scholarly journals Isotropic Matroids I: Multimatroids and Neighborhoods

10.37236/5222 ◽  
2016 ◽  
Vol 23 (4) ◽  
Author(s):  
Robert Brijder ◽  
Lorenzo Traldi
Keyword(s):  
Direct Proof ◽  
Bipartite Graphs ◽  
Simple Graph ◽  
Local Equivalence

Several properties of the isotropic matroid of a looped simple graph are presented. Results include a characterization of the multimatroids that are associated with isotropic matroids and several ways in which the isotropic matroid of $G$ incorporates information about graphs locally equivalent to $G$. Specific results of the latter type include a characterization of graphs that are locally equivalent to bipartite graphs, a direct proof that two forests are isomorphic if and only if their isotropic matroids are isomorphic, and a way to express local equivalence indirectly, using only edge pivots.

Emission of volatile organic compounds from petunia flowers is facilitated by an ABC transporter

Science ◽  
2017 ◽  
Vol 356 (6345) ◽  
pp. 1386-1388 ◽  
Author(s):  
Funmilayo Adebesin ◽  
Joshua R. Widhalm ◽  
Benoît Boachon ◽  
François Lefèvre ◽  
Baptiste Pierman ◽  
...  
Keyword(s):  
Plasma Membrane ◽  
Rna Interference ◽  
Abc Transporter ◽  
Atmospheric Chemistry ◽  
Petunia Hybrida ◽  
Direct Proof ◽  
Plant Volatile ◽  
Volatile Organic ◽  
Default Assumption

Plants synthesize a diversity of volatile molecules that are important for reproduction and defense, serve as practical products for humans, and influence atmospheric chemistry and climate. Despite progress in deciphering plant volatile biosynthesis, their release from the cell has been poorly understood. The default assumption has been that volatiles passively diffuse out of cells. By characterization of aPetunia hybridaadenosine triphosphate–binding cassette (ABC) transporter, PhABCG1, we demonstrate that passage of volatiles across the plasma membrane relies on active transport.PhABCG1down-regulation by RNA interference results in decreased emission of volatiles, which accumulate to toxic levels in the plasma membrane. This study provides direct proof of a biologically mediated mechanism of volatile emission.

A characterization of rings whose unitary Cayley graphs are planar

2018 ◽  
Vol 17 (09) ◽  
pp. 1850178 ◽  
Author(s):  
Huadong Su ◽  
Yiqiang Zhou
Keyword(s):  
Cayley Graph ◽  
Cayley Graphs ◽  
Simple Graph ◽  
Vertex Set ◽  
Unitary Cayley Graph

Let [Formula: see text] be a ring with identity. The unitary Cayley graph of [Formula: see text] is the simple graph with vertex set [Formula: see text], where two distinct vertices [Formula: see text] and [Formula: see text] are linked by an edge if and only if [Formula: see text] is a unit of [Formula: see text]. A graph is said to be planar if it can be drawn on the plane in such a way that its edges intersect only at their endpoints. In this paper, we completely characterize the rings whose unitary Cayley graphs are planar.

scholarly journals An Extremal Characterization of Projective Planes

10.37236/867 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Stefaan De Winter ◽  
Felix Lazebnik ◽  
Jacques Verstraëte
Keyword(s):  
Projective Plane ◽  
Projective Planes ◽  
Bipartite Graphs ◽  
Incidence Graph ◽  
Number Of Cycles

In this article, we prove that amongst all $n$ by $n$ bipartite graphs of girth at least six, where $n = q^2 + q + 1 \ge 157$, the incidence graph of a projective plane of order $q$, when it exists, has the maximum number of cycles of length eight. This characterizes projective planes as the partial planes with the maximum number of quadrilaterals.

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 A Bijection for the Boolean Numbers of Ferrers Graphs

2021 ◽  
Vol 38 (1) ◽  
Author(s):  
Beáta Bényi
Keyword(s):  
Bipartite Graphs ◽  
Bernoulli Numbers ◽  
Complete Bipartite Graphs ◽  
Ferrers Graphs

AbstractIn this note we prove a new characterization of the derangement sets of Ferrers graphs and present a bijection between the derangement sets and $$F_{\lambda }$$ F λ -Callan sequences. In particular, this connection reveals that the boolean numbers of the complete bipartite graphs are the D-relatives of poly-Bernoulli numbers.

scholarly journals Almost well-dominated bipartite graphs with minimum degree at least two

2020 ◽  
Author(s):  
Hadi Alizadeh ◽  
Didem Gözüpek
Keyword(s):  
Minimum Degree ◽  
Dominating Set ◽  
Domination Number ◽  
Bipartite Graphs ◽  
Maximum Cardinality ◽  
Minimum Cardinality ◽  
Upper Domination Number ◽  
The Difference ◽  
Two Parameters

A dominating set in a graph $G=(V,E)$ is a set $S$ such that every vertex of $G$ is either in $S$ or adjacent to a vertex in $S$. While the minimum cardinality of a dominating set in $G$ is called the domination number of $G$ denoted by $\gamma(G)$, the maximum cardinality of a minimal dominating set in $G$ is called the upper domination number of $G$ denoted by $\Gamma(G)$. We call the difference between these two parameters the \textit{domination gap} of $G$ and denote it by $\mu_d(G) = \Gamma(G) - \gamma(G)$. While a graph $G$ with $\mu_d(G)=0$ is said to be a \textit{well-dominated} graph, we call a graph $G$ with $\mu_d(G)=1$ an \textit{almost well-dominated} graph. In this work, we first establish an upper bound for the cardinality of bipartite graphs with $\mu_d(G)=k$, where $k\geq1$, and minimum degree at least two. We then provide a complete structural characterization of almost well-dominated bipartite graphs with minimum degree at least two. While the results by Finbow et al.~\cite{domination} imply that a 4-cycle is the only well-dominated bipartite graph with minimum degree at least two, we prove in this paper that there exist precisely 31 almost well-dominated bipartite graphs with minimum degree at least two.

scholarly journals Characterization of peptide-protein relationships in protein ambiguity groups via bipartite graphs

2021 ◽  
Author(s):  
Karin Schork ◽  
Michael Turewicz ◽  
Julian Uszkoreit ◽  
Jörg Rahnenführer ◽  
Martin Eisenacher
Keyword(s):  
Real Data ◽  
Bipartite Graphs ◽  
Data Sets ◽  
Complex Structures ◽  
Protein Databases ◽  
Protein Inference ◽  
The Relationship ◽  
Ambiguity Groups ◽  
Proteins And Peptides

Motivation: In bottom-up proteomics, proteins are enzymatically digested before measurement with mass spectrometry. The relationship between proteins and peptides can be represented by bipartite graphs. This representation is useful to aid protein inference and quantification, which is complex due to the occurrence of shared peptides. We conducted a comprehensive analysis of bipartite graphs using theoretical peptides from in silico digestion of protein databases as well as quantified peptides quantified from real data sets. Results: The graphs based on quantified peptides are smaller and have less complex structures compared to graphs using theoretical peptides. The proportion of protein nodes without unique peptides and of graphs that contain such proteins are considerably greater for real data. Large differences between the two analyzed organisms (mouse and yeast) on database as well as quantitative level have been observed. Insights of this analysis may be useful for the development of protein inference and quantification algorithms.

scholarly journals A Characterization of Circle Graphs in Terms of Multimatroid Representations

10.37236/6992 ◽  
2020 ◽  
Vol 27 (1) ◽  
Author(s):  
Robert Brijder ◽  
Lorenzo Traldi
Keyword(s):  
Simple Graph ◽  
Rank Function ◽  
Binary Matroid ◽  
Circle Graph ◽  
Circle Graphs ◽  
Isotropic System

The isotropic matroid $M[IAS(G)]$ of a looped simple graph $G$ is a binary matroid equivalent to the isotropic system of $G$. In general, $M[IAS(G)]$ is not regular, so it cannot be represented over fields of characteristic $\neq 2$. The ground set of $M[IAS(G)]$ is denoted $W(G)$; it is partitioned into 3-element subsets corresponding to the vertices of $G$. When the rank function of $M[IAS(G)]$ is restricted to subtransversals of this partition, the resulting structure is a multimatroid denoted $\mathcal{Z}_{3}(G)$. In this paper we prove that $G$ is a circle graph if and only if for every field $\mathbb{F}$, there is an $\mathbb{F}$-representable matroid with ground set $W(G)$, which defines $\mathcal{Z}_{3}(G)$ by restriction. We connect this characterization with several other circle graph characterizations that have appeared in the literature.

scholarly journals Some classes of integral graphs which belong to the class αKa U βKb,b

2003 ◽  
Vol 74 (88) ◽  
pp. 25-36 ◽  
Author(s):  
Mirko Lepovic
Keyword(s):  
Simple Graph ◽  
Largest Eigenvalue ◽  
Integral Graphs ◽  
The Largest Eigenvalue

Let G be a simple graph and let G denote its complement. We say that G is integral if its spectrum consists of integral values. We have recently established a characterization of integral graphs which belong to the class ?Ka U ?Kb,b, where mG denotes the m-fold union of the graph G. In this work we investigate integral graphs from the class ?Ka U ?Kb,b with ?1 = a+b where ?1 is the largest eigenvalue of ?Ka U ?Kb,b.

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