Universality of causal graph dynamics

Intrinsic Universality of Causal Graph Dynamics

Causal graph dynamics are transformations over graphs that capture two important symmetries of physics, namely causality and homogeneity. They can be equivalently defined as continuous and translation invariant transformations or functions induced by a local rule applied simultaneously on every vertex of the graph. Intrinsic universality is the ability of an instance of a model to simulate every other instance of the model while preserving the structure of the computation at every step of the simulation. In this work we present the construction of a family of intrinsically universal instances …

Universality of intervals of line graph order

We prove that for every $d\geq 3$ the homomorphism order of the class of line graphs of finite graphs with maximal degree $d$ is universal. This means that every finite or countably infinite partially ordered set may be represented by line graphs of graphs with maximal degree $d$ ordered by the existence of a homomorphism.

Universality for Barycentric subdivision

The spectrum of the Laplacian of successive Barycentric subdivisions of a graph converges exponentially fast to a limit which only depends on the clique number of the initial graph and not on the graph itself. The proof uses an explicit linear operator mapping the clique vector of a graph to the clique vector of the Barycentric refinement. The eigenvectors of its transpose produce integral geometric invariants for which Euler characteristic is one example.

特征多面体的普适性与图形设计

Eigenpolytope Universality and Graphical Designs

本文证明特征多面体具有普适性，并通过 Gale 对偶建立图形设计与特征多面体面的双射，进而得出最小图形设计的存在性、上界及三个复杂性结果。

Reversible Causal Graph Dynamics

Causal Graph Dynamics extend Cellular Automata to arbitrary, bounded-degree, time-varying graphs. The whole graph evolves in discrete time steps, and this global evolution is required to have a number of physics-like symmetries: shift-invariance (it acts everywhere the same) and causality (information has a bounded speed of propagation). We add a further physics-like symmetry, namely reversibility. KEYWORDS: Bijective, invertible, injective, surjective, one-to-one, onto, Cayley graphs, Hedlund, Block representation, Lattice-gas automaton, Reversible Cellular Automata.

德拉汉定理诱导普遍性的新证明

A new proof of Delahan's induced-universality result

本文通过引入生成索引集的概念，给出了德拉汉定理诱导普遍性结果的简短自洽证明。