scholarly journals Properties of $\theta$-super positive graphs

10.37236/2041 ◽  
2012 ◽  
Vol 19 (1) ◽  
Author(s):  
Cheng Yeaw Ku ◽  
Kok Bin Wong
Keyword(s):  
Disjoint Union ◽  
Perfect Matching ◽  
Matching Theory ◽  
Matching Polynomial

Let the matching polynomial of a graph $G$ be denoted by $\mu (G,x)$. A graph $G$ is said to be $\theta$-super positive if  $\mu(G,\theta)\neq 0$ and $\mu(G\setminus v,\theta)=0$ for all $v\in V(G)$. In particular, $G$ is $0$-super positive if and only if $G$ has a perfect matching. While much is known about $0$-super positive graphs, almost nothing is known about $\theta$-super positive graphs for $\theta \neq 0$. This motivates us to investigate the structure of $\theta$-super positive graphs in this paper. Though a $0$-super positive graph need not contain any cycle, we show that a $\theta$-super positive graph with $\theta \neq 0$ must contain a cycle. We introduce two important types of $\theta$-super positive graphs, namely $\theta$-elementary and $\theta$-base graphs. One of our main results is that any $\theta$-super positive graph $G$ can be constructed by adding certain type of edges to a disjoint union of $\theta$-base graphs; moreover, these $\theta$-base graphs are uniquely determined by $G$. We also give a characterization of $\theta$-elementary graphs: a graph $G$ is $\theta$-elementary if and only if the set of all its $\theta$-barrier sets form a partition of $V(G)$. Here, $\theta$-elementary graphs and $\theta$-barrier sets can be regarded as $\theta$-analogue of elementary graphs and Tutte sets in classical matching theory.

Impedance Matching between Signal Generator and Piezoelectric Transducer

2012 ◽  
Vol 472-475 ◽  
pp. 1488-1491
Author(s):  
Zhen Jiang Tan ◽  
Ming Zhou
Keyword(s):  
High Power ◽  
Piezoelectric Transducer ◽  
Perfect Matching ◽  
Impedance Matching ◽  
Signal Generator ◽  
Matching Theory ◽  
Signal Source ◽  
Key Technology ◽  
Power Pulse ◽  
High Power Pulse

In order to acquire enough energy, high power pulse must be used to drive a piezoelectric transducer. At the same time, it is a key technology to design an impedance matching circuit between a signal generator and a piezoelectric transducer so that signal source can transmit energy to the piezoelectric transducer effectively. In the paper, the impedance matching theory is analyzed. And, a method of using an oscilloscope to measure resonance frequency of the piezoelectric transducer is tested, through which, a perfect matching circuit is designed.

scholarly journals Face-Width of Pfaffian Braces and Polyhex Graphs on Surfaces

10.37236/3540 ◽  
2014 ◽  
Vol 21 (4) ◽  
Author(s):  
Dong Ye ◽  
Heping Zhang
Keyword(s):  
Bipartite Graph ◽  
Polynomial Time ◽  
Perfect Matching ◽  
Cartesian Product ◽  
Perfect Matchings ◽  
Face Width ◽  
Graphs On Surfaces ◽  
Positive Genus ◽  
Pfaffian Graph

A graph $G$ with a perfect matching is Pfaffian if it admits an orientation $D$ such that every central cycle $C$ (i.e. $C$ is of even size and $G-V(C)$ has a perfect matching) has an odd number of edges oriented in either direction of the cycle. It is known that the number of perfect matchings of a Pfaffian graph can be computed in polynomial time. In this paper, we show that every embedding of a Pfaffian brace (i.e. 2-extendable bipartite graph)  on a surface with a positive genus has face-width at most 3.  Further, we study Pfaffian cubic braces and obtain a characterization of Pfaffian polyhex graphs: a polyhex graph is Pfaffian if and only if it is either non-bipartite or isomorphic to the cube, or the Heawood graph, or the Cartesian product $C_k\times K_2$ for even integers $k\ge 6$.

scholarly journals Commuting decomposition of Kn1,n2,...,nk through realization of the product A(G)A(GPk )

2018 ◽  
Vol 6 (1) ◽  
pp. 343-356
Author(s):  
K. Arathi Bhat ◽  
G. Sudhakara
Keyword(s):  
Chromatic Number ◽  
Perfect Matching ◽  
Domination Number ◽  
Perfect Matchings ◽  
Factor Graph ◽  
Vertex Set ◽  
Graph Parameters

Abstract In this paper, we introduce the notion of perfect matching property for a k-partition of vertex set of given graph. We consider nontrivial graphs G and GPk , the k-complement of graph G with respect to a kpartition of V(G), to prove that A(G)A(GPk ) is realizable as a graph if and only if P satis_es perfect matching property. For A(G)A(GPk ) = A(Γ) for some graph Γ, we obtain graph parameters such as chromatic number, domination number etc., for those graphs and characterization of P is given for which GPk and Γ are isomorphic. Given a 1-factor graph G with 2n vertices, we propose a partition P for which GPk is a graph of rank r and A(G)A(GPk ) is graphical, where n ≤ r ≤ 2n. Motivated by the result of characterizing decomposable Kn,n into commuting perfect matchings [2], we characterize complete k-partite graph Kn1,n2,...,nk which has a commuting decomposition into a perfect matching and its k-complement.

scholarly journals A characterization of the existence of a Souslin line

1982 ◽  
Vol 25 (3) ◽  
pp. 425-431
Author(s):  
Nobuyuki Kemoto
Keyword(s):  
Disjoint Union ◽  
Chain Condition ◽  
First Category ◽  
Countable Chain Condition ◽  
Isolated Points ◽  
Countable Chain

The main purpose of this paper is to show that there exists a Souslin line if and only if there exists a countable chain condition space which is not weak-separable but has a generic π-base. If I is the closure of the isolated points in a space X, then X is said to be weak-separable if a first category set is dense in X – I. A π-base is said to be generic if, whenever a member of is included in the disjoint union of members of it is included in one of them.

scholarly journals Upper paired domination versus upper domination

2021 ◽  
Vol vol. 23, no. 3 (Graph Theory) ◽  
Author(s):  
Hadi Alizadeh ◽  
Didem Gözüpek
Keyword(s):  
Perfect Matching ◽  
Dominating Set ◽  
Domination Number ◽  
Graph Classes ◽  
Paired Domination Number ◽  
Upper Domination Number ◽  
Paired Domination ◽  
Cactus Graphs ◽  
Open Question

A paired dominating set $P$ is a dominating set with the additional property that $P$ has a perfect matching. While the maximum cardainality of a minimal dominating set in a graph $G$ is called the upper domination number of $G$, denoted by $\Gamma(G)$, the maximum cardinality of a minimal paired dominating set in $G$ is called the upper paired domination number of $G$, denoted by $\Gamma_{pr}(G)$. By Henning and Pradhan (2019), we know that $\Gamma_{pr}(G)\leq 2\Gamma(G)$ for any graph $G$ without isolated vertices. We focus on the graphs satisfying the equality $\Gamma_{pr}(G)= 2\Gamma(G)$. We give characterizations for two special graph classes: bipartite and unicyclic graphs with $\Gamma_{pr}(G)= 2\Gamma(G)$ by using the results of Ulatowski (2015). Besides, we study the graphs with $\Gamma_{pr}(G)= 2\Gamma(G)$ and a restricted girth. In this context, we provide two characterizations: one for graphs with $\Gamma_{pr}(G)= 2\Gamma(G)$ and girth at least 6 and the other for $C_3$-free cactus graphs with $\Gamma_{pr}(G)= 2\Gamma(G)$. We also pose the characterization of the general case of $C_3$-free graphs with $\Gamma_{pr}(G)= 2\Gamma(G)$ as an open question.

scholarly journals Semigroups with atomistic congruence lattices

1992 ◽  
Vol 52 (1) ◽  
pp. 88-102 ◽  
Author(s):  
Karl Auinger
Keyword(s):  
Congruence Lattice ◽  
Disjoint Union ◽  
Partition Lattice ◽  
Congruence Lattices

AbstractIn this note a characterization of semigroups with atomistic consruence lattices, given for weakly reductive semigroups, is generalized to arbitrary semigroups. Also, it is shown that there is a complete congruence on the congruence lattice of such a semigroup that decomposes it into a disjoint union of intervals of the partition lattice.

On The Inverse Of A Class Of Bipartite Graphs With Unique Perfect Matchings

2015 ◽  
Vol 29 ◽  
pp. 89-101 ◽  
Author(s):  
Swarup Panda ◽  
Dr. Sukanta Pati
Keyword(s):  
Open Problem ◽  
Perfect Matching ◽  
Undirected Graph ◽  
Nonnegative Matrix ◽  
Weighted Graph ◽  
Weighted Graphs ◽  
Positive Weight ◽  
Weighted Trees ◽  
The Matrix

Let G be a simple, undirected graph and Gw be the weighted graph obtained from G by giving weights to its edges using a positive weight function w. A weighted graph Gw is said to be nonsingular if its adjacency matrix A(Gw) is nonsingular. In [9], Godsil has given a class $\mathcal{G }$of connected, unweighted, bipartite, nonsingular graphs G with a unique perfect matching, such that A(G)−1 is signature similar to a nonnegative matrix, that is, there exists a diagonal matrix D with diagonal entries ±1 such that DA(G)−1D is nonnegative. The graph associated to the matrix DA(G)−1D is called the inverse of G and it is denoted by G+. The graph G+ is an undirected, weighted, connected, bipartite graph with a unique perfect matching. Nonsingular, unweighted trees are contained inside the class G. We first give a constructive characterization of the class of weighted graphs Hw that can occur as the inverse of some graph G∈\mathcal{ G}. This generalizes Theorem 2.6 of Neumann and Pati[13], where the authors have characterized graphs that occur as inverses of nonsingular, unweighted trees. A weighted graph Gw is said to have the property (R) if for each eigenvalue λ of A(Gw), 1⁄λ is also an eigenvalue of A(Gw). If further, the multiplicity of λ and 1⁄λ are the same, then Gw is said to have property (SR). A characterization of the class of nonsingular, weighted trees Tw with at least 8 vertices that have property (R) was given in [13] under some restriction on the weights. It is natural to ask for such a characterization for the whole of G, possibly with some weaker restrictions on the weights. We supply such a characterization. In particular, for trees it settles an open problem raised in [13].

A CHARACTERIZATION OF PM-COMPACT CLAW-FREE CUBIC GRAPHS

2014 ◽  
Vol 06 (02) ◽  
pp. 1450025 ◽  
Author(s):  
XIUMEI WANG ◽  
WEIPING SHANG ◽  
YIXUN LIN ◽  
MARCELO H. CARVALHO
Keyword(s):  
Convex Hull ◽  
Perfect Matching ◽  
Perfect Matchings ◽  
Cubic Graphs ◽  
Matching Polytopes ◽  
Matching Polytope

The perfect matching polytope of a graph G is the convex hull of the incidence vectors of all perfect matchings in G. This paper characterizes claw-free cubic graphs whose 1-skeleton graphs of perfect matching polytopes have diameter 1.

scholarly journals THE TYPICAL STRUCTURE OF MAXIMAL TRIANGLE-FREE GRAPHS

2015 ◽  
Vol 3 ◽  
Author(s):  
JÓZSEF BALOGH ◽  
HONG LIU ◽  
ŠÁRKA PETŘÍČKOVÁ ◽  
MARYAM SHARIFZADEH
Keyword(s):  
Perfect Matching ◽  
Independent Set ◽  
Independent Sets ◽  
Free Graph ◽  
Typical Structure ◽  
Vertex Partition ◽  
Maximal Independent Sets ◽  
Vertex Disjoint ◽  
Removal Lemma

Recently, settling a question of Erdős, Balogh, and Petříčková showed that there are at most $2^{n^{2}/8+o(n^{2})}$$n$-vertex maximal triangle-free graphs, matching the previously known lower bound. Here, we characterize the typical structure of maximal triangle-free graphs. We show that almost every maximal triangle-free graph $G$ admits a vertex partition $X\cup Y$ such that $G[X]$ is a perfect matching and $Y$ is an independent set.Our proof uses the Ruzsa–Szemerédi removal lemma, the Erdős–Simonovits stability theorem, and recent results of Balogh, Morris, and Samotij and Saxton and Thomason on characterization of the structure of independent sets in hypergraphs. The proof also relies on a new bound on the number of maximal independent sets in triangle-free graphs with many vertex-disjoint $P_{3}$s, which is of independent interest.

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