Non-connected Graphs Research Articles

Permutation-equivariant quantum convolutional neural networks

Abstract The Symmetric group $S_{n}$ manifests itself in large classes of quantum systems as the invariance of certain characteristics of a quantum state with respect to permuting the qubits. Subgroups of $S_{n}$ arise, among many other contexts, to describe label symmetry of classical images with respect to spatial transformations, such as reflection or rotation. Equipped with the formalism of geometric quantum machine learning, in this study we propose the architectures of equivariant quantum convolutional neural networks (EQCNNs) adherent to $S_{n}$ and its subgroups. We demonstrate that a careful choice of pixel-to-qubit embedding order can facilitate easy construction of EQCNNs for small subgroups of $S_{n}$. Our novel EQCNN architecture corresponding to the full permutation group $S_{n}$ is built by applying all possible QCNNs with equal probability, which can also be conceptualized as a dropout strategy in quantum neural networks. For subgroups of $S_{n}$, our numerical results using MNIST datasets show better classification accuracy than non-equivariant QCNNs. The $S_{n}$-equivariant QCNN architecture shows significantly improved training and test performance than non-equivariant QCNN for classification of connected and non-connected graphs. When trained with sufficiently large number of data, the $S_{n}$-equivariant QCNN shows better average performance compared to $S_{n}$-equivariant QNN . These results contribute towards building powerful quantum machine learning architectures in permutation-symmetric systems.

Graph Transformer GANs With Graph Masked Modeling for Architectural Layout Generation.

We present a novel graph Transformer generative adversarial network (GTGAN) to learn effective graph node relations in an end-to-end fashion for challenging graph-constrained architectural layout generation tasks. The proposed graph-Transformer-based generator includes a novel graph Transformer encoder that combines graph convolutions and self-attentions in a Transformer to model both local and global interactions across connected and non-connected graph nodes. Specifically, the proposed connected node attention (CNA) and non-connected node attention (NNA) aim to capture the global relations across connected nodes and non-connected nodes in the input graph, respectively. The proposed graph modeling block (GMB) aims to exploit local vertex interactions based on a house layout topology. Moreover, we propose a new node classification-based discriminator to preserve the high-level semantic and discriminative node features for different house components. To maintain the relative spatial relationships between ground truth and predicted graphs, we also propose a novel graph-based cycle-consistency loss. Finally, we propose a novel self-guided pre-training method for graph representation learning. This approach involves simultaneous masking of nodes and edges at an elevated mask ratio (i.e., 40%) and their subsequent reconstruction using an asymmetric graph-centric autoencoder architecture. This method markedly improves the model's learning proficiency and expediency. Experiments on three challenging graph-constrained architectural layout generation tasks (i.e., house layout generation, house roof generation, and building layout generation) with three public datasets demonstrate the effectiveness of the proposed method in terms of objective quantitative scores and subjective visual realism. New state-of-the-art results are established by large margins on these three tasks.

Large Graph Sampling Algorithm for Frequent Subgraph Mining

Large graph networks frequently appear in the latest applications. Their graph structures are very large, and the interaction among the vertices makes it difficult to split the structures into separate multiple structures, thus increasing the difficulty of frequent subgraph mining. The process of calculating subgraph isomorphism often requires many calculations. Reducing the unessential structure of the graph is an effective method to improve the efficiency. Therefore, we propose a large graph sampling algorithm (RASI) based on random areas selection sampling and incorporate graph induction techniques to reduce the structure of the original graph. In addition, we find that constraining the weight of the number of vertices in the entire graph is essential to reduce the calculation of subgraph isomorphisms. This parameter is constrained in the sampling process to improve the efficiency of frequent subgraph mining. Experimental results show that RASI has more stable performance and performs better than other sampling algorithms in non-connected graphs. Mining frequent subgraphs by graph sampling can significantly improve the efficiency of mining, and the number of subgraphs before and after sampling is very similar.

Skeleton matching polytope: Realization and isomorphism

Let G be a simple graph. A set of pairwise disjoint edges is a matching of G. The matching polytope of G, denoted M(G), is the convex hull of the set of the incidence vectors of the matchings of G. The graph G(M(G)), whose vertices and edges are the vertices and edges of M(G), is the skeleton of the matching polytope of G, SMP(G) or, even simpler, SMP. In this paper, we consider the following questions: (1) Which graphs realize SMPs? (2) Given two non-isomorphic graphs, are their skeletons non-isomorphic as well? Concerning the first question, it is well-known (see Naddef and Pulleyblank (1984) and Balinski (1961) [3]) that neither non-connected graphs nor non-Hamiltonian graphs realize SMPs. Among other simple results, we characterize both planar graphs and bipartite graphs that occur as SMPs. The second question is completely solved here. For every G and H, other than G=G′∪i=1kGi and H=H′∪i=1kHi where, for i=1,…,k, Gi and Hi are isomorphic to K3 or S1,3 and G′≃H′, we prove that their skeleton matching polytopes, G(M(G)) and G(M(H)), are isomorphic graphs if and only if G and H are also isomorphic.

Semi-complement graph of lattice modules

Let L be a C-lattice and M be a lattice module over L. In this paper, we introduce the semi-complement graph of M denoted by $$\Gamma (M)$$ that is the undirected graph with all semi-complement elements of M as a vertex set, and two vertices X and Y are adjacent if and only if $$X\vee Y$$ is a semi-complement element. In this paper, we investigate some properties of $$\Gamma (M)$$ under some condition on M. For instance, we characterize non-connected graph $$\Gamma (M)$$ of a principally generated comultiplication lattice module M over a C-lattice L.

The odd origin of Gerstenhaber brackets, Batalin-Vilkovisky operators, and master equations

Using five basic principles, we treat Gerstenhaber/Lie brackets, Batalin-Vilkovisky (BV) operators, and master equations appearing in mathematical and physical contexts in a unified way. The different contexts for this are given by the different types of (Feynman) graphs that underlie the particular situation. Two of the maxims we bring forth are (1) that extending to the non-connected graphs gives a commutative multiplication forming a part of the BV structure and (2) that there is a universal odd twist that unifies and explains seemingly ad hoc choices of signs and is responsible for the BV operator being a differential. Our treatment results in uniform, general theorems. These allow us to prove new results and recover and connect many constructions that have appeared independently throughout the literature. The more general point of view also allows us to disentangle the necessary from the circumstantial.

What is Ramsey-equivalent to a clique?

A graph G is Ramsey for H if every two-colouring of the edges of G contains a monochromatic copy of H. Two graphs H and H′ are Ramsey-equivalent if every graph G is Ramsey for H if and only if it is Ramsey for H′. In this paper, we study the problem of determining which graphs are Ramsey-equivalent to the complete graph Kk. A famous theorem of Nešetřil and Rödl implies that any graph H which is Ramsey-equivalent to Kk must contain Kk. We prove that the only connected graph which is Ramsey-equivalent to Kk is itself. This gives a negative answer to the question of Szabó, Zumstein, and Zürcher on whether Kk is Ramsey-equivalent to Kk⋅K2, the graph on k+1 vertices consisting of Kk with a pendent edge.In fact, we prove a stronger result. A graph G is Ramsey minimal for a graph H if it is Ramsey for H but no proper subgraph of G is Ramsey for H. Let s(H) be the smallest minimum degree over all Ramsey minimal graphs for H. The study of s(H) was introduced by Burr, Erdős, and Lovász, where they show that s(Kk)=(k−1)2. We prove that s(Kk⋅K2)=k−1, and hence Kk and Kk⋅K2 are not Ramsey-equivalent.We also address the question of which non-connected graphs are Ramsey-equivalent to Kk. Let f(k,t) be the maximum f such that the graph H=Kk+fKt, consisting of Kk and f disjoint copies of Kt, is Ramsey-equivalent to Kk. Szabó, Zumstein, and Zürcher gave a lower bound on f(k,t). We prove an upper bound on f(k,t) which is roughly within a factor 2 of the lower bound.

ON A GRAPH RELATED TO THE MAXIMAL SUBGROUPS OF A GROUP

AbstractLet G be a finitely generated group. We investigate the graph ΓM(G), whose vertices are the maximal subgroups of G and where two vertices M1 and M2 are joined by an edge whenever M1∩M2≠1. We show that if G is a finite simple group then the graph ΓM(G) is connected and its diameter is 62 at most. We also show that if G is a finite group, then ΓM(G) either is connected or has at least two vertices and no edges. Finite groups G with a nonconnected graph ΓM(G) are classified. They are all solvable groups, and if G is a finite solvable group with a connected graph ΓM(G), then the diameter of ΓM(G) is at most 2. In the infinite case, we determine the structure of finitely generated infinite nonsimple groups G with a nonconnected graph ΓM(G). In particular, we show that if G is a finitely generated locally graded group with a nonconnected graph ΓM(G), then G must be finite.

Equivariant discrete Morse theory

In this paper, we study Forman’s discrete Morse theory in the case where a group acts on the underlying complex. We generalize the notion of a Morse matching, and obtain a theory that can be used to simplify the description of the G -homotopy type of a simplicial complex. As an application, we determine the C 2 × S n − 2 -homotopy type of the complex of non-connected graphs on n nodes.

Non-connected toric Hilbert schemes

We construct small (50 and 26 points, respectively) point sets in dimension 5 whose graphs of triangulations are not connected. These examples improve our construction in J. Amer. Math. Soc.13:3 (2000), 611–637 not only in size, but also in that the associated toric Hilbert schemes are not connected either, a question left open in that article. Additionally, the point sets can easily be put into convex position, providing examples of 5-dimensional polytopes with non-connected graph of triangulations.

All Ramsey Numbers r(K 3, G) for Connected Graphs of Order 7 and 8

In this paper we will give all Ramsey numbers r(K3, G) for connected graphs G of order 7 and 8. For smaller orders and non-connected graphs of order up to 8, the number of graphs with given Ramsey number is listed for all possible values.