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It is proved that any n-vertex, k-valent undirected simple graph, G, contains a spanning tree with at least $$\frac{{{\text{(}}k - {\text{2)}}n}}{{{\text{(5}}k - {\text{8}}{\text{.5)}}}}$$ leaves. As a result of this it is shown that any such graph contains an independent set, of size at least n/5k, with the property that deleting the vertices in this set and their incident edges does not disconnect G. This latter result is applied by giving an improved upper bound on the area required to embed arbitrary graphs into grid graphs.

Recently, learning embeddings of nodes in graphs has attracted increasing research attention. There are two main kinds of graph embedding methods, i.e., transductive embedding methods and inductive embedding methods. The former focuses on directly optimizing the embedding vectors, and the latter tries to learn a mapping function for the given nodes and features. However, little work has focused on applying the learned model from one graph to another, which is a pervasive idea in Computer Vision or Natural Language Processing. Although some of the graph neural networks (GNNs) present a similar motivation, none of them considers both the structure bias and the feature bias between graphs. In this paper, we present a novel graph embedding problem called Adaptive Task (AT), and propose a unified framework for the adaptive task, which introduces two types of alignment to learn adaptive node embeddings across graphs. Then, based on the proposed framework, a novel Graph Adaptive Embedding network (GraphAE) is designed to address the adaptive task. Furthermore, we extend GraphAE to a multi-graph version to consider a more complex adaptive situation. The extensive experimental results demonstrate that our model significantly outperforms the state-of-the-art methods, and also show that our framework can make a great improvement over a number of existing GNNs.

Research on graph representation learning (a.k.a. embedding) has received great attention in recent years and shows effective results for various types of networks. Nevertheless, few initiatives have been focused on the particular case of embeddings for bipartite graphs. In this paper, we first define the graph embedding problem in the case of bipartite graphs. Next, we propose a taxonomy of approaches used to tackle this problem and draw a description of state-of-the-art methods. Then, we establish their pros and cons with respect to conventional network embeddings. Finally, we provide a description of available resources to lead experiments on the subject.

The state-assignment problem of finite-state machines (FSMs) is addressed. State assignment is a mapping from the set of states (symbolic names) of an FSM to the set of binary codes with the objective of minimising the area of the combinational circuit required to realise the FSM. It is one of the most important optimisation problems in the automatic synthesis of sequential circuits since it has a major impact on the area, speed, power and testability of the circuits. The problem of finding an optimal state assignment is NP-hard. A new scheme is presented based on mean-field annealing (MFA) to solve the graph-embedding problem which is the main step in the state-assignment process. The MFA algorithm combines the characteristics of the simulated annealing and the Hopfield neural network. To solve the problem by MFA, the graph-embedding problem is mapped into a neural network and an energy function is formulated. Experiments over the MCNC FSM benchmarks demonstrate that the proposed MFA algorithm can produce superior results compared with the specialised methods such as the MUSTANG, NOVA and genetic algorithm.