Related papers: The lattice of cycles of an undirected graph
We initiate the algorithmic study of retracting a graph into a cycle in the graph, which seeks a mapping of the graph vertices to the cycle vertices, so as to minimize the maximum stretch of any edge, subject to the constraint that the…
Let L be a lattice ordered effect algebra. We prove that the lattice uniformities on L which make uniformly continuous the operations $\ominus$ and $\oplus$ of L are uniquely determined by their system of neighbourhoods of 0 and form a…
We give formulas, in terms of graph theoretical invariants, for the minimum distance and the generalized Hamming weights of the linear code generated by the rows of the incidence matrix of a signed graph over a finite field, and for those…
A unicellular map is the embedding of a connected graph in a surface in such a way that the complement of the graph is a topological disk. In this paper we present a bijective link between unicellular maps on a non-orientable surface and…
After recalling the definition of Grassmann algebra and elements of Grassmann--Berezin calculus, we use the expression of Pfaffians as Grassmann integrals to generalize a series of formulas relating generating functions of paths in digraphs…
We give a simple proof of Tutte's theorem stating that the cycle space of a 3--connected graph is generated by the set of non-separating circuits of the graph. Keywords: graph, cycle, circuit, cycle space, non-separating circuit, strong…
The notion of clause set cycle abstracts a family of methods for automated inductive theorem proving based on the detection of cyclic dependencies between clause sets. By discerning the underlying logical features of clause set cycles, we…
Given a finite abelian group $G$, consider the complete graph on the set of all elements of $G$. Find a Hamiltonian cycle in this graph and for each pair of consecutive vertices along the cycle compute their sum. What are the smallest and…
Graphs are used in many disciplines to model the relationships that exist between objects in a complex discrete system. Researchers may wish to compare a network of interest to a "typical" graph from a family (or ensemble) of graphs which…
Simple cycles, also known as self-avoiding polygons, are cycles on graphs which are not allowed to visit any vertex more than once. We present an exact formula for enumerating the simple cycles of any length on any directed graph involving…
We give necessary and sufficient conditions under which the Jacobian of a graph is generated by a divisor that is the difference of two vertices. This answers a question posed by Becker and Glass and allows us to prove various other…
An arithmetical structure on a finite, connected graph without loops is given by an assignment of positive integers to the vertices such that, at each vertex, the integer there is a divisor of the sum of the integers at adjacent vertices,…
We analyze the problem of discovering long cycles inside a graph. We propose and test two algorithms for this task. The first one is based on recent advances in statistical mechanics and relies on a message passing procedure. The second…
This paper investigates the independence polynomials arising from iterated strong products of cycle graphs, examining their algebraic symmetries and combinatorial structures. Leveraging modular arithmetic and Galois theory, we establish…
We consider the problem of structure learning for linear causal models based on observational data. We treat models given by possibly cyclic mixed graphs, which allow for feedback loops and effects of latent confounders. Generalizing…
A visualized graph is a powerful tool for data analysis and synthesis tasks. In this case, the task of visualization constitutes not only in displaying vertices and edges according to the graph representation, but also in ensuring that the…
It has been an open problem to find the Moore-Penrose inverses of the incidence, Laplacian, and signless Laplacian matrices of families of graphs except trees and unicyclic graphs. Since the inverse formulas for an odd unicyclic graph and…
The manuscript considers mathematical models for creating a topological drawing of a graph based on the methods of G. Ringel's vertex rotation theory. An algorithm is presented for generating a topological drawing of a flat part of a graph…
For the distributions of finitely many binary random variables, we study the interaction of restrictions of the supports with conditional independence constraints. We prove a generalization of the Hammersley-Clifford theorem for…
Arithmetical structures on a graph were introduced by Lorenzini as some intersection matrices that arise in the study of degenerating curves in algebraic geometry. In this article we study these arithmetical structures, in particular we are…