On equivalence of Markov properties over undirected graphs

1992 ◽  
Vol 29 (3) ◽  
pp. 745-749 ◽  
Author(s):  
F. Matúš
Keyword(s):  
Conditional Independence ◽  
Undirected Graph ◽  
Undirected Graphs ◽  
Markov Properties

The dependence of coincidence of the global, local and pairwise Markov properties on the underlying undirected graph is examined. The pairs of these properties are found to be equivalent for graphs with some small excluded subgraphs. Probabilistic representations of the corresponding conditional independence structures are discussed.

On equivalence of Markov properties over undirected graphs

1992 ◽  
Vol 29 (03) ◽  
pp. 745-749 ◽  
Author(s):  
F. Matúš
Keyword(s):  
Conditional Independence ◽  
Undirected Graph ◽  
Undirected Graphs ◽  
Markov Properties

The dependence of coincidence of the global, local and pairwise Markov properties on the underlying undirected graph is examined. The pairs of these properties are found to be equivalent for graphs with some small excluded subgraphs. Probabilistic representations of the corresponding conditional independence structures are discussed.

Undirected Graphs Realizable as Graphs of Modular Lattices

1965 ◽  
Vol 17 ◽  
pp. 923-932 ◽  
Author(s):  
Laurence R. Alvarez
Keyword(s):  
Distributive Lattice ◽  
Partial Order ◽  
Directed Graph ◽  
Undirected Graph ◽  
Single Edge ◽  
Undirected Graphs ◽  
Modular Lattices

If (L, ≥) is a lattice or partial order we may think of its Hesse diagram as a directed graph, G, containing the single edge E(c, d) if and only if c covers d in (L, ≥). This graph we shall call the graph of (L, ≥). Strictly speaking it is the basis graph of (L, ≥) with the loops at each vertex removed; see (3, p. 170).We shall say that an undirected graph Gu can be realized as the graph of a (modular) (distributive) lattice if and only if there is some (modular) (distributive) lattice whose graph has Gu as its associated undirected graph.

scholarly journals Space-Efficient Fully Dynamic DFS in Undirected Graphs †

Algorithms ◽  
2019 ◽  
Vol 12 (3) ◽  
pp. 52 ◽  
Author(s):  
Kengo Nakamura ◽  
Kunihiko Sadakane
Keyword(s):  
Undirected Graph ◽  
Undirected Graphs ◽  
Edge Connectivity ◽  
Worst Case ◽  
Depth First Search ◽  
Insertions And Deletions ◽  
Adjacency List ◽  
Graph Traversal ◽  
Dynamic Connectivity ◽  
Traversal Algorithm

Depth-first search (DFS) is a well-known graph traversal algorithm and can be performed in O ( n + m ) time for a graph with n vertices and m edges. We consider the dynamic DFS problem, that is, to maintain a DFS tree of an undirected graph G under the condition that edges and vertices are gradually inserted into or deleted from G. We present an algorithm for this problem, which takes worst-case O ( m n · polylog ( n ) ) time per update and requires only ( 3 m + o ( m ) ) log n bits of space. This algorithm reduces the space usage of dynamic DFS algorithm to only 1.5 times as much space as that of the adjacency list of the graph. We also show applications of our dynamic DFS algorithm to dynamic connectivity, biconnectivity, and 2-edge-connectivity problems under vertex insertions and deletions.

scholarly journals Graphs of Morphisms of Graphs

10.37236/919 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
R. Brown ◽  
I. Morris ◽  
J. Shrimpton ◽  
C. D. Wensley
Keyword(s):  
Undirected Graph ◽  
Directed Graphs ◽  
Undirected Graphs ◽  
Single Vertex ◽  
Monoid Structure

This is an account for the combinatorially minded reader of various categories of directed and undirected graphs, and their analogies with the category of sets. As an application, the endomorphisms of a graph are in this context not only composable, giving a monoid structure, but also have a notion of adjacency, so that the set of endomorphisms is both a monoid and a graph. We extend Shrimpton's (unpublished) investigations on the morphism digraphs of reflexive digraphs to the undirected case by using an equivalence between a category of reflexive, undirected graphs and the category of reflexive, directed graphs with reversal. In so doing, we emphasise a picture of the elements of an undirected graph, as involving two types of edges with a single vertex, namely 'bands' and 'loops'. Such edges are distinguished by the behaviour of morphisms with respect to these elements.

Conditional Independence and Markov Properties

1995 ◽  
pp. 99-113
Author(s):  
H. H. Andersen ◽  
M. Højbjerre ◽  
D. Sørensen ◽  
P. S. Eriksen
Keyword(s):  
Conditional Independence ◽  
Markov Properties

Random graph languages

2017 ◽  
Vol 09 (02) ◽  
pp. 1750020
Author(s):  
Haizhong Shi ◽  
Yue Shi
Keyword(s):  
Graph Theory ◽  
Random Graph ◽  
Undirected Graph ◽  
Regular Languages ◽  
Undirected Graphs ◽  
Natural Languages ◽  
R Language ◽  
Graph Language ◽  
Random Graph Theory ◽  
Graph Languages

There tend to be no related researches regarding the relationships between graph theory and languages ever since the concept of graph-semigroup was first proposed in 1991. In 2011, after finding out the inner co-relations among digraphs, undirected graphs and languages, we proposed certain concepts including undirected graph language and digraph language; moreover, in 2014, we proposed a broaden concept–(V,R)-language and proved: (1) both undirected graph language and digraph language are (V,R)-languages; (2) both undirected graph language and digraph language are regular languages; (3) natural languages are regular languages. In this paper, we propose a new concept–Random Graph Language and build the relationships between random graph and language, which provides researchers with the possibility to do research about languages by using random graph theory.

On Rigid Undirected Graphs

1966 ◽  
Vol 18 ◽  
pp. 1237-1242 ◽  
Author(s):  
Z. Hedrlín ◽  
A. Pultr
Keyword(s):  
Undirected Graph ◽  
Identity Mapping ◽  
Undirected Graphs ◽  
Probability Methods ◽  
Almost All

By an undirected graph we mean a couple (X, R), where X is a set and R is a subset of X × X such that (x, y) ∈ R implies (y, x) ∈ R. The cardinal of X, denoted by |X|, will be called the cardinal of the graph.A mapping f:X → X is called an endomorphism of (X, R) if (x, y) ∈ R implies that (f(x), f(y)) ∈ R for all x, y ∈ R.An undirected graph (X, R) is called rigid if there is only one endomorphism of (X, R), namely the identity mapping of X.P. Erdös communicated orally that, using probability methods, it is possible to prove that almost all finite undirected graphs are rigid.

scholarly journals On 1-sum flows in undirected graphs

2016 ◽  
Vol 31 ◽  
pp. 646-665
Author(s):  
Saieed Akbari ◽  
Shmuel Friedland ◽  
Klas Markstrom ◽  
Sanaz Zare
Keyword(s):  
Linear Algebra ◽  
Regular Graph ◽  
Undirected Graph ◽  
Bipartite Graphs ◽  
Real Numbers ◽  
Undirected Graphs

Let $G=(V, E)$ be a simple undirected graph. For a given set $L\subset \mathbb{R}$, a function $\omega: E \longrightarrow L$ is called an $L$-flow. Given a vector $\gamma \in \mathbb{R}^V$, $\omega$ is a $\gamma$-$L$-flow if for each $v\in V$, the sum of the values on the edges incident to $v$ is $\gamma(v)$. If $\gamma(v)=c$, for all $v\in V$, then the $\gamma$-$L$-flow is called a $c$-sum $L$-flow. In this paper, the existence of $\gamma$-$L$-flows for various choices of sets $L$ of real numbers is studied, with an emphasis on 1-sum flows. Let $L$ be a subset of real numbers containing $0$ and denote $L^*:=L\setminus \{0\}$. Answering a question from S. Akbari, M. Kano, and S. Zare. A generalization of $0$-sum flows in graphs. \emph{Linear Algebra Appl.}, 438:3629--3634, 2013.], the bipartite graphs which admit a $1$-sum $\mathbb{R}^*$-flow or a $1$-sum $\mathbb{Z}^*$-flow are characterized. It is also shown that every $k$-regular graph, with $k$ either odd or congruent to 2 modulo 4, admits a $1$-sum $\{-1, 0, 1\}$-flow.

scholarly journals Graph Self-Transformation Model Based on the Operation of Change the End of the Edge

2020 ◽  
Vol 23 (3) ◽  
pp. 315-335
Author(s):  
Igor Borisovich Burdonov
Keyword(s):  
Connected Graph ◽  
Undirected Graph ◽  
Wiener Index ◽  
Transformation Model ◽  
Common Vertex ◽  
Distributed Network ◽  
Undirected Graphs ◽  
Limited Degree ◽  
Multiple Edges ◽  
Root Tree

We consider a distributed network whose topology is described by an undirected graph. The network itself can change its topology, using special “commands” provided by its nodes. The work proposes an extremely local atomic transformation acb of a change the end c of the edge ac, “moving” along the edge cb from vertex c to vertex b. As a result of this operation, the edge ac is removed, and the edge ab is added. Such a transformation is performed by a “command” from a common vertex c of two adjacent edges ac and cb. It is shown that from any tree you can get any other tree with the same set of vertices using only atomic transformations. If the degrees of the tree vertices are bounded by the number d (d 3), then the transformation does not violate this restriction. As an example of the purpose of such a transformation, the problems of maximizing and minimizing the Wiener index of a tree with a limited degree of vertices without changing the set of its vertices are considered. The Wiener index is the sum of pairwise distances between the vertices of a graph. The maximum Wiener index has a linear tree (a tree with two leaf vertices). For a root tree with a minimum Wiener index, its type and method for calculating the number of vertices in the branches of the neighbors of the root are determined. Two distributed algorithms are proposed: transforming a tree into a linear tree and transforming a linear tree into a tree with a minimum Wiener index. It is proved that both algorithms have complexity no higher than 2n–2, where n is the number of tree vertices. We also consider the transformation of arbitrary undirected graphs, in which there can be cycles, multiple edges and loops, without restricting the degree of the vertices. It is shown that any connected graph with n vertices can be transformed into any other connected graph with k vertices and the same number of edges in no more than 2(n+k)–2.

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