scholarly journals The Smith and Critical Groups of the Square Rook's Graph and its Complement

10.37236/5442 ◽  
2016 ◽  
Vol 23 (4) ◽  
Author(s):  
Joshua E. Ducey ◽  
Jonathan Gerhard ◽  
Noah Watson
Keyword(s):  
Normal Form ◽  
Bipartite Graph ◽  
Complete Bipartite Graph ◽  
Line Graph ◽  
Cartesian Product ◽  
Laplacian Matrix ◽  
Critical Group ◽  
Critical Groups ◽  
Complete Graphs ◽  
Vertex Set

Let $R_{n}$ denote the graph with vertex set consisting of the squares of an $n \times n$ grid, with two squares of the grid adjacent when they lie in the same row or column.  This is the square rook's graph, and can also be thought of as the Cartesian product of two complete graphs of order $n$, or the line graph of the complete bipartite graph $K_{n,n}$.  In this paper we compute the Smith group and critical group of the graph $R_{n}$ and its complement.  This is equivalent to determining the Smith normal form of both the adjacency and Laplacian matrix of each of these graphs.  In doing so we verify a 1986 conjecture of Rushanan.

The Critical Group of the Vertex Corona of Cycles and Complete Graphs in Industry

2013 ◽  
Vol 357-360 ◽  
pp. 2802-2805
Author(s):  
Xiang Hua Tan ◽  
Jian Guo Zhu ◽  
Cheng Wen Cai ◽  
Kai Rong Yu ◽  
Hao Yun Qian
Keyword(s):  
Normal Form ◽  
Spanning Trees ◽  
Laplacian Matrix ◽  
Graph Laplacian ◽  
Smith Normal Form ◽  
Critical Group ◽  
Complete Graphs ◽  
Self Organized ◽  
Self Organized Criticality ◽  
Number Of Spanning Trees

Self-organized criticality is an important theory widely used in various domain such as a variety of industrial accidents, power system, punctuated equilibrium in biology etc.. The Critical Group of the graph is mainly focused on the Abelian sandpile model of self-organized criticality, whose order is the number of spanning trees in the graph, and which is closely connected with the graph Laplacian matrix. In this paper, the main tools will be the computation for Smith normal form of an integer matrix, which can be achieved by the implementation of a series of row and column operations in the ring Ζ of integers. Hence, the structure of the critical group on the vertex corona is determined and it is shown that the Smith normal form is the direct sum of n (m-1)+1cyclic groups. Furthermore, it follows from Kirchooffs Matrix Tree Theorem that the number of spanning trees of the Graph is n (m+1)n (m-1).

scholarly journals Partitioning the vertex set of $G$ to make $G\,\Box\, H$ an efficient open domination graph

2016 ◽  
Vol Vol. 18 no. 3 (Graph Theory) ◽  
Author(s):  
Tadeja Kraner Šumenjak ◽  
Iztok Peterin ◽  
Douglas F. Rall ◽  
Aleksandra Tepeh
Keyword(s):  
Bipartite Graph ◽  
Complete Graph ◽  
Complete Bipartite Graph ◽  
Cartesian Product ◽  
Cartesian Products ◽  
Vertex Set

A graph is an efficient open domination graph if there exists a subset of vertices whose open neighborhoods partition its vertex set. We characterize those graphs $G$ for which the Cartesian product $G \Box H$ is an efficient open domination graph when $H$ is a complete graph of order at least 3 or a complete bipartite graph. The characterization is based on the existence of a certain type of weak partition of $V(G)$. For the class of trees when $H$ is complete of order at least 3, the characterization is constructive. In addition, a special type of efficient open domination graph is characterized among Cartesian products $G \Box H$ when $H$ is a 5-cycle or a 4-cycle. Comment: 16 pages, 2 figures

scholarly journals Grundy Number of Some Chordal Graphs

2018 ◽  
Vol 7 (4.10) ◽  
pp. 64
Author(s):  
R. Nagarathinam ◽  
N. Parvathi ◽  
. .
Keyword(s):  
Line Graph ◽  
Cartesian Product ◽  
Chordal Graphs ◽  
Complete Graphs ◽  
Proper Coloring ◽  
Coloring Problem ◽  
Grundy Number

For a given graph G and integer k, the Coloring problem is that of testing whether G has a k-coloring, that is, whether there exists a vertex mapping c : V → {1, 2, . . .} such that c(u) 12≠"> c(v) for every edge uv ∈ E. For proper coloring, colors assigned must be minimum, but for Grundy coloring which should be maximum. In this instance, Grundy numbers of chordal graphs like Cartesian product of two path graphs, join of the path and complete graphs and the line graph of tadpole have been executed 

scholarly journals The bichromaticity of a lattice-graph

1977 ◽  
Vol 23 (3) ◽  
pp. 354-359 ◽  
Author(s):  
Frank Harary ◽  
Derbiau Hsu ◽  
Zevi Miller
Keyword(s):  
Bipartite Graph ◽  
Complete Bipartite Graph ◽  
Cartesian Product ◽  
Maximum Order ◽  
Lattice Graph

AbstractThe bichromaticity β(B) of a bipartite graph B has been defined as the maximum order of a complete bipartite graph onto which B is homomorphic. This number was previously determined for trees and even cycles. It is now shown that for a lattice-graph Pm × Pm the cartesian product of two paths, the bichromaticity is 2 + {mn/2}.

Graham’s pebbling conjecture holds for the product of a graph and a sufficiently large complete bipartite graph

2019 ◽  
Vol 11 (06) ◽  
pp. 1950068
Author(s):  
Nopparat Pleanmani
Keyword(s):  
Bipartite Graph ◽  
Network Optimization ◽  
Connected Graph ◽  
Complete Bipartite Graph ◽  
Fixed Number ◽  
Arbitrary Vertex ◽  
Graph Pebbling ◽  
Vertex Set ◽  
Graham’S Conjecture ◽  
Pebbling Number

A graph pebbling is a network optimization model for the transmission of consumable resources. A pebbling move on a connected graph [Formula: see text] is the process of removing two pebbles from a vertex and placing one of them on an adjacent vertex after configuration of a fixed number of pebbles on the vertex set of [Formula: see text]. The pebbling number of [Formula: see text], denoted by [Formula: see text], is defined to be the least number of pebbles to guarantee that for any configuration of pebbles on [Formula: see text] and arbitrary vertex [Formula: see text], there is a sequence of pebbling movement that places at least one pebble on [Formula: see text]. For connected graphs [Formula: see text] and [Formula: see text], Graham’s conjecture asserted that [Formula: see text]. In this paper, we show that such conjecture holds when [Formula: see text] is a complete bipartite graph with sufficiently large order in terms of [Formula: see text] and the order of [Formula: see text].

When the annihilator-ideal graph is planar or complete bipartite graph

2019 ◽  
Vol 12 (02) ◽  
pp. 1950024
Author(s):  
M. J. Nikmehr ◽  
S. M. Hosseini
Keyword(s):  
Commutative Ring ◽  
Bipartite Graph ◽  
Complete Bipartite Graph ◽  
Simple Graph ◽  
Annihilator Ideal ◽  
Vertex Set

Let [Formula: see text] be a commutative ring with identity and [Formula: see text] be the set of ideals of [Formula: see text] with nonzero annihilator. The annihilator-ideal graph of [Formula: see text], denoted by [Formula: see text], is a simple graph with the vertex set [Formula: see text], and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text]. In this paper, we present some results on the bipartite, complete bipartite, outer planar and unicyclic of the annihilator-ideal graphs of a commutative ring. Among other results, bipartite annihilator-ideal graphs of rings are characterized. Also, we investigate planarity of the annihilator-ideal graph and classify rings whose annihilator-ideal graph is planar.

scholarly journals On Consensus of Star-Composed Networks with an Application of Laplacian Spectrum

2017 ◽  
Vol 2017 ◽  
pp. 1-13
Author(s):  
Da Huang ◽  
Haijun Jiang ◽  
Zhiyong Yu ◽  
Qiongxiang Huang ◽  
Xing Chen
Keyword(s):  
Communication Network ◽  
Bipartite Graph ◽  
Complete Bipartite Graph ◽  
Laplacian Matrix ◽  
Convergence Speed ◽  
Laplacian Spectrum ◽  
Graph Spectra ◽  
Nonzero Eigenvalue ◽  
Insight Into

In this paper, we mainly study the performance of star-composed networks which can achieve consensus. Specifically, we investigate the convergence speed and robustness of the consensus of the networks, which can be measured by the smallest nonzero eigenvalue λ2 of the Laplacian matrix and the H2 norm of the graph, respectively. In particular, we introduce the notion of the corona of two graphs to construct star-composed networks and apply the Laplacian spectrum to discuss the convergence speed and robustness for the communication network. Finally, the performances of the star-composed networks have been compared, and we find that the network in which the centers construct a balanced complete bipartite graph has the most advantages of performance. Our research would provide a new insight into the combination between the field of consensus study and the theory of graph spectra.

scholarly journals On the Adjacency, Laplacian, and Signless Laplacian Spectrum of Coalescence of Complete Graphs

2016 ◽  
Vol 2016 ◽  
pp. 1-11 ◽  
Author(s):  
S. R. Jog ◽  
Raju Kotambari
Keyword(s):  
Line Graph ◽  
Laplacian Matrix ◽  
Spectral Graph Theory ◽  
Complete Graphs ◽  
Laplacian Spectrum ◽  
Spectral Graph ◽  
Signless Laplacian Spectrum ◽  
Subdivision Graph ◽  
Laplacian Energy ◽  
Signless Laplacian

Coalescence as one of the operations on a pair of graphs is significant due to its simple form of chromatic polynomial. The adjacency matrix, Laplacian matrix, and signless Laplacian matrix are common matrices usually considered for discussion under spectral graph theory. In this paper, we compute adjacency, Laplacian, and signless Laplacian energy (Qenergy) of coalescence of pair of complete graphs. Also, as an application, we obtain the adjacency energy of subdivision graph and line graph of coalescence from itsQenergy.

scholarly journals BILANGAN KROMATIK LOKASI DARI GRAF P m P n ; K m P n ; DAN K , m K n

2013 ◽  
Vol 2 (1) ◽  
pp. 14
Author(s):  
Mariza Wenni
Keyword(s):  
Natural Number ◽  
Chromatic Number ◽  
Cartesian Product ◽  
Complete Graphs ◽  
Chromatic Numbers ◽  
Color Code ◽  
Connected Graphs ◽  
Vertex Set ◽  
Distinct Color

Let G and H be two connected graphs. Let c be a vertex k-coloring of aconnected graph G and let = fCg be a partition of V (G) into the resultingcolor classes. For each v 2 V (G), the color code of v is dened to be k-vector: c1; C2; :::; Ck(v) =(d(v; C1); d(v; C2); :::; d(v; Ck)), where d(v; Ci) = minfd(v; x) j x 2 Cg, 1 i k. Ifdistinct vertices have distinct color codes with respect to , then c is called a locatingcoloring of G. The locating chromatic number of G is the smallest natural number ksuch that there are locating coloring with k colors in G. The Cartesian product of graphG and H is a graph with vertex set V (G) V (H), where two vertices (a; b) and (a)are adjacent whenever a = a0and bb02 E(H), or aa0i2 E(G) and b = b, denotedby GH. In this paper, we will study about the locating chromatic numbers of thecartesian product of two paths, the cartesian product of paths and complete graphs, andthe cartesian product of two complete graphs.

Some Results on Annihilating-ideal Graphs

2016 ◽  
Vol 59 (3) ◽  
pp. 641-651
Author(s):  
Farzad Shaveisi
Keyword(s):  
Direct Summand ◽  
Commutative Ring ◽  
Bipartite Graph ◽  
Maximum Degree ◽  
Complete Bipartite Graph ◽  
Independence Number ◽  
Prime Ideals ◽  
Reduced Ring ◽  
Vertex Set ◽  
Minimal Prime

AbstractThe annihilating-ideal graph of a commutative ring R, denoted by 𝔸𝔾(R), is a graph whose vertex set consists of all non-zero annihilating ideals and two distinct vertices I and J are adjacent if and only if IJ = (0). Here we show that if R is a reduced ring and the independence number of 𝔸𝔾(R) is finite, then the edge chromatic number of 𝔸𝔾(R) equals its maximum degree and this number equals 2|Min(R)|−1 also, it is proved that the independence number of 𝔸𝔾(R) equals 2|Min(R)|−1, where Min(R) denotes the set of minimal prime ideals of R. Then we give some criteria for a graph to be isomorphic with an annihilating-ideal graph of a ring. For example, it is shown that every bipartite annihilating-ideal graph is a complete bipartite graph with at most two horns. Among other results, it is shown that a ûnite graph 𝔸𝔾(R) is not Eulerian, and that it is Hamiltonian if and only if R contains no Gorenstain ring as its direct summand.

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