Coloring, location and domination of corona graphs

A vertex coloring of a graph $G$ is an assignment of colors to the vertices of $G$ such that every two adjacent vertices of $G$ have different colors. A coloring related property of a graphs is also an assignment of colors or labels to the vertices of a graph, in which the process of labeling is done according to an extra condition. A set $S$ of vertices of a graph $G$ is a dominating set in $G$ if every vertex outside of $S$ is adjacent to at least one vertex belonging to $S$. A domination parameter of $G$ is related to those structures of a graph satisfying some domination property together with other conditions on the vertices of $G$. In this article we study several mathematical properties related to coloring, domination and location of corona graphs. We investigate the distance-$k$ colorings of corona graphs. Particularly, we obtain tight bounds for the distance-2 chromatic number and distance-3 chromatic number of corona graphs, throughout some relationships between the distance-$k$ chromatic number of corona graphs and the distance-$k$ chromatic number of its factors. Moreover, we give the exact value of the distance-$k$ chromatic number of the corona of a path and an arbitrary graph. On the other hand, we obtain bounds for the Roman dominating number and the locating-domination number of corona graphs. We give closed formulaes for the $k$-domination number, the distance-$k$ domination number, the independence domination number, the domatic number and the idomatic number of corona graphs.Comment: 18 page

Recent Conceptual Consequences of Loop Quantum Gravity. Part I: Foundational Aspects

Conceptual consequences of recent results in loop quantum gravity are collected and discussed here in view of their implications for a modern philosophy of science which is mainly understood as one that totalizes scientific insight so as to eventually achieve a consistent model of what may be called fundamental heuristics on an onto-epistemic background which is part of recently proposed transcendental materialism. This enterprise is being understood as a serious attempt of answering recent appeals to philosophy so as to provide a conceptual foundation for what is going on in modern physics, and of bridging the obvious gap between physics and philosophy. This present first part of the paper deals with foundational aspects of this enterprise, a second part will deal with its holistic aspects.Comment: 25 page

The Absolute Relativity Theory

This paper is a first presentation of a new approach of physics that we propose to refer as the Absolute Relativity Theory (ART) since it refutes the idea of a pre-existing space-time. It includes an algebraic definition of particles, interactions and Lagrangians. It proposed also a purely algebraic explanation of the passing of time phenomenon that leads to see usual Euler-Lagrange equations as the continuous version of the Knizhnik-Zamolodchikov monodromy. The identification of this monodromy with the local ones of the Lorentzian manifolds gives the Einstein equation algebraically explained in a quantized context. A fact that could lead to the unification of physics. By giving an algebraic classification of particles and interactions, the ART also proposes a new branch of physics, namely the Mass Quantification Theory, that provides a general method to calculate the characteristics of particles and interactions. Some examples are provided. The MQT also predicts the existence of as of today not yet observed particles that could be part of the dark matter. By giving a new interpretation of the weak interaction, it also suggests an interpretation of the so-called dark energy

VQGraph: Graph Vector-Quantization for Bridging GNNs and MLPs

Graph Neural Networks (GNNs) conduct message passing which aggregates local neighbors to update node representations. Such message passing leads to scalability issues in practical latency-constrained applications. To address this issue, recent methods adopt knowledge distillation (KD) to learn computationally-efficient multi-layer perceptron (MLP) by mimicking the output of GNN. However, the existing GNN representation space may not be expressive enough for representing diverse local structures of the underlying graph, which limits the knowledge transfer from GNN to MLP. Here we present a novel framework VQGraph to learn a powerful graph representation space for bridging GNNs and MLPs. We adopt the encoder of a variant of a vector-quantized variational autoencoder (VQ-VAE) as a structure-aware graph tokenizer, which explicitly represents the nodes of diverse local structures as numerous discrete tokens and constitutes a meaningful codebook. Equipped with the learned codebook, we propose a new token-based distillation objective based on soft token assignments to sufficiently transfer the structural knowledge from GNN to MLP. Extensive experiments and analyses demonstrate the strong performance of VQGraph, where we achieve new state-of-the-art performance on GNN-MLP distillation in both transductive and inductive settings across seven graph datasets. We show that VQGraph with better performance infers faster than GNNs by 828x, and also achieves accuracy improvement over GNNs and stand-alone MLPs by 3.90% and 28.05% on average, respectively. Code: https://github.com/YangLing0818/VQGraph.Comment: arXiv admin note: text overlap with arXiv:1906.00446 by other author

Loops and Knots as Topoi of Substance. Spinoza Revisited

The relationship between modern philosophy and physics is discussed. It is shown that the latter develops some need for a modernized metaphysics which shows up as an ultima philosophia of considerable heuristic value, rather than as the prima philosophia in the Aristotelian sense as it had been intended, in the first place. It is shown then, that it is the philosophy of Spinoza in fact, that can still serve as a paradigm for such an approach. In particular, Spinoza's concept of infinite substance is compared with the philosophical implications of the foundational aspects of modern physical theory. Various connotations of sub-stance are discussed within pre-geometric theories, especially with a view to the role of spin networks within quantum gravity. It is found to be useful to intro-duce a separation into physics then, so as to differ between foundational and empirical theories, respectively. This leads to a straightforward connection bet-ween foundational theories and speculative philosophy on the one hand, and between empirical theories and sceptical philosophy on the other. This might help in the end, to clarify some recent problems, such as the absence of time and causality at a fundamental level. It is implied that recent results relating to topos theory might open the way towards eventually deriving logic from physics, and also towards a possible transition from logic to hermeneutic.Comment: 42 page