10 research outputs found
Towards a characterization of convergent sequences of -line graphs
Let and be graphs such that has at least 3 vertices and is
connected. The -line graph of , denoted by , is that graph whose
vertices are the edges of and where two vertices of are adjacent if
they are adjacent in and lie in a common copy of . For each nonnegative
integer , let denote the -th iteration of the -line graph
of . We say that the sequence converges if there exists a
positive integer such that , and for
we set as the set of all graphs whose sequence
converges when . The sets and
have been characterized. To progress towards the characterization of
in general, this paper defines and studies the following property:
a graph is minimally -convergent if but no proper
subgraph of is in . In addition, prove conditions that imply
divergence, and use these results to develop some of the properties of
minimally -convergent graphs.Comment: 11 pages, 11 figure
On coloring digraphs with forbidden induced subgraphs
This thesis mainly focuses on the structural properties of digraphs with high dichromatic number. The dichromatic number of a digraph , denoted by \dichi(D), is designed to be the directed analog of the chromatic number of a graph , denoted by . The field of -boundedness studies the induced subgraphs that need to be present in a graph with high chromatic number. In this thesis, we study the equivalent of -boundedness but with dichromatic number instead. In particular, we study the induced subgraphs of digraphs with high dichromatic number from two different perspectives which we describe below.
First, we present results in the area of heroes. A digraph is a hero of a class of digraphs if there exists a constant such that every -free digraph has \dichi(D)\leq c. It is already known that when is the family of -free digraphs for some digraph , the existence of heroes that are not transitive tournaments implies that is the disjoint union of oriented stars. In this thesis, we narrow down the characterization of the digraphs which have heroes that are not transitive tournaments to the disjoint union of oriented stars of degree at most 4. Furthermore, we provide a big step towards the characterization of heroes in -free digraphs, where . We achieve the latter by developing mathematical tools for proving that a hero in -free digraphs is also a hero in -free digraphs.
Second, we present results in the area of \dichi-boundedness. In this area, we try to determine the classes of digraphs for which transitive tournaments are heroes. In particular, we ask whether, for a given class of digraphs , there exists a function such that, for every , \dichi(D)\leq f(k) whenever and is -free. We provide a comprehensive literature review of the subject and outline the -boundedness results that have an equivalent result in \dichi-boundedness. We conclude by generalizing a key lemma in the literature and using it to prove -free digraphs are \dichi-bounded, where and are small brooms whose orientations are related and have an additional particular property
A counterexample to a conjecture about triangle-free induced subgraphs of graphs with large chromatic number
We prove that for every , there is a graph with and
such that every induced subgraph of with satisfies .
This disproves a well-known conjecture. Our construction is a digraph with
bounded clique number, large dichromatic number, and no induced directed cycles
of odd length at least 5.Comment: Moving one of the results to a different paper, where it fits bette
Digraphs with All Induced Directed Cycles of the Same Length are not → χ -Bounded
For t > 2, let us call a digraph D t-chordal if all induced directed cycles in D have length equal to t. In an earlier paper, we asked for which t it is true that t-chordal graphs with bounded clique number have bounded dichromatic number. Recently, Aboulker, Bousquet, and de Verclos answered this in the negative for t = 3, that is, they gave a construction of 3-chordal digraphs with clique number at most 3 and arbitrarily large dichromatic number. In this paper, we extend their result, giving for each t > 3 a construction of t-chordal digraphs with clique number at most 3 and arbitrarily large dichromatic number, thus answering our question in the negative. On the other hand, we show that a more restricted class, digraphs with no induced directed cycle of length less than t, and no induced directed t-vertex path, have bounded dichromatic number if their clique number is bounded. We also show the following complexity result: for fixed t > 2, the problem of determining whether a digraph is t-chordal is coNP-complete.This research was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC), RGPIN-2020-03912
On the importance of catalyst-adsorbate 3D interactions for relaxed energy predictions
The use of machine learning for material property prediction and discovery
has traditionally centered on graph neural networks that incorporate the
geometric configuration of all atoms. However, in practice not all this
information may be readily available, e.g.~when evaluating the potentially
unknown binding of adsorbates to catalyst. In this paper, we investigate
whether it is possible to predict a system's relaxed energy in the OC20 dataset
while ignoring the relative position of the adsorbate with respect to the
electro-catalyst. We consider SchNet, DimeNet++ and FAENet as base
architectures and measure the impact of four modifications on model
performance: removing edges in the input graph, pooling independent
representations, not sharing the backbone weights and using an attention
mechanism to propagate non-geometric relative information. We find that while
removing binding site information impairs accuracy as expected, modified models
are able to predict relaxed energies with remarkably decent MAE. Our work
suggests future research directions in accelerated materials discovery where
information on reactant configurations can be reduced or altogether omitted
Spatiotemporal Characteristics of the Largest HIV-1 CRF02_AG Outbreak in Spain: Evidence for Onward Transmissions
Background and Aim: The circulating recombinant form 02_AG (CRF02_AG) is the predominant clade among the human immunodeficiency virus type-1 (HIV-1) non-Bs with a prevalence of 5.97% (95% Confidence Interval-CI: 5.41–6.57%) across Spain. Our aim was to estimate the levels of regional clustering for CRF02_AG and the spatiotemporal characteristics of the largest CRF02_AG subepidemic in Spain.Methods: We studied 396 CRF02_AG sequences obtained from HIV-1 diagnosed patients during 2000–2014 from 10 autonomous communities of Spain. Phylogenetic analysis was performed on the 391 CRF02_AG sequences along with all globally sampled CRF02_AG sequences (N = 3,302) as references. Phylodynamic and phylogeographic analysis was performed to the largest CRF02_AG monophyletic cluster by a Bayesian method in BEAST v1.8.0 and by reconstructing ancestral states using the criterion of parsimony in Mesquite v3.4, respectively.Results: The HIV-1 CRF02_AG prevalence differed across Spanish autonomous communities we sampled from (p < 0.001). Phylogenetic analysis revealed that 52.7% of the CRF02_AG sequences formed 56 monophyletic clusters, with a range of 2–79 sequences. The CRF02_AG regional dispersal differed across Spain (p = 0.003), as suggested by monophyletic clustering. For the largest monophyletic cluster (subepidemic) (N = 79), 49.4% of the clustered sequences originated from Madrid, while most sequences (51.9%) had been obtained from men having sex with men (MSM). Molecular clock analysis suggested that the origin (tMRCA) of the CRF02_AG subepidemic was in 2002 (median estimate; 95% Highest Posterior Density-HPD interval: 1999–2004). Additionally, we found significant clustering within the CRF02_AG subepidemic according to the ethnic origin.Conclusion: CRF02_AG has been introduced as a result of multiple introductions in Spain, following regional dispersal in several cases. We showed that CRF02_AG transmissions were mostly due to regional dispersal in Spain. The hot-spot for the largest CRF02_AG regional subepidemic in Spain was in Madrid associated with MSM transmission risk group. The existence of subepidemics suggest that several spillovers occurred from Madrid to other areas. CRF02_AG sequences from Hispanics were clustered in a separate subclade suggesting no linkage between the local and Hispanic subepidemics
Crowns in Linear -Graphs of Minimum Degree
A 3-graph is a pair H = (V, E) of sets, where elements of V are called points or vertices and E contains some 3-element subsets of V, called edges. A 3-graph is called linear if any two distinct edges intersect in at most one vertex. There is a recent interest in extremal properties of 3-graphs containing no crown, three pairwise disjoint edges and a fourth edge which intersects all of them. We show that every linear 3-graph with minimum degree 4 contains a crown. This is not true if 4 is replaced by 3
Removing Symmetry in Circulant Graphs and Point-Block Incidence Graphs
An automorphism of a graph is a mapping of the vertices onto themselves such that connections between respective edges are preserved. A vertex v in a graph G is fixed if it is mapped to itself under every automorphism of G. The fixing number of a graph G is the minimum number of vertices, when fixed, fixes all of the vertices in G. The determination of fixing numbers is important as it can be useful in determining the group of automorphisms of a graph-a famous and difficult problem. Fixing numbers were introduced and initially studied by Gibbons and Laison, Erwin and Harary and Boutin. In this paper, we investigate fixing numbers for graphs with an underlying cyclic structure, which provides an inherent presence of symmetry. We first determine fixing numbers for circulant graphs, showing in many cases the fixing number is 2. However, we also show that circulant graphs with twins, which are pairs of vertices with the same neighbourhoods, have considerably higher fixing numbers. This is the first paper that investigates fixing numbers of point-block incidence graphs, which lie at the intersection of graph theory and combinatorial design theory. We also present a surprising result-identifying infinite families of graphs in which fixing any vertex fixes every vertex, thus removing all symmetries from the graph
On the importance of catalyst-adsorbate 3D interactions for relaxed energy predictions
International audienceThe use of machine learning for material property prediction and discovery has traditionally centered on graph neural networks that incorporate the geometric configuration of all atoms. However, in practice not all this information may be readily available, e.g. when evaluating the potentially unknown binding of adsorbates to catalyst. In this paper, we investigate whether it is possible to predict a system's relaxed energy in the OC20 dataset while ignoring the relative position of the adsorbate with respect to the electro-catalyst. We consider SchNet, DimeNet++ and FAENet as base architectures and measure the impact of four modifications on model performance: removing edges in the input graph, pooling independent representations, not sharing the backbone weights and using an attention mechanism to propagate non-geometric relative information. We find that while removing binding site information impairs accuracy as expected, modified models are able to predict relaxed energies with remarkably decent MAE. Our work suggests future research directions in accelerated materials discovery where information on reactant configurations can be reduced or altogether omitted