scholarly journals Kernels of Directed Graph Laplacians

10.37236/1065 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
J. S. Caughman ◽  
J. J. P. Veerman
Keyword(s):  
Directed Graph ◽  
Directed Graphs ◽  
Connected Components ◽  
General Context ◽  
Natural Basis ◽  
Theoretic Property ◽  
Graph Theoretic ◽  
Negative Edge ◽  
Graph Laplacians ◽  
Kirchhoff Matrix

Let $G$ denote a directed graph with adjacency matrix $Q$ and in-degree matrix $D$. We consider the Kirchhoff matrix $L=D-Q$, sometimes referred to as the directed Laplacian. A classical result of Kirchhoff asserts that when $G$ is undirected, the multiplicity of the eigenvalue 0 equals the number of connected components of $G$. This fact has a meaningful generalization to directed graphs, as was recently observed by Chebotarev and Agaev in 2005. Since this result has many important applications in the sciences, we offer an independent and self-contained proof of their theorem, showing in this paper that the algebraic and geometric multiplicities of 0 are equal, and that a graph-theoretic property determines the dimension of this eigenspace – namely, the number of reaches of the directed graph. We also extend their results by deriving a natural basis for the corresponding eigenspace. The results are proved in the general context of stochastic matrices, and apply equally well to directed graphs with non-negative edge weights.

Decomposable Probabilistic Influence Diagrams

1991 ◽  
Vol 5 (2) ◽  
pp. 229-243 ◽  
Author(s):  
C. C. Chyu
Keyword(s):  
Stochastic Modeling ◽  
Directed Graph ◽  
Efficient Algorithm ◽  
Conditional Independence ◽  
Directed Graphs ◽  
Influence Diagrams ◽  
Influence Diagram ◽  
Modeling Tool ◽  
Graph Theoretic ◽  
Independence Properties

Probabilistic influence diagrams are a useful stochastic modeling tool. To calculate probabilities of interest relative to a probabilistic influence diagram efficiently, it will be helpful for us to use an associated decomposable-directed graph. We first explore and discuss some graph-theoretic and conditional independence properties of decomposable probabilistic influence diagrams. These properties are helpful in providing an efficient algorithm for obtaining a posterior decomposable probabilistic influence diagram given the state of one or more observed nodes. The connection between Shachter's “sequential creation of conditionally barren nodes” concept and Lauritzen and Spiegeihalter's “moralization and triangulation” algorithm for calculating probabilities relative to a probabilistic influence diagram is made explicit. We also discuss how to use wisely the concepts of “sequential creation of conditionally barren nodes” and “merging nodes” together with the graph-theoretic properties of decomposable directed graphs to compute probabilities relative to probabilistic influence diagrams.

A Graph-Theoretic Approach to Comparing and Integrating Genetic, Physical and Sequence-Based Maps

Genetics ◽  
2003 ◽  
Vol 165 (4) ◽  
pp. 2235-2247
Author(s):  
Immanuel V Yap ◽  
David Schneider ◽  
Jon Kleinberg ◽  
David Matthews ◽  
Samuel Cartinhour ◽  
...  
Keyword(s):  
Directed Graph ◽  
Source Material ◽  
Complete Picture ◽  
Theoretic Approach ◽  
Research Groups ◽  
Genome Coverage ◽  
Graph Theoretic ◽  
Novel Approach ◽  
Graph Theoretic Approach ◽  
Locus Order

AbstractFor many species, multiple maps are available, often constructed independently by different research groups using different sets of markers and different source material. Integration of these maps provides a higher density of markers and greater genome coverage than is possible using a single study. In this article, we describe a novel approach to comparing and integrating maps by using abstract graphs. A map is modeled as a directed graph in which nodes represent mapped markers and edges define the order of adjacent markers. Independently constructed graphs representing corresponding maps from different studies are merged on the basis of their common loci. Absence of a path between two nodes indicates that their order is undetermined. A cycle indicates inconsistency among the mapping studies with regard to the order of the loci involved. The integrated graph thus produced represents a complete picture of all of the mapping studies that comprise it, including all of the ambiguities and inconsistencies among them. The objective of this representation is to guide additional research aimed at interpreting these ambiguities and inconsistencies in locus order rather than presenting a “consensus order” that ignores these problems.

scholarly journals On Directed Versions of the Hajnal–Szemerédi Theorem

2015 ◽  
Vol 24 (6) ◽  
pp. 873-928 ◽  
Author(s):  
ANDREW TREGLOWN
Keyword(s):  
Directed Graph ◽  
Minimum Degree ◽  
Directed Graphs ◽  
Perfect Matchings ◽  
Large Order ◽  
Vertex Disjoint ◽  
Removal Lemma

We say that a (di)graph G has a perfect H-packing if there exists a set of vertex-disjoint copies of H which cover all the vertices in G. The seminal Hajnal–Szemerédi theorem characterizes the minimum degree that ensures a graph G contains a perfect Kr-packing. In this paper we prove the following analogue for directed graphs: Suppose that T is a tournament on r vertices and G is a digraph of sufficiently large order n where r divides n. If G has minimum in- and outdegree at least (1−1/r)n then G contains a perfect T-packing.In the case when T is a cyclic triangle, this result verifies a recent conjecture of Czygrinow, Kierstead and Molla [4] (for large digraphs). Furthermore, in the case when T is transitive we conjecture that it suffices for every vertex in G to have sufficiently large indegree or outdegree. We prove this conjecture for transitive triangles and asymptotically for all r ⩾ 3. Our approach makes use of a result of Keevash and Mycroft [10] concerning almost perfect matchings in hypergraphs as well as the Directed Graph Removal Lemma [1, 6].

scholarly journals Convert a Strongly Connected Directed Graph to a Black-and-White 3-SAT Problem by the Balatonboglár Model

2020 ◽  
Author(s):  
Gábor Kusper ◽  
Csaba Biró
Keyword(s):  
Directed Graph ◽  
Directed Graphs ◽  
Deep Insight ◽  
Communication Graph ◽  
Sat Problem ◽  
Weak Model ◽  
Know How ◽  
Black And White ◽  
Communication Graphs ◽  
Strongly Connected

In a previous paper we defined the Black-and-White SAT problem which has exactly two solutions, where each variable is either true or false. We showed that Black-and-White $2$-SAT problems represent strongly connected directed graphs. We presented also the strong model of communication graphs. In this work we introduce two new models, the weak model, and the Balatonbogl\'{a}r model of communication graphs. A communication graph is a directed graph, where no self loops are allowed. In this work we show that the weak model of a strongly connected communication graph is a Black-and-White SAT problem. We prove a powerful theorem, the so called Transitions Theorem. This theorem states that for any model which is between the strong and the weak model, we have that this model represents strongly connected communication graphs as Blask-and-White SAT problems. We show that the Balatonbogl\'{a}r model is between the strong and the weak model, and it generates $3$-SAT problems, so the Balatonbogl\'{a}r model represents strongly connected communication graphs as Black-and-White $3$-SAT problems. Our motivation to study these models is the following: The strong model generates a $2$-SAT problem from the input directed graph, so it does not give us a deep insight how to convert a general SAT problem into a directed graph. The weak model generates huge models, because it represents all cycles, even non-simple cycles, of the input directed graph. We need something between them to gain more experience. From the Balatonbogl\'{a}r model we learned that it is enough to have a subset of a clause, which represents a cycle in the weak model, to make the Balatonbogl\'{a}r model more compact. We still do not know how to represent a SAT problem as a directed graph, but this work gives a strong link between two prominent fields of formal methods: SAT problem and directed graphs.

scholarly journals Active Semantic Relations in Layered Enterprise Architecture Development

2021 ◽  
pp. 3-16
Author(s):  
Matt Baxter ◽  
Simon Polovina ◽  
Wim Laurier ◽  
Mark von Rosing
Keyword(s):  
Directed Graph ◽  
Enterprise Architecture ◽  
Directed Graphs ◽  
Building Blocks ◽  
Semantic Relations ◽  
Formal Concept ◽  
General Applicability ◽  
Conceptual Graph ◽  
Architecture Development

AbstractEnterprise Architecture (EA) metamodels align an organisation’s business, information and technology resources so that these assets best meet the organisation’s purpose. The Layered EA Development (LEAD) Ontology enhances EA practices by a metamodel with layered metaobjects as its building blocks interconnected by semantic relations. Each metaobject connects to another metaobject by two semantic relations in opposing directions, thus highlighting how each metaobject views other metaobjects from its perspective. While the resulting two directed graphs reveal all the multiple pathways in the metamodel, more desirable would be to have one directed graph that focusses on the dependencies in the pathways. Towards this aim, using CG-FCA (where CG refers to Conceptual Graph and FCA to Formal Concept Analysis) and a LEAD case study, we determine an algorithm that elicits the active as opposed to the passive semantic relations between the metaobjects resulting in one directed graph metamodel. We also identified the general applicability of our algorithm to any metamodel that consists of triples of objects with active and passive relations.

scholarly journals Parameterized Complexity of Directed Spanner Problems

Algorithmica ◽  
2021 ◽  
Author(s):  
Fedor V. Fomin ◽  
Petr A. Golovach ◽  
William Lochet ◽  
Pranabendu Misra ◽  
Saket Saurabh ◽  
...  
Keyword(s):  
Directed Graph ◽  
Parameterized Complexity ◽  
Directed Graphs ◽  
Directed Acyclic Graphs ◽  
Polynomial Kernel ◽  
Distortion Parameter ◽  
Spanning Subgraph ◽  
Acyclic Graphs ◽  
Positive Integers ◽  
Additive Distortion

AbstractWe initiate the parameterized complexity study of minimum t-spanner problems on directed graphs. For a positive integer t, a multiplicative t-spanner of a (directed) graph G is a spanning subgraph H such that the distance between any two vertices in H is at most t times the distance between these vertices in G, that is, H keeps the distances in G up to the distortion (or stretch) factor t. An additive t-spanner is defined as a spanning subgraph that keeps the distances up to the additive distortion parameter t, that is, the distances in H and G differ by at most t. The task of Directed Multiplicative Spanner is, given a directed graph G with m arcs and positive integers t and k, decide whether G has a multiplicative t-spanner with at most $$m-k$$ m - k arcs. Similarly, Directed Additive Spanner asks whether G has an additive t-spanner with at most $$m-k$$ m - k arcs. We show that (i) Directed Multiplicative Spanner admits a polynomial kernel of size $$\mathcal {O}(k^4t^5)$$ O ( k 4 t 5 ) and can be solved in randomized $$(4t)^k\cdot n^{\mathcal {O}(1)}$$ ( 4 t ) k · n O ( 1 ) time, (ii) the weighted variant of Directed Multiplicative Spanner can be solved in $$k^{2k}\cdot n^{\mathcal {O}(1)}$$ k 2 k · n O ( 1 ) time on directed acyclic graphs, (iii) Directed Additive Spanner is $${{\,\mathrm{\mathsf{W}}\,}}[1]$$ W [ 1 ] -hard when parameterized by k for every fixed $$t\ge 1$$ t ≥ 1 even when the input graphs are restricted to be directed acyclic graphs. The latter claim contrasts with the recent result of Kobayashi from STACS 2020 that the problem for undirected graphs is $${{\,\mathrm{\mathsf{FPT}}\,}}$$ FPT when parameterized by t and k.

scholarly journals Semi-Degree Threshold for Anti-Directed Hamiltonian Cycles

10.37236/3610 ◽  
2015 ◽  
Vol 22 (4) ◽  
Author(s):  
Louis DeBiasio ◽  
Theodore Molla
Keyword(s):  
Directed Graph ◽  
Hamiltonian Cycle ◽  
Minimum Degree ◽  
Directed Graphs ◽  
Hamiltonian Cycles ◽  
Alternate Direction ◽  
Degree Conditions

In 1960 Ghouila-Houri extended Dirac's theorem to directed graphs by proving that if $D$ is a directed graph on $n$ vertices with minimum out-degree and in-degree at least $n/2$, then $D$ contains a directed Hamiltonian cycle. For directed graphs one may ask for other orientations of a Hamiltonian cycle and in 1980 Grant initiated the problem of determining minimum degree conditions for a directed graph $D$ to contain an anti-directed Hamiltonian cycle (an orientation in which consecutive edges alternate direction). We prove that for sufficiently large even $n$, if $D$ is a directed graph on $n$ vertices with minimum out-degree and in-degree at least $\frac{n}{2}+1$, then $D$ contains an anti-directed Hamiltonian cycle. In fact, we prove the stronger result that $\frac{n}{2}$ is sufficient unless $D$ is one of two counterexamples. This result is sharp.

Distributed Constrained Attitude and Position Control Using Graph Laplacians

2010 ◽  
Author(s):  
Innocent Okoloko ◽  
Yoonsoo Kim
Keyword(s):  
Attitude Control ◽  
Position Control ◽  
Dimensional Space ◽  
Stochastic Matrix ◽  
Three Dimensional ◽  
Laplacian Matrix ◽  
Semidefinite Program ◽  
Graph Theoretic ◽  
Graph Laplacians ◽  
Quadratically Constrained

We present a graph theoretic and optimization based method for attitude and position consensus of a team of communicating vehicles navigating in three dimensional space. Coordinated control of such vehicles has applications in planetary scale mobile sensor networks, and multiple vehicle navigation in general. Using the Laplacian matrix of the communication graph, and attitude quaternions, a synthesis of the optimal stochastic matrix that drives the attitudes to consensus, is done, by solving a constrained semidefinite program. This novel methodology attempts to extend quadratically constrained attitude control (Q-CAC), to the consensus framework. The solutions obtained are used to realize coordinated rendezvous, and formation acquisition, in the presence of static and dynamic obstacles.

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