-transposition graphs.     

关于线图和全图的原子键连通性指数

连通图G的原子键连通性(ABC)指数定义为: ABC(G)=\sum\limits_{uv\in E(G)} \sqrt{\frac{d(u)+d(v)-2}{d(u)d(v)}} , 其中E(G)为图G的边集, d(u) 和d(v)为顶点u和v的度数. 原子键连通性指数是化学图论中比较重要的连通度指数, 最近的研究表明它可以用来研究烷烃的能量信息. 给出了线图和全图的ABC指数的上界和下界, 并且证明了这些界是可达的.     

Topological efficiency under graph operations

In this paper we present explicit formulas for computing the topological efficiency index of the most important graph operations such as the Cartesian product, composition, corona, join and hierarchical product of two graphs. We apply our results to compute this distance-related invariant for some important classes of molecular graphs and nano-structures by specializing components of these graph operations.     

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.     

二树的ABC指标

The atom-bond connectivity(ABC) index of a graph G, introduced by Estrada,Torres, Rodr′?guez and Gutman in 1998, is defined as the sum of the weights(1/di+1/dj-2/didj )~(1/2) of all edges vivj of G, where di denotes the degree of the vertex vi in G. In this paper, we give an upper bound of the ABC index of a two-tree G with n vertices, that is, ABC(G) ≤(2n- 4)2~(1/2)/2+(2n-4)~(1/2)/n-1. We also determine the two-trees with the maximum and the second maximum ABC index.