Related papers: On compositions of numbers and graphs
A solution is proposed for the problem of composition of ordinary generating functions. A new class of functions that provides a composition of ordinary generating functions is introduced; main theorems are presented; compositae are written…
A graph is a data structure composed of dots (i.e. vertices) and lines (i.e. edges). The dots and lines of a graph can be organized into intricate arrangements. The ability for a graph to denote objects and their relationships to one…
This report concerns the information content of a graph, optionally conditional on one or more background, "common knowledge" graphs. It describes an algorithm to estimate this information content, and includes some examples based on…
In this text I present some problems which led to the introduction of special kinds of graphs as tools for studying singular points of algebraic surfaces. I explain how such graphs were first described using words, and how several…
Knopfmacher et all [1] was introduced the graph compositions` notion. In this note we add to these a new construction of tree-like graphs where nodes are graphs themselves. The first examples of these tree-like compositions, a corresponding…
A critical review is presented on the most recent attempt to generally explain the notion of "statistical symmetry". This particular explanation, however, is incomplete and misses one important and essential aspect. The aim of this short…
Motivated by the remarkable interplay between (chordal) graphs and matrix algebra, we associate to each graph a so-called completion number that might encode some aspects of that interplay. We show that this number is not trivial, and we…
Sampling algorithms, hypergraph degree sequences, and polytopes play a crucial role in statistical analysis of network data. This article offers a brief overview of open problems in this area of discrete mathematics from the point of view…
This book collects the lectures about graph theory and its applications which were given to students of mathematical departments of Moscow State University and Peking University. Graph theory is a very wide field with a lot of applications…
The analysis of solutions to algebraic equations is further simplified. A couple of functions and their analytic continuation or root findings are required.
We survey two decades of work on the (sequential) topological complexity of configuration spaces of graphs (ordered and unordered), aiming to give an account that is unifying, elementary, and self-contained. We discuss the traditional…
The point of view of these notes on the topic is to bring out the flavour that Representation Theory is an extension of the first course on Group Theory. We also emphasize the importance of the base field. These notes cover completely the…
In this paper we discuss reconstruction problems for graphs. We develop some new ideas like isomorphic extension of isomorphic graphs, partitioning of vertex sets into sets of equivalent points, subdeck property, etc. and develop an…
This is a research monograph on constructions of and group actions on countable homogeneous graphs, concentrating particularly on the simple random graph and its edge-coloured variants. We study various aspects of the graphs, but the…
The Union Closed Sets Conjecture is one of the most renowned problems in combinatorics. Its appeal lies in the simplicity of its statement contrasted with the potential complexity of its resolution. The conjecture posits that, in any union…
These informal notes are concerned with spaces of functions in various situations, including continuous functions on topological spaces, holomorphic functions of one or more complex variables, and so on.
The purpose of this note is twofold. First, we survey results on the construction of large class groups of number fields by specialization of finite covers of curves. Then we give examples of applications of these techniques.
This note describes necessary and sufficient conditions for a sequence of positive integers to be the degree sequence of a connected simple graph. Conditions are also given under which a sequence is necessarily connected i.e. the sequence…
The main purpose of this survey is to introduce an inexperienced reader to additive prime number theory and some related branches of analytic number theory. We state the main problems in the field, sketch their history and the basic…
The theory of graphons has proven to be a powerful tool in many areas of graph theory. In this paper, we introduce several foundational aspects of the theory of digraphons -- asymmetric two-variable functions that arise as limits of…