Research on the Hereditary Properties of the Cartesian Product Operation of Graphs

2012 ◽  
Vol 241-244 ◽  
pp. 2802-2806
Author(s):  
Hua Dong Wang ◽  
Bin Wang ◽  
Yan Zhong Hu
Keyword(s):  
Euler Characteristic ◽  
Cartesian Product ◽  
Hamilton Cycle ◽  
Hereditary Property ◽  
Constant Property ◽  
Product Graph ◽  
Product Operation ◽  
Graph Operation ◽  
Hereditary Properties

This paper defined the hereditary property (or constant property) concerning graph operation, and discussed various forms of the hereditary property under the circumstance of Cartesian product graph operation. The main conclusions include: The non-planarity and Hamiltonicity of graph are hereditary concerning the Cartesian product, but planarity of graph is not, Euler characteristic and non-hamiltonicity of graph are not hereditary as well. Therefore, when we applied this principle into practice, we testified that Hamilton cycle does exist in hypercube.

On the Hamilton Cycle of the Hypercube

2011 ◽  
Vol 480-481 ◽  
pp. 922-927 ◽  
Author(s):  
Yan Zhong Hu ◽  
Hua Dong Wang
Keyword(s):  
Euler Characteristic ◽  
Interconnection Networks ◽  
Cartesian Product ◽  
Hamilton Cycle ◽  
The Other ◽  
Product Graph ◽  
Product Graphs ◽  
Cartesian Product Graphs ◽  
Isomorphic Graphs ◽  
The Relationship

Hypercube is one of the basic types of interconnection networks. In this paper, we use the concept of the Cartesian product graph to define the hypercube Qn, we study the relationship between the isomorphic graphs and the Cartesian product graphs, and we get the result that there exists a Hamilton cycle in the hypercube Qn. Meanwhile, the other properties of the hypercube Qn, such as Euler characteristic and bipartite characteristic are also introduced.

Minimal Characteristic Algebras for Rectangular k-Normal Identities

2011 ◽  
Vol 18 (04) ◽  
pp. 611-628
Author(s):  
K. Hambrook ◽  
S. L. Wismath
Keyword(s):  
Hereditary Property ◽  
Fixed Type ◽  
Hereditary Properties

A characteristic algebra for a hereditary property of identities of a fixed type τ is an algebra [Formula: see text] such that for any variety V of type τ, we have [Formula: see text] if and only if every identity satisfied by V has the property p. This is equivalent to [Formula: see text] being a generator for the variety determined by all identities of type τ which have property p. Płonka has produced minimal (smallest cardinality) characteristic algebras for a number of hereditary properties, including regularity, normality, uniformity, biregularity, right- and leftmost, outermost, and external-compatibility. In this paper, we use a construction of Płonka to study minimal characteristic algebras for the property of rectangular k-normality. In particular, we construct minimal characteristic algebras of type (2) for k-normality and rectangularity for 1 ≤ k ≤ 3.

scholarly journals Dimensi Metrik Kuat Lokal Graf Hasil Operasi Kali Kartesian

2020 ◽  
Vol 1 (2) ◽  
pp. 64
Author(s):  
Nurma Ariska Sutardji ◽  
Liliek Susilowati ◽  
Utami Dyah Purwati
Keyword(s):  
Graph Theory ◽  
Cartesian Product ◽  
Metric Dimension ◽  
Star Graph ◽  
Product Graph ◽  
Path Graph ◽  
Strong Metric Dimension ◽  
Metric Dimensions ◽  
Development Result ◽  
Corona Product

The strong local metric dimension is the development result of a strong metric dimension study, one of the study topics in graph theory. Some of graphs that have been discovered about strong local metric dimension are path graph, star graph, complete graph, cycle graphs, and the result corona product graph. In the previous study have been built about strong local metric dimensions of corona product graph. The purpose of this research is to determine the strong local metric dimension of cartesian product graph between any connected graph G and H, denoted by dimsl (G x H). In this research, local metric dimension of G x H is influenced by local strong metric dimension of graph G and local strong metric dimension of graph H. Graph G and graph H has at least two order.

scholarly journals The Edit Distance Function and Symmetrization

10.37236/2262 ◽  
2013 ◽  
Vol 20 (3) ◽  
Author(s):  
Ryan R. Martin
Keyword(s):  
Distance Function ◽  
Edit Distance ◽  
Symmetric Difference ◽  
Hereditary Property ◽  
Vertex Set ◽  
Hereditary Properties

The edit distance between two graphs on the same labeled vertex set is the size of the symmetric difference of the edge sets.  The distance between a graph, G, and a hereditary property, ℋ, is the minimum of the distance between G and each G'∈ℋ.  The edit distance function of ℋ is a function of p∈[0,1] and is the limit of the maximum normalized distance between a graph of density p and ℋ.This paper utilizes a method due to Sidorenko [Combinatorica 13(1), pp. 109-120], called "symmetrization", for computing the edit distance function of various hereditary properties.  For any graph H, Forb(H) denotes the property of not having an induced copy of H.  This paper gives some results regarding estimation of the function for an arbitrary hereditary property. This paper also gives the edit distance function for Forb(H), where H is a cycle on 9 or fewer vertices.

scholarly journals A Note on the Speed of Hereditary Graph Properties

10.37236/644 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Vadim V. Lozin ◽  
Colin Mayhill ◽  
Victor Zamaraev
Keyword(s):  
Practical Importance ◽  
Bipartite Graphs ◽  
Hereditary Property ◽  
Induced Subgraphs ◽  
Forbidden Induced Subgraphs ◽  
Graph Properties ◽  
Graph Property ◽  
Vertex Set ◽  
Split Graphs ◽  
Hereditary Properties

For a graph property $X$, let $X_n$ be the number of graphs with vertex set $\{1,\ldots,n\}$ having property $X$, also known as the speed of $X$. A property $X$ is called factorial if $X$ is hereditary (i.e. closed under taking induced subgraphs) and $n^{c_1n}\le X_n\le n^{c_2n}$ for some positive constants $c_1$ and $c_2$. Hereditary properties with the speed slower than factorial are surprisingly well structured. The situation with factorial properties is more complicated and less explored, although this family includes many properties of theoretical or practical importance, such as planar graphs or graphs of bounded vertex degree. To simplify the study of factorial properties, we propose the following conjecture: the speed of a hereditary property $X$ is factorial if and only if the fastest of the following three properties is factorial: bipartite graphs in $X$, co-bipartite graphs in $X$ and split graphs in $X$. In this note, we verify the conjecture for hereditary properties defined by forbidden induced subgraphs with at most 4 vertices.

THE ISOPERIMETRIC NUMBER OF d–DIMENSIONAL k–ARY ARRAYS

1999 ◽  
Vol 10 (03) ◽  
pp. 289-300 ◽  
Author(s):  
M. CEMIL AZIZOĞLU ◽  
ÖMER EĞECIOĞLU
Keyword(s):  
Cartesian Product ◽  
Product Graph ◽  
Path Graph ◽  
Number Formula ◽  
Isoperimetric Number

The d–dimensional k-ary array [Formula: see text] is the d–fold Cartesian product graph of the path graph Pk with k vertices. We show that the (edge) isoperimetric number [Formula: see text] of [Formula: see text] is given by [Formula: see text] and identify the cardinalities and the structure of the isoperimetric sets. For odd k, the cardinalities of isoperimetric sets in [Formula: see text] are [Formula: see text], whereas every isoperimetric set for k even has cardinality [Formula: see text].

Edge neighbor connectivity of Cartesian product graph G×K2

2011 ◽  
Vol 217 (12) ◽  
pp. 5508-5511 ◽  
Author(s):  
Xuebin Zhao ◽  
Zhao Zhang ◽  
Qin Ren
Keyword(s):  
Cartesian Product ◽  
Product Graph ◽  
Neighbor Connectivity

scholarly journals Some Extremal Problems for Hereditary Properties of Graphs

10.37236/3419 ◽  
2014 ◽  
Vol 21 (1) ◽  
Author(s):  
Vladimir Nikiforov
Keyword(s):  
Real Number ◽  
Extremal Problems ◽  
Hereditary Property ◽  
Hereditary Properties ◽  
Extremal Parameter ◽  
Stone Theorem

Given an infinite hereditary property of graphs $\mathcal{P}$, the principal extremal parameter of $\mathcal{P}$ is the value\[ \pi\left( \mathcal{P}\right) =\lim_{n\rightarrow\infty}\binom{n}{2}^{-1}\max\{e\left( G\right) :\text{ }G\in\mathcal{P}\text{ and }v\left(G\right) =n\}.\]The Erdős-Stone theorem gives $\pi\left( \mathcal{P}\right) $ if $\mathcal{P}$ is monotone, but this result does not apply to hereditary $\mathcal{P}$. Thus, one of the results of this note is to establish $\pi\left( \mathcal{P}\right) $ for any hereditary property $\mathcal{P}.$Similar questions are studied for the parameter $\lambda^{\left( p\right)}\left( G\right)$, defined for every real number $p\geq1$ and every graph $G$ of order $n$ as\[\lambda^{\left( p\right) }\left( G\right) =\max_{\left\vert x_{1}\right\vert^{p}\text{ }+\text{ }\cdots\text{ }+\text{ }\left\vert x_{n}\right\vert ^{p} \text{ }=\text{ }1}2\sum_{\{u,v\}\in E\left( G\right) }x_{u}x_{v}.\]It is shown that the limit\[ \lambda^{\left( p\right) }\left( \mathcal{P}\right) =\lim_{n\rightarrow\infty}n^{2/p-2}\max\{\lambda^{\left( p\right) }\left( G\right) :\text{ }G\in \mathcal{P}\text{ and }v\left( G\right) =n\}\]exists for every hereditary property $\mathcal{P}$.A key result of the note is the equality \[\lambda^{(p)}\left( \mathcal{P}\right) =\pi\left( \mathcal{P}\right) ,\]which holds for all $p>1.$ In particular, edge extremal problems andspectral extremal problems for graphs are asymptotically equivalent.

scholarly journals The magnitude of a graph

2017 ◽  
Vol 166 (2) ◽  
pp. 247-264
Author(s):  
TOM LEINSTER
Keyword(s):  
Power Series ◽  
Rational Function ◽  
Euler Characteristic ◽  
Cartesian Product ◽  
Tutte Polynomial ◽  
Graph Invariants ◽  
Geometric Measure ◽  
Cycle Matroid

AbstractThe magnitude of a graph is one of a family of cardinality-like invariants extending across mathematics; it is a cousin to Euler characteristic and geometric measure. Among its cardinality-like properties are multiplicativity with respect to cartesian product and an inclusion-exclusion formula for the magnitude of a union. Formally, the magnitude of a graph is both a rational function over ℚ and a power series over ℤ. It shares features with one of the most important of all graph invariants, the Tutte polynomial; for instance, magnitude is invariant under Whitney twists when the points of identification are adjacent. Nevertheless, the magnitude of a graph is not determined by its Tutte polynomial, nor even by its cycle matroid, and it therefore carries information that they do not.

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