FINITENESS OF HIGHER CODIMENSIONAL DISJOINT MINIMAL GRAPHS

We estimate the number of disjoint open subsets in Rn, which can support area-decreasing minimal graphs. This result generalizes the related results of Li-Wang and Tkachev for minimal hypersurfaces to higher codimensional case.

Bernstein type theorem of minimal Lagrangian graphs in quaternion Euclidean space H^n

In this paper, a Bernstein type theorem for minimal Lagrangian submanifolds in quaternion Euclidean space H^n is studied.

Automatic generation of min-weighted persistent formations

This paper researched into some methods for generating min-weighted rigid graphs and min-weighted persistent graphs. Rigidity and persistence are currently used in various studies on coordination and control of autonomous multi-agent formations. To minimize the communication complexity of formations and reduce energy consumption, this paper introduces the rigidity matrix and presents three algorithms for generating rain-weighted rigid and min- weighted persistent graphs. First, the existence of a min-weighted rigid graph is proved by using the rigidity matrix, and algorithm 1 is presented to generate the min-weighted rigid graphs. Second, the algorithm 2 based on the rigidity matrix is presented to direct the edges of min-weighted rigid graphs to generate min-weighted persistent graphs. Third, the formations with range constraints are considered, and algorithm 3 is presented to find whether a framework can form a min-weighted persistent formation. Finally, some simulations are given to show the efficiency of our research.

Generation of minimally persistent circle formation for a multi-agent system

In this paper, two methods of generating minimally persistent circle formation are presented. The proposed methods adopt a leader-follower strategy and all followers are firstly motivated to move into the leader＇s interaction range. Based on the information about relative angle and relative distance, two numbering schemes are proposed to generate minimally persistent circle formation. Distributed control laws are also designed to maintain the desired relative distance between agents. The distinctive features of the proposed methods are as follows. First, only 2n - 3 unilateral communication links for n agents are needed during the circle formation process and thus the communication complexity can be reduced. In addition, the formation topology is kept fixed for the whole motion and achieves a self-stability property. Finally, each follower keeps a regualr interval with its neighbors and the formation converges to a uniform circle formation. Simulation results are also provided to demonstrate the effectiveness of the proposed methods.

On the Least Eigenvalue of Graphs with Cut Vertices

Let ψ be a certain set of graphs.A graph is called a minimizing graph in the set ψ if its least eigenvalue attains the minimum among all graphs in ψ.In this paper,we determine the unique minimizing graph in ψn,where ψn denotes the set of connected graphs of order n with cut vertices.

Boundary Regularity for Minimal Graphs of Higher Codimensions

In this paper,the authors derive H¨older gradient estimates for graphic functions of minimal graphs of arbitrary codimensions over bounded open sets of Euclidean space under some suitable conditions.

Unique Factorization of Compositive Hereditary Graph Properties

A graph property is any class of graphs that is closed under isomorphisms, A graph property P is hereditary if it is closed under taking subgraphs; it is compositive if for any graphs

Optimal global regularity for minimal graphs over convex domains in hyperbolic space

We use the concept of the inside-(a,η,h)domain to construct a subsolution to the Dirichlet problem for minimal graphs over convex domains in hyperbolic space.As an application,we prove that the Hölder exponent max{1/a,1/(n+1)}for the problem is optimal for any a∈[2,+∞].

Bipartite double cover and perfect 2-matching covered graph with its algorithm

Let B（G） denote the bipartite double cover of a non-bipartite graph G with v ≥ 2 vertices and s edges. We prove that G is a perfect 2-matching covered graph if and only if B（G） is a 1-extendable graph. Furthermore, we prove that B（G） is a minimally l-extendable graph if and only if G is a minimally perfect 2-matching covered graph and for each e = xy ∈ E（G）, there is an independent set S in G such that ｜ГG（S）｜ = ｜S｜ ＋ 1, x ∈ S and ｜ГG-xy（S）｜ = ｜S｜ Then, we construct a digraph D from B（G） or G and show that D is a strongly connected digraph if and only if G is a perfect 2-matching covered graph. So we design an algorithm in O（x√vε） time that determines whether G is a perfect 2-matching covered graph or not.

On Minimal Graph Evolutions in the Hyperbolic Space

We study here minmal graph evolutions in the hyperbolic space and prove that there exists a unique smooth solution.

Social Choice Meets Graph Drawing: How to Get Subexponential Time Algorithms for Ranking and Drawing Problems

We analyze a common feature of p-Kemeny AGGregation（p-KAGG） and p-One-Sided Crossing Minimization（p-OSCM） to provide new insights and findings of interest to both the graph drawing community and the social choice community. We obtain parameterized subexponential-time algorithms for p-KAGG—a problem in social choice theory—and for p-OSCM—a problem in graph drawing. These algorithms run in time O＊（2O（√k log k））,where k is the parameter, and significantly improve the previous best algorithms with running times O.1.403k/and O.1.4656k/, respectively. We also study natural ＂above-guarantee＂ versions of these problems and show them to be fixed parameter tractable. In fact, we show that the above-guarantee versions of these problems are equivalent to a weighted variant of p-directed feedback arc set. Our results for the above-guarantee version of p-KAGG reveal an interesting contrast. We show that when the number of ＂votes＂ in the input to p-KAGG is odd the above guarantee version can still be solved in time O＊（2O（√k log k））, while if it is even then the problem cannot have a subexponential time algorithm unless the exponential time hypothesis fails（equivalently, unless FPT D M[1]）.