scholarly journals Computing Persistent Homology of Directed Flag Complexes

Algorithms ◽  
2020 ◽  
Vol 13 (1) ◽  
pp. 19
Author(s):  
Daniel Lütgehetmann ◽  
Dejan Govc ◽  
Jason P. Smith ◽  
Ran Levi
Keyword(s):  
Performance Analysis ◽  
Directed Graph ◽  
Persistent Homology ◽  
Undirected Graphs ◽  
Flag Complex ◽  
Data Collections ◽  
Flag Complexes ◽  
Compute Time

We present a new computing package Flagser, designed to construct the directed flag complex of a finite directed graph, and compute persistent homology for flexibly defined filtrations on the graph and the resulting complex. The persistent homology computation part of Flagser is based on the program Ripser by U. Bauer, but is optimised specifically for large computations. The construction of the directed flag complex is done in a way that allows easy parallelisation by arbitrarily many cores. Flagser also has the option of working with undirected graphs. For homology computations Flagser has an approximate option, which shortens compute time with remarkable accuracy. We demonstrate the power of Flagser by applying it to the construction of the directed flag complex of digital reconstructions of brain microcircuitry by the Blue Brain Project and several other examples. In some instances we perform computation of homology. For a more complete performance analysis, we also apply Flagser to some other data collections. In all cases the hardware used in the computation, the use of memory and the compute time are recorded.

scholarly journals Computing Persistent Homology of Directed Flag Complexes

2019 ◽  
Author(s):  
Daniel Luetgehetmann ◽  
Dejan Govc ◽  
Jason P. Smith ◽  
Ran Levi
Keyword(s):  
Performance Analysis ◽  
Directed Graph ◽  
Persistent Homology ◽  
Undirected Graphs ◽  
Flag Complex ◽  
Data Collections ◽  
Flag Complexes ◽  
Compute Time

We present a new computing package Flagser, designed to construct the directed flag complex of a finite directed graph, and compute persistent homology for flexibly defined filtrations on the graph and the resulting complex. The persistent homology computation part of Flagser is based on the program Ripser [2], but is optimised specifically for large computations. The construction of the directed flag complex is done in a way that allows easy parallelisation by arbitrarily many cores. Flagser also has the option of working with undirected graphs. For homology computations Flagser has an Approximate option, which shortens compute time with remarkable accuracy. We demonstrate the power of Flagser by applying it to the construction of the directed flag complex of digital reconstructions of brain microcircuitry by the Blue Brain Project and several other examples. In some instances we perform computation of homology. For a more complete performance analysis, we also apply Flagser to some other data collections. In all cases the hardware used in the computation, the use of memory and the compute time are recorded.

scholarly journals Intrinsic linking and knotting in tournaments

2019 ◽  
Vol 28 (12) ◽  
pp. 1950076
Author(s):  
Thomas Fleming ◽  
Joel Foisy
Keyword(s):  
Directed Graph ◽  
Upper Bound ◽  
Directed Edge ◽  
Undirected Graphs ◽  
Oriented Cycle ◽  
Minimum Number ◽  
The Difference

A directed graph [Formula: see text] is intrinsically linked if every embedding of that graph contains a nonsplit link [Formula: see text], where each component of [Formula: see text] is a consistently oriented cycle in [Formula: see text]. A tournament is a directed graph where each pair of vertices is connected by exactly one directed edge. We consider intrinsic linking and knotting in tournaments, and study the minimum number of vertices required for a tournament to have various intrinsic linking or knotting properties. We produce the following bounds: intrinsically linked ([Formula: see text]), intrinsically knotted ([Formula: see text]), intrinsically 3-linked ([Formula: see text]), intrinsically 4-linked ([Formula: see text]), intrinsically 5-linked ([Formula: see text]), intrinsically [Formula: see text]-linked ([Formula: see text]), intrinsically linked with knotted components ([Formula: see text]), and the disjoint linking property ([Formula: see text]). We also introduce the consistency gap, which measures the difference in the order of a graph required for intrinsic [Formula: see text]-linking in tournaments versus undirected graphs. We conjecture the consistency gap to be nondecreasing in [Formula: see text], and provide an upper bound at each [Formula: see text].

scholarly journals The Diameter of Almost Eulerian Digraphs

10.37236/429 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Peter Dankelmann ◽  
L. Volkmann
Keyword(s):  
Graph Theory ◽  
Directed Graph ◽  
Upper Bound ◽  
Minimum Degree ◽  
Additive Constant ◽  
Undirected Graphs

Soares [J. Graph Theory 1992] showed that the well known upper bound $\frac{3}{\delta+1}n+O(1)$ on the diameter of undirected graphs of order $n$ and minimum degree $\delta$ also holds for digraphs, provided they are eulerian. In this paper we investigate if similar bounds can be given for digraphs that are, in some sense, close to being eulerian. In particular we show that a directed graph of order $n$ and minimum degree $\delta$ whose arc set can be partitioned into $s$ trails, where $s\leq \delta-2$, has diameter at most $3 ( \delta+1 - \frac{s}{3})^{-1}n+O(1)$. If $s$ also divides $\delta-2$, then we show the diameter to be at most $3(\delta+1 - \frac{(\delta-2)s}{3(\delta-2)+s} )^{-1}n+O(1)$. The latter bound is sharp, apart from an additive constant. As a corollary we obtain the sharp upper bound $3( \delta+1 - \frac{\delta-2}{3\delta-5})^{-1} n + O(1)$ on the diameter of digraphs that have an eulerian trail.

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 Digraphs with eulerian chains

1986 ◽  
Vol 41 (3) ◽  
pp. 298-303
Author(s):  
D. K. Skilton
Keyword(s):  
Directed Graph ◽  
Undirected Graphs ◽  
Survey Paper ◽  
Similar Work

AbstractAn eulerian chain in a directed graph is a continuous directed route which traces every arc of the digraph exactly once. Such a route may be finite or infinite, and may have 0, 1 or 2 end vertices. For each kind of eulerian chain, there is a characterization of those diagraphs possessing such a route. In this survey paper we strealine these characterizations, and then synthesize them into a single description of all digraphs having some eulerian chain. Similar work has been done for eulerian chains in undirected graphs, so we are able to compare corresponding results for graphs and digraphs.

scholarly journals Quantum Symmetries of Graph C*-algebras

2018 ◽  
Vol 61 (4) ◽  
pp. 848-864 ◽  
Author(s):  
Simon Schmidt ◽  
Moritz Weber
Keyword(s):  
Automorphism Group ◽  
Directed Graph ◽  
Operator Algebras ◽  
Automorphism Groups ◽  
Undirected Graphs ◽  
C Algebras ◽  
Quantum Symmetry ◽  
Quantum Symmetries ◽  
Multiple Edges ◽  
Definition Of

AbstractThe study of graph C*-algebras has a long history in operator algebras. Surprisingly, their quantum symmetries have not yet been computed. We close this gap by proving that the quantum automorphism group of a finite, directed graph without multiple edges acts maximally on the corresponding graph C*-algebra. This shows that the quantum symmetry of a graph coincides with the quantum symmetry of the graph C*-algebra. In our result, we use the definition of quantum automorphism groups of graphs as given by Banica in 2005. Note that Bichon gave a different definition in 2003; our action is inspired from his work. We review and compare these two definitions and we give a complete table of quantum automorphism groups (with respect to either of the two definitions) for undirected graphs on four vertices.

scholarly journals A Construction for the Hat Problem on a Directed Graph

10.37236/1994 ◽  
2012 ◽  
Vol 19 (1) ◽  
Author(s):  
Rani Hod ◽  
Marcin Krzywkowski
Keyword(s):  
Directed Graph ◽  
Directed Graphs ◽  
Maximum Clique ◽  
Clique Number ◽  
Bipartite Graphs ◽  
The Other ◽  
Complete Graphs ◽  
Undirected Graphs ◽  
Asymptotically Optimal

A team of $n$ players plays the following game. After a strategy session, each player is randomly fitted with a blue or red hat. Then, without further communication, everybody can try to guess simultaneously his own hat color by looking at the hat colors of the other players. Visibility is defined by a directed graph; that is, vertices correspond to players, and a player can see each player to whom he is connected by an arc. The team wins if at least one player guesses his hat color correctly, and no one guesses his hat color wrong; otherwise the team loses. The team aims to maximize the probability of a win, and this maximum is called the hat number of the graph.Previous works focused on the hat problem on complete graphs and on undirected graphs. Some cases were solved, e.g., complete graphs of certain orders, trees, cycles, and bipartite graphs. These led Uriel Feige to conjecture that the hat number of any graph is equal to the hat number of its maximum clique.We show that the conjecture does not hold for directed graphs. Moreover, for every value of the maximum clique size, we provide a tight characterization of the range of possible values of the hat number. We construct families of directed graphs with a fixed clique number the hat number of which is asymptotically optimal. We also determine the hat number of tournaments to be one half.

scholarly journals Simplicial Complexes are Game Complexes

10.37236/6958 ◽  
2019 ◽  
Vol 26 (3) ◽  
Author(s):  
Sara Faridi ◽  
Svenja Huntemann ◽  
Richard J. Nowakowski
Keyword(s):  
Simplicial Complex ◽  
Simplicial Complexes ◽  
Combinatorial Games ◽  
Natural Question ◽  
Flag Complex ◽  
Game Trees ◽  
Game Value ◽  
Flag Complexes

Strong placement games (SP-games) are a class of combinatorial games whose structure allows one to describe the game via simplicial complexes. A natural question is whether well-known parameters of combinatorial games, such as "game value", appear as invariants of the simplicial complexes. This paper is the first step in that direction. We show that every simplicial complex encodes a certain type of SP-game (called an "invariant SP-game") whose ruleset is independent of the board it is played on. We also show that in the class of SP-games isomorphic simplicial complexes correspond to isomorphic game trees, and hence equal game values. We also study a subclass of SP-games corresponding to flag complexes, showing that there is always a game whose corresponding complex is a flag complex no matter which board it is played on.

A Note on Undirected Graphs Realizable as P.O. Sets

1970 ◽  
Vol 13 (3) ◽  
pp. 371-374 ◽  
Author(s):  
C. E. Haff ◽  
U. S. R. Murty ◽  
R. C. Wilton
Keyword(s):  
Directed Graph ◽  
Undirected Graphs ◽  
Vertex Set ◽  
Multiple Edges

Let (P, ≥) be a p.o. set. The basis graph of (P, ≥) is defined to be the directed graph whose vertex set is P and in which the ordered pair 〈a, b〉 is an edge if and only if b covers a in (P, ≥).Let D be a directed graph. All graphs considered in this note are finite and are free of loops and multiple edges.

scholarly journals The homotopy theory of polyhedral products associated with flag complexes

2018 ◽  
Vol 155 (1) ◽  
pp. 206-228
Author(s):  
Taras Panov ◽  
Stephen Theriault
Keyword(s):  
Simplicial Complex ◽  
Homotopy Theory ◽  
Chordal Graph ◽  
Polyhedral Product ◽  
Flag Complex ◽  
Vertex Set ◽  
Flag Complexes

If $K$ is a simplicial complex on $m$ vertices, the flagification of $K$ is the minimal flag complex $K^{f}$ on the same vertex set that contains $K$. Letting $L$ be the set of vertices, there is a sequence of simplicial inclusions $L\stackrel{}{\longrightarrow }K\stackrel{}{\longrightarrow }K^{f}$. This induces a sequence of maps of polyhedral products $(\text{}\underline{X},\text{}\underline{A})^{L}\stackrel{g}{\longrightarrow }(\text{}\underline{X},\text{}\underline{A})^{K}\stackrel{f}{\longrightarrow }(\text{}\underline{X},\text{}\underline{A})^{K^{f}}$. We show that $\unicode[STIX]{x1D6FA}f$ and $\unicode[STIX]{x1D6FA}f\circ \unicode[STIX]{x1D6FA}g$ have right homotopy inverses and draw consequences. For a flag complex $K$ the polyhedral product of the form $(\text{}\underline{CY},\text{}\underline{Y})^{K}$ is a co-$H$-space if and only if the 1-skeleton of $K$ is a chordal graph, and we deduce that the maps $f$ and $f\circ g$ have right homotopy inverses in this case.

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