Related papers: Integer sequence discovery from small graphs
Graph inference plays an essential role in machine learning, pattern recognition, and classification. Signal processing based approaches in literature generally assume some variational property of the observed data on the graph. We make a…
We find an asymptotic enumeration formula for the number of simple $r$-uniform hypergraphs with a given degree sequence, when the number of edges is sufficiently large. The formula is given in terms of the solution of a system of equations.…
This is a collection of 1031 formulas that were generated by a computer program in 1992. The set is the database of integer sequences as of 1992 which contained 4568 sequences. These sequences were later published in the Encyclopedia of…
The three subgraphs of a connected graph induced by the center, annulus and periphery are called its metric subgraphs. The main results are as follows. (1) There exists a graph of order $n$ whose metric subgraphs are all paths if and only…
We describe recent achievements in the theory of weight systems, which are functions on chord diagrams satisfying so-called $4$-term relations. Our main attention is devoted to constructions of weight systems. The two main sources of these…
Analyzing interconnection structures among underlying entities or objects in a dataset through the use of graph analytics has been shown to provide tremendous value in many application domains. However, graphs are not the primary…
In this note we define and study graph invariants generalizing to higher dimension the maximum degree of a vertex and the vertex-connectivity (our $0$-dimensional cases). These are known to coincide almost surely in any regime for…
Labeled infinite trees provide combinatorial interpretations for many integer sequences generated by nested recurrence relations. Typically, such sequences are monotone increasing. Several of these sequences also have straightforward…
We consider the set of all graphs on n labeled vertices with prescribed degrees D=(d_1, ..., d_n). For a wide class of tame degree sequences D we prove a computationally efficient asymptotic formula approximating the number of graphs within…
We investigate the maximum size of graph families on a common vertex set of cardinality $n$ such that the symmetric difference of the edge sets of any two members of the family satisfies some prescribed condition. We solve the problem…
We discuss the problem of extending data mining approaches to cases in which data points arise in the form of individual graphs. Being able to find the intrinsic low-dimensionality in ensembles of graphs can be useful in a variety of…
This document is an exposition of an assortment of open problems arising from the exact enumeration of (perfect) matchings of finite graphs. Roughly half have been solved at the time of this writing; see the document "Twenty Open Problems…
Predicting missing links in complex networks requires algorithms that are able to explore statistical regularities in the existing data. Here we investigate the interplay between algorithm efficiency and network structures through the…
Summarization of long sequences into a concise statement is a core problem in natural language processing, requiring non-trivial understanding of the input. Based on the promising results of graph neural networks on highly structured data,…
Given a query graph that represents a pattern of interest, the emerging pattern detection problem can be viewed as a continuous query problem on a dynamic graph. We present an incremental algorithm for continuous query processing on dynamic…
A set of independence statements may define the independence structure of interest in a family of joint probability distributions. This structure is often captured by a graph that consists of nodes representing the random variables and of…
We present enumeration results on the number of connected graphs up to 10 vertices for which there is at least one other graph with the same spectrum (a cospectral mate), or at least one other graph with the same Smith normal form…
A linear inference is a valid inequality of Boolean algebra in which each variable occurs at most once on each side. In this work we leverage recently developed graphical representations of linear formulae to build an implementation that is…
Computers and algorithms play an ever-increasing role in obtaining new results in graph theory. In this survey, we present a broad range of techniques used in computer-assisted graph theory, including the exhaustive generation of all…
Directed graphs are widely used in modelling of nonsymmetric relations in various sciences and engineering disciplines. We discuss invariants of strongly connected directed graphs - minimal number of vertices or edges necessary to remove to…