A NEW CLASS OF LOCALLY CONVEX SPACES AND THE GENERALIZATION OF KALTON'S CLOSED GRAPH THEOREM

By introducing the notions of L-spaces and L_r-spaces, a complete generalization of Kalton's closed graph theorem is obtained. It points out the class of L_r-spaces is the maximal class of range spaces for the closed graph theorem when the class of domain spaces is the class of Mackey spaces with weakly * sequentially complete dual.Some examples are constructed showing that the class of L_r-spaces is strictly larger than the class of separable B_r-complete spaces.Some properties of L-spaces and L_r-spaces are discussed and the relations between B-complete (resp. B_r-complete) spaces and L-spaces (resp. L_r-spaces) are given.

Stability and Generalization of Hypergraph Collaborative Networks

Graph neural networks have been shown to be very effective in utilizing pairwise relationships across samples.Recently,there have been several successful proposals to generalize graph neural networks to hypergraph neural networks to exploit more com-plex relationships.In particular,the hypergraph collaborative networks yield superior results compared to other hypergraph neural net-works for various semi-supervised learning tasks.The collaborative network can provide high quality vertex embeddings and hyperedge embeddings together by formulating them as a joint optimization problem and by using their consistency in reconstructing the given hy-pergraph.In this paper,we aim to establish the algorithmic stability of the core layer of the collaborative network and provide generaliz--ation guarantees.The analysis sheds light on the design of hypergraph filters in collaborative networks,for instance,how the data and hypergraph filters should be scaled to achieve uniform stability of the learning process.Some experimental results on real-world datasets are presented to illustrate the theory.

A Note on General Third Geometric-arithmetic Index of Special Chemical Molecular Structures

In theoretical chemistry, the geometric-arithmetic indices were introduced to measure the stability of alkanes and the strain energy of cycloalkanes. In this note, we report the general third geometric-arithmetic index of unilateral polyomino chain and unilateral hexagonal chain. Also, the third geometric-arithmetic index of these chemical structures are presented.

Generalization performance of graph-based semisupervised classification

Semi-supervised learning has been of growing interest over the past few years and many methods have been proposed. Although various algorithms are provided to implement semi-supervised learning,there are still gaps in our understanding of the dependence of generalization error on the numbers of labeled and unlabeled data. In this paper,we consider a graph-based semi-supervised classification algorithm and establish its generalization error bounds. Our results show the close relations between the generalization performance and the structural invariants of data graph.

GENERAL THEORETICAL RESULTS ON RECTILINEAR EMBEDABILITY OF GRAPHS

In the design of certain kinds of electronic circuits the following question arises:given a non-negative integer k, what graphs admit of a plane embedding such that every edge is a broken lineformed by horizontal and vertical segments and having at mort k bends? Any such graph is said tobe k--rectilinear. No matter what k is, an obvious necessary condition for k-rectilinearity is that thedegree of each vertex does not exceed four.Our main result is that every planar graph H satisfying this condition is 3--rectilinear:in fact,it is 2--rectilinear with the only exception of the octahedron. We also outline a polynomial-timealgorithm which actually constructs a plane embedding of H with at most 2 bends (3 bends if H isthe octahedron) on each edge. The resulting embedding has the property that the total number ofbends does not exceed 2n, where n is the number of vertices of H.

On the ErdÖs Distance Conjecture in Geometry

ErdÖs asks if it is possible to have n points in general position in the plane (no three on a line or four on a circle) such that for every i (1≤i≤n-1 ) there is a distance determined by the points that occur exactly i times. So far some examples have been discovered for 2≤n≤8 [1] [2]. A solution for the 8 point is provided by I. Palasti [3]. Here two other possible solutions for the 8 point case as well as all possible answers to 4 - 7 point cases are provided and finally a brief discussion on the generalization of the problem to higher dimensions is given.

Super Edge-connectivity and Zeroth-order General Randi Index for -1≤α< 0

Let G be a connected graph with order n,minimum degree δ = δ（G） and edge-connectivity λ =λ（G）. A graph G is maximally edge-connected if λ = δ, and super edge-connected if every minimum edgecut consists of edges incident with a vertex of minimum degree. Define the zeroth-order general Randic index R_α-0（G） =Σ x∈V（G） d_G-α（x）, where dG（x） denotes the degree of the vertex x. In this paper, we present two sufficient conditions for graphs and triangle-free graphs to be super edge-connected in terms of the zeroth-order general Randic index for -1 ≤α 〈 0, respectively.

General Induced Matching Extendability of G^3

A graph G is induced matching extendable if every induced matching of G is included in a perfect matching of G. A graph G is generalized induced matching extendable if every induced matching of G is included in a maximum matching of G. A graph G is claw-free, if G dose not contain any induced subgraph isomorphic to K1,3. The k-th power of G, denoted by Gu, is the graph with vertex set V（G） in which two vertices are adjacent if and only if the distance between them is at most k in G. In this paper we show that, if the maximum matchings of G and G3 have the same cardinality, then G3 is generalized induced matching extendable. We also show that this result is best possible. As a result, we show that if G is a connected claw-flee graph, then G3 is generalized induced matching extendable.