Ordering trees by their spectral radii

Let T be a tree with n vertices and let A（T） be the adjacency matrix of T. Spectral radius of T is the largest eigenvalue of A（T）. Wu et al. [Wu, B.F., Yuan, X.Y, and Xiao, E.L. On the spectral radii of trees, Journal of East China Normal University （Natural Science）, 3：22-28 （2004）] determined the first seven trees of order n with the smallest spectral radius. In this paper, we extend this ordering by determining the trees with the eighth to the tenth smallest spectral radius among all trees with n vertices.     

On the ordering of the Kirchhoff indices of the complements of trees and unicyclic graphs

The Kirchhoff index Kf(G) of a graph G is defined to be the sum of the resistance distances between all pairs of vertices of G. In this paper, we develop a novel method for ordering the Kirchhoff indices of the complements of trees and unicyclic graphs. With this method, we determine the first five maximum values of Kf■ and the first four maximum values of Kf(ū),where ■ and ū are the complements of a tree T and unicyclic graph U, respectively.     

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.     

On the maximal eccentric connectivity indices of graphs

For a connected simple graph G, the eccentricity ec(v) of a vertex v in G is the distance from v to a vertex farthest from v, and d(v) denotes the degree of a vertex v. The eccentric connectivity index of G, denoted by ξc(G), is defined as v∈V(G)d(v)ec(v). In this paper, we will determine the graphs with maximal eccentric connectivity index among the connected graphs with n vertices and m edges(n ≤ m ≤ n + 4), and propose a conjecture on the graphs with maximal eccentric connectivity index among the connected graphs with n vertices and m edges(m ≥ n + 5).     

Ordering Trees with Nearly Perfect Matchings by Algebraic Connectivity

Let T2k＋1 be the set of trees on 2k＋1 vertices with nearly perfect matchings and α（T） be the algebraic connectivity of a tree T. The authors determine the largest twelve values of the algebraic connectivity of the trees in T2k＋1. Specifically, 10 trees T2,T3,... ,T11 and two classes of trees T（1） and T（12） in T2k＋1 are introduced. It is shown in this paper that for each tree T^′1,T^″1∈T（1）and T^′12,T^″12∈T（12） and each i,j with 2≤i〈j≤11,α（T^′1）=α（T^″1）〉α（Tj）〉α（T^′12）=α（T^″12）.It is also shown that for each tree T with T∈T2k＋1／（T（1）∪{T2,T3,…,T11}∪T（12））,α（T^′12）〉α（T）.     

双圈图的无符号拉普拉斯谱半径和谱展

The signless Laplacian spread of a graph is defined to be the difference between the largest eigenvalue and the smallest eigenvalue of its signless Laplacian matrix. In this paper, we determine the first to llth largest signless Laplacian spectral radii in the class of bicyclic graphs with n vertices. Moreover, the unique bicyclic graph with the largest or the second largest signless Laplacian spread among the class of connected bicyclic graphs of order n is determined, respectively.     

The product of the restrained domination numbers of a graph and its complement

Let G=(V,E) be a graph.A set S■V is a restrained dominating set if every vertex in V-S is adjacent to a vertex in S and to a vertex in V-S.The restrained domination number of G,denoted γr(G),is the smallest cardinality of a restrained dominating set of G.In this paper,we show that if G is a graph of order n≥4,then γr(G)γr(G)≤2n.We also characterize the graphs achieving the upper bound.     

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.     

双圈图的Laplace谱跨度

The Laplacian spread of a graph is defined to be the difference between the largest eigenvalue and the second smallest eigenvalue of the Laplacian matrix of the graph. In our recent work, we have determined the graphs with maximal Laplacian spreads among all trees of fixed order and among all unicyclic graphs of fixed order, respectively. In this paper, we continue the work on Laplacian spread of graphs, and prove that there exist exactly two bicyclic graphs with maximal Laplacian spread among all bicyclic graphs of fixed order, which are obtained from a star by adding two incident edges and by adding two nonincident edges between the pendant vertices of the star, respectively.     

实分片代数曲线的拓扑结构

The piecewise algebraic curve is a kind generalization of the classical algebraic curve.By analyzing the topology of real algebraic curves on the triangles,a practi-caUy algrithm for analyzing the topology of piecewise algebraic curves is given.The algrithm produces a planar graph which is topologically equivalent to the piecewise algebraic curve.     

The Algebraic Connectivity of Graphs with Given Matching Number

The algebraic connectivity of a graph is the second smallest eigenvalue of the associated Laplacian matrix. In this paper, we not only characterize the extremal graphs with the maximal algebraic connectivity among all graphs of order n with given matching number, but also determine the extremal tree with the maximal algebraic connectivity among all trees of order n with given matching number.     

The smallest values of algebraic connectivity for unicyclic graphs

The algebraic connectivity of G is the second smallest eigenvalue of its Laplacian matrix. Let U_n be the set of all unicyclic graphs of order n. In this paper, we will provide the ordering of unicyclic graphs in U_n up to the last seven graphs according to their algebraic connectivities when n≥13. This extends the results of Liu and Liu [Y. Liu, Y. Liu, The ordering of unicyclic graphs with the smallest algebraic connectivity, Discrete Math. 309 (2009) 4315-4325] and Guo [J.-M. Guo, A conjecture on the algebraic connectivity of connected graphs with fixed girth, Discrete Math. 308 (2008) 5702-5711].     

A note on growing binary trees

The class ? of binary search trees is studied. A leaf is a vertex of degree 0; ?_n is the subset of ? consisting of trees with n leaves. We grow trees in ?_n from ?_n ? 1 thereby inducing a probability measure on ?_n. We will show that the expected value of the average leaf distance of t ∈ ?_n is asymptotic to log₂n as n → ∞.     

The six classes of trees with the largest algebraic connectivity

In this paper, we study the algebraic connectivity α(T) of a tree T. We introduce six Classes (C1)-(C6) of trees of order n, and prove that if T is a tree of order n?15, then if and only if , where the equality holds if and only if T is a tree in the Class (C6). At the same time we give a complete ordering of the trees in these six classes by their algebraic connectivity. In particular, we show that α(T_i)>α(T_j) if 1?i<j?6 and T_i is any tree in the Class (Ci) and T_j is any tree in the Class (Cj). We also give the values of the algebraic connectivity of the trees in these six classes. As a technique used in the proofs of the above mentioned results, we also give a complete characterization of the equality case of a well-known relation between the algebraic connectivity of a tree T and the Perron value of the bottleneck matrix of a Perron branch of T.     

A sharp Markov brothers-type inequality in the spaces L ∞ and L 1 on the segment

The inequality between the uniform norm of the derivative of order ? of an algebraic polynomial of degree n and the L ₁-norm of the polynomial itself on a segment are studied. For all ? ≥ (n ? 1)/3, the exact constant and the extremal polynomial are written out.     

Counting graphs with different numbers of spanning trees through the counting of prime partitions

Let A _n (n ? 1) be the set of all integers x such that there exists a connected graph on n vertices with precisely x spanning trees. It was shown by Sedlá?ek that |A _n | grows faster than the linear function. In this paper, we show that |A _n | grows faster than \(\sqrt n e^{(2\pi /\sqrt 3 )\sqrt {n/\log n} } \) by making use of some asymptotic results for prime partitions. The result settles a question posed in J. Sedlá?ek, On the number of spanning trees of finite graphs, ?as. Pěst. Mat., 94 (1969), 217–221.     

The smallest graphs with certain adjacency properties

A graph is said to have property P₁,_n if for every sequence of n + 1 points, there is another point adjacent only to the first point. It has previously been shown that almost all graphs have property P₁,_n. It is easy to verify that for each n, there is a cube with this property. A more delicate question asks for the construction of the smallest graphs having property P₁,_n. We find that this problem is intimately related with the discovery of the highly symmetric graphs known as cages, and are thereby enabled to resolve this question for 1?n?6.     

Sums of roots of unity in cyclotomic fields

Let ζ_n denote a primitive nth root of unity, n ≥ 4. For any integer k, 2 ≤ k ≤ n ? 2 it is shown that the diophantine equation 1 + ζ_n + ? + ζ_n^k?1 = ?α^q has no solutions with ?, α in $\text{Q}$(ζ_n), ? a root of unity, α an algebraic integer, and q an integer ≥ 2, except when n = 10, 12, or 30, where the solutions are completely determined.     

On the average order of a class of arithmetical functions

We consider a class of arithmetical functions generated by Dirichlet series that satisfy a functional equation with multiple gamma factors. We prove one- and two-sided omega theorems for the error terms associated with summatory functions of the type Σ_λn≤xa(n)(x ? λ_n)^?, where ? ≥ 0. In particular, we improve results of Hardy for the circle and Dirichlet divisor problems and results of Szegö and Walfisz for the Piltz divisor problem in algebraic number fields.     

Nombres de ramsey dans le cas orienté

Given two directed graphs G₁, G₂, the Ramsey number R(G₁,G₂) is the smallest integer n such that for any partition {U₁,U₂} of the arcs of the complete symmetric directed graph K¹_n, there exists an integer i such that the partial graph generated by U_i contains G_i as a subgraph. In this article, we determine R(P?_m,D?_n) and R(D?_m,D?_n) for some values of m and n, where P?_m denotes the directed path having m vertices and D?_m is obtained from P?_m by adding an arc from the initial vertex of P?_m to the terminal vertex.