SOME RESULTS ON BUNDLE SYSTEMS FOR A COUNTABLE DISCRETE AMENABLE GROUP ACTION

We consider the style number, independence number and entropy for a frame bundle dynamical system. The base system of which is a countable discrete amenable group action on a compact metric space. We obtain the existence of cover measures, an ergodic theorem about mean linear independence and the style number, and a variational principle for style numbers and independence numbers. We also study the relationship between the entropy of base systems and that of their bundle systems.

A POLYNOMIAL ALGORITHM FOR FINDING THEMINIMUM FEEDBACK VERTEX SET OF A3-REGULAR SIMPLE GRAPH

A subset of the vertex set of a graph is a feedback vertex set of the graph if the resulting graph is a forest after removed the vertex subset from the graph. A polynomial algorithm for finding a minimum feedback vertex set of a 3-regular simple graph is provided.

ON k-DIAMETER OF k-CONNECTED GRAPHS

Let G be a k-connected simple graph with order n. The k-diameter, combining connectivity with diameter, of G is the minimum integer d k(G) for which between any two vertices in G there are at least k internally vertex-disjoint paths of length at most d k(G). For a fixed positive integer d, some conditions to insure d k(G)≤d are given in this paper. In particular, if d≥3 and the sum of degrees of any s (s =2 or 3) nonadjacent vertices is at least n+(s-1)k+1-d, then d k(G)≤d. Furthermore, these conditions are sharp and the upper bound d of k-diameter is best possible.

RELATIONS AMONG SOME PARAMETERS OF HYPERGRAPHS

The relations among the dominating number, independence number and covering number of hypergraphs are investigated. Main results are as follows：Dv（H）≤min{α≤（H）, p（H）, p（H）, T（H）}; De（H）≤min{v（H）, T（H）, p（H）}; DT（H） ≤αT（H）; S（H）≤ Dv （H） ＋ α（H）≤n; 2≤ Dv （H） ＋ T（H） ≤n; 2 〈 Dv （H） ＋ v（H）≤n/2 ＋ [n/r]; Dv （H） ＋ p（H） 〈_n;2≤De（H） ＋ Dv（H）≤n/2 ＋ [n/r];α（H） ＋ De（H）≤n;2 ≤ De（H） ＋ v（H）≤2[n/r]; 2 De（H） ＋ p（H）≤n-r ＋ 2.

The Independence-Separation Problem on the 3-D Rook’s Graph

Both independence and independence-separation problems on chessboard graphs have been studied in detail, with hundreds of papers in the broader independence category, and several on the independence-separation problem variant for chessboard graphs. In this paper, the inde-pendence-separation problem is considered on the d-dimensional rook’s graph. A lower bound of k, for , is found for the independence-separation number on the d-dimensional rook’s graph, denoted by . For the case where , it is found that when n is odd and , . Conjecture and discussion are added.

A Method for the Solution of the 2D-Oswatitsch Equations

Corresponding to Oswatitsch’s Mach number independence principle the Mach number of hypersonic inviscid flows, , does not affect components of various non-dimensional formulations such as velocity and density, pressure coefficients and Mach number behind a strong shock. On this account, the principle is significant in the development process for hypersonic vehicles. Oswatitsch deduced a system of partial differential equations which describes hypersonic flow. These equations are the basic gasdynamic equations as well as Crocco’s theorem which are reduced for the case of very high Mach numbers, . Their numerical solution can not only result in simplified algorithms prospectively utilized to describe the flow around bodies flying mainly in the lower stratosphere with very high Mach numbers. It also offers a deeper understanding of similarity effects for hypersonic flows. In this paper, a solution method for Oswatisch’s equations for perfect gas, based on a 4-step Runge-Kutta-algorithm, is presented including a fast shock-fitting procedure. An analysis of numerical stability is followed by a detailed comparison of results for different Mach numbers and ratios of the specific heats.

具有特定独立的罗马2-控制数的树

For a graph G=(V,E),a Roman{2}-dominating function f:V→{0,1,2}has the property that for every vertex v∈V with f(v)=0,either v is adjacent to at least one vertex u for which f(u)=2,or at least two vertices u1 and u2 for which f(u1)=f(u2)=1.A Roman{2}-dominating function f=(V0,V1,V2)is called independent if V1∪V2 is an independent set.The weight of an independent Roman{2}-dominating function f is the valueω(f)=Σv∈V f(v),and the independent Roman{2}-domination number i{R2}(G)is the minimum weight of an independent Roman{2}-dominating function on G.In this paper,we characterize all trees with i{R2}(T)=γ(T)+1,and give a linear time algorithm to compute the value of i{R2}(T)for any tree T.

Sharp Bounds on the A_(α)-index of Graphs in Terms of the Independence Number

Given a graph G,the adjacency matrix and degree diagonal matrix of G are denoted by A(G)and D(G),respectively.In 2017,Nikiforov~([24])proposed the A_(α)-matrix:A_α(G)=αD(G)+(1-α)A(G),whereα∈[0,1].The largest eigenvalue of this novel matrix is called the A_(α)-index of G.In this paper,we characterize the graphs with minimum A_(α)-index among n-vertex graphs with independence number i forα∈[0,1),where i=1,[n/2],[n/2],[n/2]+1,n-3,n-2,n-1,whereas for i=2 we consider the same problem forα∈[0,3/4].Furthermore,we determine the unique graph(resp.tree)on n vertices with given independence number having the maximum A_(α)-index withα∈[0,1),whereas for the n-vertex bipartite graphs with given independence number,we characterize the unique graph having the maximum A_α-index withα∈[1/2,1).

Conflict-free Connection Number and Independence Number of a Graph

An edge-colored graph G is conflict-free connected if any two of its vertices are connected by a path,which contains a color used on exactly one of its edges.The conflict-free connection number of a connected graph G,denoted by cf c(G),is defined as the minimum number of colors that are required in order to make G conflict-free connected.In this paper,we investigate the relation between the conflict-free connection number and the independence number of a graph.We firstly show that cf c(G)≤α(G)for any connected graph G,and give an example to show that the bound is sharp.With this result,we prove that if T is a tree with?(T)≥(α(T)+2)/2,then cf c(T)=?(T).

Structure of Independent Sets in Direct Products of Some Vertex-transitive Graphs

Let Circ（r, n） be a circular graph. It is well known that its independence number α（Circ（r, n）） = r. In this paper we prove that for every vertex transitive graph H, and describe the structure of maximum independent sets in Circ（r, n） × H. As consequences, we prove for G being Kneser graphs, and the graphs defined by permutations and partial permutations, respectively. The structure of maximum independent sets in these direct products is also described.

A Note on Algebraic Independence of Numbers

1.IntroductionWe denote the set of all positive integers by N,and the fields of all rational,algebraic numbers by Q,A,respectively.Let Z~* [Z1,…,Z2] be the set of all non-zero polynomials in variables Z1,…,Z2,with integer coefficients.For α∈A we denotemaxand multiply from i=1 to α~((i)) by and N(a),respectively;where d is the degree of α,and

Binding Numbers for Fractional ID-k-factor-critical Graphs

LetG be a graph,and k≥2 be a positive integer.A graph G is fractional independentset-deletable k-factor-critical（in short,fractional ID-k-factor-critical）,if G I has a fractional k-factor for every independent set I of G.The binding number bind（G）of a graph G is defined as bind（G）=min｜NG（X）｜｜X｜：=X V（G）,NG（X）=V（G）.In this paper,it is proved that a graph G is fractional ID-k-factor-critical if n≥6k 9 and bind（G）〉（3k 1）（n 1）kn 2k＋2.

On the Relations of Graph Parameters and Its Total Parameters

In this paper we get some relations between α（G）, α＇（G）, β（G）, β＇（G） and αT（G）, βT（G）. And all bounds in these relations are best possible, where α（G）, α＇（G）,/3（G）, β（G）, αT（G） and βT（G） are the covering number, edge-covering number, independent number, edge-independent number （or matching number）, total covering number and total independent number, respectively.

Some Results on Spanning Trees

Some structures of spanning trees with many or less leaves in a connected graph are determined.We show（1） a connected graph G has a spanning tree T with minimum leaves such that T contains a longest path,and（2） a connected graph G on n vertices contains a spanning tree T with the maximum leaves such that Δ（G） =Δ（T） and the number of leaves of T is not greater than n D（G）＋1,where D（G） is the diameter of G.

Comparison between Merrifield–Simmons Index and Wiener Index of Graphs

The Merrifield–Simmons indexσis the total number of independent vertex sets(including the empty set)of the graph G.The Wiener index W is the sum of distances in all unordered pairs of vertices of G.We construct some new graphs satisfyingσ>W and W>σ,respectively.In particular,infinite graphs satisfying W>σare invented with graphs with diameter 2 and infinite ones satisfyingσ>W are discovered with so-called universally diametrical graphs.