Related papers: The lattice of cycles of an undirected graph
In this paper we determine, under some mild restrictions, the lattice of submodules $\gL$ of a module $M$ all of whose composition factors have multiplicity one. Such a lattice is distributive, and hence determined by its poset of down-sets…
We study the properties of finite graphs in which the ball of radius $r$ around each vertex induces a graph isomorphic to some fixed graph $F$. This is a natural extension of the study of regular graphs, and of the study of graphs of…
Mutually unbiased bases that can be cyclically generated by a single unitary operator are of special interest, since they can be readily implemented in practice. We show that, for a system of qubits, finding such a generator can be cast as…
We construct a dynamical lattice model based on a crossed module of possibly non-abelian finite groups. Its degrees of freedom are defined on links and plaquettes, while gauge transformations are based on vertices and links of the…
In this paper, we introduce a family of graphs which is a generalization of zero-divisor graphs and compute an upper-bound for the diameter of such graphs. We also investigate their cycles and cores.
We consider Laplacians on $\Z^2$-periodic discrete graphs. The following results are obtained: 1) The Floquet-Bloch decomposition is constructed and basic properties are derived. 2) The estimates of the Lebesgue measure of the spectrum in…
Complex bases, along with direct-sums defined by rings of imaginary quadratic integers, induce algebraic lattices. In this work, we study such lattices and their reduction algorithms. Firstly, when the lattice is spanned over a two…
Consider a family of graphs having a fixed girth and a large size. We give an optimal lower asymptotic bound on the number of even cycles of any constant length, as the order of the graphs tends to infinity.
This work is devoted to the study of the relationships between graph theory and the qualitative analysis of ordinary differential equations, with a special focus on two-dimensional systems. In particular, we reinterpret classical results…
In order to find Hamiltonian cycle, algorithm should find edges that creates a Hamiltonian cycle. Higher number of edges creates more possibilities to check to solve the problem. Algorithm rests on analysis of original graph and opposite…
In this paper, we define a class of auxiliary graphs associated with simple undirected graphs. This class of auxiliary graphs is based on the set of spanning trees of the original graph and the edges constituting those spanning trees. A…
Matroids generalize the familiar notion of linear dependence from linear algebra. Following a brief discussion of founding work in computability and matroids, we use the techniques of reverse mathematics to determine the logical strength of…
We study recursively defined functions associated with directed graphs on the k dimensional nonnegative integral lattice. The existence of certain combinatorial structures associated with these function classes are shown to be independent…
Rough sets are efficient for data pre-processing in data mining. Matroids are based on linear algebra and graph theory, and have a variety of applications in many fields. Both rough sets and matroids are closely related to lattices. For a…
A biased graph consists of a graph $G$ together with a collection of distinguished cycles of $G$, called balanced cycles, with the property that no theta subgraph contains exactly two balanced cycles. Perhaps the most natural biased graphs…
In this work we obtain basis for the null space of unicyclic graphs. We extend the null decomposition of trees from [11] for unicyclic graphs. As an application, we obtain closed formulas for the independence and matching numbers of…
The critical group of a graph is a finite abelian group whose order is the number of spanning forests of the graph. This paper provides three basic structural results on the critical group of a line graph. The first deals with connected…
A set $B$ is a basis for a vector space $V$ if every element of $V$ can be uniquely written as a linear combination of the elements of $B$. There is a similar definition of a basis for a finite group. We show that certain semidirect…
The Laplacian matrix and its pseudo-inverse for a strongly connected directed graph is fundamental in computing many properties of a directed graph. Examples include random-walk centrality and betweenness measures, average hitting and…
We develop the theory of linear evolution equations associated with the adjacency matrix of a graph, focusing in particular on infinite graphs of two kinds: uniformly locally finite graphs as well as locally finite line graphs. We discuss…