scholarly journals LIGHTS OUT! on graph products over the ring of integers modulo k

2021 ◽  
Vol 37 ◽  
pp. 416-424
Author(s):  
Ryan Munter ◽  
Travis Peters
Keyword(s):  
Cartesian Product ◽  
Simple Graph ◽  
Graph Products ◽  
Ring Of Integers ◽  
Product Graph ◽  
Turn On ◽  
Strong Product ◽  
Product Graphs ◽  
Traditional Game ◽  
Do So

LIGHTS OUT! is a game played on a finite, simple graph. The vertices of the graph are the lights, which may be on or off, and the edges of the graph determine how neighboring vertices turn on or off when a vertex is pressed. Given an initial configuration of vertices that are on, the object of the game is to turn all the lights out. The traditional game is played over $\mathbb{Z}_2$, where the vertices are either lit or unlit, but the game can be generalized to $\mathbb{Z}_k$, where the lights have different colors. Previously, the game was investigated on Cartesian product graphs over $\mathbb{Z}_2$. We extend this work to $\mathbb{Z}_k$ and investigate two other fundamental graph products, the direct (or tensor) product and the strong product. We provide conditions for which the direct product graph and the strong product graph are solvable based on the factor graphs, and we do so using both open and closed neighborhood switching over $\mathbb{Z}_k$.

scholarly journals On the Laplacian spectra of product graphs

2015 ◽  
Vol 9 (1) ◽  
pp. 39-58 ◽  
Author(s):  
S. Barik ◽  
R.B. Bapat ◽  
S. Pati
Keyword(s):  
Cartesian Product ◽  
Complete Characterization ◽  
Graph Products ◽  
Lexicographic Product ◽  
Laplacian Spectrum ◽  
Strong Product ◽  
Laplacian Spectra ◽  
Product Graphs ◽  
Characteristic Sets

Graph products and their structural properties have been studied extensively by many researchers. We investigate the Laplacian eigenvalues and eigenvectors of the product graphs for the four standard products, namely, the Cartesian product, the direct product, the strong product and the lexicographic product. A complete characterization of Laplacian spectrum of the Cartesian product of two graphs has been done by Merris. We give an explicit complete characterization of the Laplacian spectrum of the lexicographic product of two graphs using the Laplacian spectra of the factors. For the other two products, we describe the complete spectrum of the product graphs in some particular cases. We supply some new results relating to the algebraic connectivity of the product graphs. We describe the characteristic sets for the Cartesian product and for the lexicographic product of two graphs. As an application we construct new classes of Laplacian integral graphs.

scholarly journals The partition dimension of strong product graphs and Cartesian product graphs

2014 ◽  
Vol 331 ◽  
pp. 43-52 ◽  
Author(s):  
Ismael González Yero ◽  
Marko Jakovac ◽  
Dorota Kuziak ◽  
Andrej Taranenko
Keyword(s):  
Cartesian Product ◽  
Strong Product ◽  
Product Graphs ◽  
Partition Dimension ◽  
Cartesian Product Graphs

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.

scholarly journals Gromov Hyperbolicity in Strong Product Graphs

10.37236/3271 ◽  
2013 ◽  
Vol 20 (3) ◽  
Author(s):  
Walter Carballosa ◽  
Rocío M. Casablanca ◽  
Amauris De la Cruz ◽  
José M. Rodríguez
Keyword(s):  
Metric Space ◽  
The Other ◽  
Gromov Hyperbolicity ◽  
Geodesic Triangle ◽  
Product Graph ◽  
Strong Product ◽  
Geodesic Metric Space ◽  
Product Graphs ◽  
Geodesic Metric ◽  
Hyperbolicity Constant

If X is a geodesic metric space and $x_1,x_2,x_3\in X$, a geodesic triangle $T=\{x_1,x_2,x_3\}$ is the union of the three geodesics $[x_1x_2]$, $[x_2x_3]$ and $[x_3x_1]$ in $X$. The space $X$ is $\delta$-hyperbolic $($in the Gromov sense$)$ if any side of $T$ is contained in a $\delta$-neighborhood of the union of the two other sides, for every geodesic triangle $T$ in $X$. If $X$ is hyperbolic, we denote by $\delta (X)$ the sharp hyperbolicity constant of $X$, i.e., $\delta (X)=\inf\{\delta\geq 0: \, X \, \text{ is $\delta$-hyperbolic}\,\}\,.$ In this paper we characterize the strong product of two graphs $G_1\boxtimes G_2$ which are hyperbolic, in terms of $G_1$ and $G_2$: the strong product graph $G_1\boxtimes G_2$ is hyperbolic if and only if one of the factors is hyperbolic and the other one is bounded. We also prove some sharp relations between $\delta (G_1\boxtimes G_2)$, $\delta (G_1)$, $\delta (G_2)$ and the diameters of $G_1$ and $G_2$ (and we find families of graphs for which the inequalities are attained). Furthermore, we obtain the exact values of the hyperbolicity constant for many strong product graphs.

scholarly journals Steiner Degree Distance of Two Graph Products

2019 ◽  
Vol 27 (2) ◽  
pp. 83-99 ◽  
Author(s):  
Yaping Mao ◽  
Zhao Wang ◽  
Kinkar Ch. Das
Keyword(s):  
Connected Graph ◽  
Cartesian Product ◽  
Graph Products ◽  
Product Graphs ◽  
Degree Distance ◽  
Cartesian Product Graphs

AbstractThe degree distance DD(G) of a connected graph G was invented by Dobrynin and Kochetova in 1994. Recently, one of the present authors introduced the concept of k-center Steiner degree distance defined as SDD_k (G) = \sum\limits_{\mathop {S \subseteq V(G)}\limits_{\left| S \right| = k} } {\left[ {\sum\limits_{v \in S} {{\it deg} _G (v)} } \right]d_G (S),} where dG(S) is the Steiner k-distance of S and degG(v) is the degree of the vertex v in G. In this paper, we investigate the Steiner degree distance of complete and Cartesian product graphs.

scholarly journals Roman domination in Cartesian product graphs and strong product graphs

2013 ◽  
Vol 7 (2) ◽  
pp. 262-274 ◽  
Author(s):  
Yero González ◽  
Juan Rodríguez-Velázquez
Keyword(s):  
Minimum Weight ◽  
Domination Number ◽  
Cartesian Product ◽  
Roman Domination ◽  
Roman Domination Number ◽  
Strong Product ◽  
Product Graphs ◽  
Roman Dominating Function ◽  
Cartesian Product Graphs ◽  
Dominating Function

A map f : V ? {0, 1, 2} is a Roman dominating function for G if for every vertex v with f(v) = 0, there exists a vertex u, adjacent to v, with f(u) = 2. The weight of a Roman dominating function is f(V ) = ?u?v f(u). The minimum weight of a Roman dominating function on G is the Roman domination number of G. In this article we study the Roman domination number of Cartesian product graphs and strong product graphs.

THE DIAMETER VARIABILITY OF THE CARTESIAN PRODUCT OF GRAPHS

2014 ◽  
Vol 06 (01) ◽  
pp. 1450001 ◽  
Author(s):  
M. R. CHITHRA ◽  
A. VIJAYAKUMAR
Keyword(s):  
Interconnection Networks ◽  
Cartesian Product ◽  
Graph Products ◽  
Single Edge ◽  
Cartesian Product Of Graphs ◽  
Product Of Graphs

The diameter of a graph can be affected by the addition or deletion of edges. In this paper, we examine the Cartesian product of graphs whose diameter increases (decreases) by the deletion (addition) of a single edge. The problems of minimality and maximality of the Cartesian product of graphs with respect to its diameter are also solved. These problems are motivated by the fact that most of the interconnection networks are graph products and a good network must be hard to disrupt and the transmissions must remain connected even if some vertices or edges fail.

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