Related papers: Defining Recursive Predicates in Graph Orders
A numbering $f$ of a graph $G$ of order $n$ is a labeling that assigns distinct elements of the set $\left\{ 1,2,\ldots ,n\right\} $ to the vertices of $G$, where each $uv\in E\left( G\right) $ is labeled $f\left( u\right) +f\left( v\right)…
In-context learning is the ability of a pretrained model to adapt to novel and diverse downstream tasks by conditioning on prompt examples, without optimizing any parameters. While large language models have demonstrated this ability, how…
In an earlier paper the authors proved that limits of convergent graph sequences can be described by various structures, including certain 2-variable real functions called graphons, random graph models satisfying certain consistency…
Learning a graph topology to reveal the underlying relationship between data entities plays an important role in various machine learning and data analysis tasks. Under the assumption that structured data vary smoothly over a graph, the…
In this paper we study prime graphs of finite groups. The prime graph of a finite group $G$, also known as the Gruenberg-Kegel graph, is the graph with vertex set {primes dividing $|G|$} and an edge $p$-$q$ if and only if there exists an…
This paper is the continuation of the research of the author and his colleagues of the {\it canonical} decomposition of graphs. The idea of the canonical decomposition is to define the binary operation on the set of graphs and to represent…
Any graph which is not vertex transitive has a proper induced subgraph which is unique due to its structure or the way of its connection to the rest of the graph. We have called such subgraph as an anchor. Using an anchor which, in fact, is…
Let $G=(V(G),E(G))$ be a simple, finite and undirected graph of order $p$ and size $q$. For $k\ge 1$, a bijection $f: V(G)\cup E(G) \to \{k, k+1, k+2, \ldots, k+p+q-1\}$ such that $f(uv)= |f(u) - f(v)|$ for every edge $uv\in E(G)$ is said…
Deep learning-based graph generation approaches have remarkable capacities for graph data modeling, allowing them to solve a wide range of real-world problems. Making these methods able to consider different conditions during the generation…
Predictive coding is a message-passing framework initially developed to model information processing in the brain, and now also topic of research in machine learning due to some interesting properties. One of such properties is the natural…
Recursive distinctioning (RD) is a name coined by Joel Isaacson in his original patent document describing how fundamental patterns of process arise from the systematic application of operations of distinction and description upon…
We investigate the relationship between recursive enumerability and elementary frame definability in first-order predicate modal logic. On the one hand, it is well-known that every first-order predicate modal logic complete with respect to…
For any graph $G$ on $n$ vertices and for any {\em symmetric} subgraph $J$ of $K_{n,n}$, we construct an infinite sequence of graphs based on the pair $(G,J)$. The First graph in the sequence is $G$, then at each stage replacing every…
In this monography, it is proposed to consider the concepts of spectra of edge cuts and edge cycles of a graph as a basic mathematical structure for solving the problem of graph isomorphism. An edge cut is defined by an edge and the…
Nowhere dense graph classes, introduced by Nesetril and Ossona de Mendez, form a large variety of classes of "sparse graphs" including the class of planar graphs, actually all classes with excluded minors, and also bounded degree graphs and…
Distributional learning provides a framework for studying the learnability of structured languages from positive data. In this paper, we extend this framework to graph languages generated by fixed-interface clause systems. We formulate…
We study deterministic constructions of graphs for which the unique completion of low rank matrices is generically possible regardless of the values of the entries. We relate the completability to the presence of some patterns (particular…
Retrieval-augmented generation (RAG) enhances large language models by grounding outputs in retrieved knowledge. However, existing RAG methods including graph- and hypergraph-based approaches treat retrieved evidence as an unordered set,…
A successor-invariant first-order formula is a formula that has access to an auxiliary successor relation on a structure's universe, but the model relation is independent of the particular interpretation of this relation. It is well known…
In this paper, based on the contributions of Tucker (1983) and Seb{\H{o}} (1992), we generalize the concept of a sequential coloring of a graph to a framework in which the algorithm may use a coloring rule-base obtained from suitable…