scholarly journals Skew Spectra of Oriented Bipartite Graphs

10.37236/3331 ◽  
2013 ◽  
Vol 20 (4) ◽  
Author(s):  
A. Anuradha ◽  
R. Balakrishnan ◽  
Xiaolin Chen ◽  
Xueliang Li ◽  
Huishu Lian ◽  
...  
Keyword(s):  
Bipartite Graph ◽  
Cartesian Product ◽  
Bipartite Graphs ◽  
Affirmative Answer ◽  
Cartesian Products ◽  
Oriented Graphs ◽  
Canonical Orientation ◽  
Product Graph ◽  
Skew Energy ◽  
Cartesian Products Of Graphs

A graph $G$ is said to have a parity-linked orientation $\phi$ if every even cycle $C_{2k}$ in $G^{\phi}$ is evenly (resp. oddly) oriented whenever $k$ is even (resp. odd). In this paper, this concept is used to provide an affirmative answer to the following conjecture of D. Cui and Y. Hou [D. Cui, Y. Hou, On the skew spectra of Cartesian products of graphs, Electronic J. Combin. 20(2):#P19, 2013]: Let $G=G(X,Y)$ be a bipartite graph. Call the $X\rightarrow Y$ orientation of $G,$ the canonical orientation. Let $\phi$ be any orientation of $G$ and let $Sp_S(G^{\phi})$ and $Sp(G)$ denote respectively the skew spectrum of $G^{\phi}$ and the spectrum of $G.$ Then $Sp_S(G^{\phi}) = {\bf{i}} Sp(G)$ if and only if $\phi$ is switching-equivalent to the canonical orientation of $G.$ Using this result, we determine the switch for a special family of oriented hypercubes $Q_d^{\phi},$ $d\geq 1.$ Moreover, we give an orientation of the Cartesian product of a bipartite graph and a graph, and then determine the skew spectrum of the resulting oriented product graph, which generalizes a result of Cui and Hou. Further this can be used to construct new families of oriented graphs with maximum skew energy.

scholarly journals On the Skew Spectra of Cartesian Products of Graphs

10.37236/2864 ◽  
2013 ◽  
Vol 20 (2) ◽  
Author(s):  
Cui Denglan ◽  
Hou Yaoping
Keyword(s):  
Directed Graph ◽  
Adjacency Matrix ◽  
Undirected Graph ◽  
Oriented Graph ◽  
Bipartite Graphs ◽  
Cartesian Products ◽  
Product Graph ◽  
Underlying Graph ◽  
Skew Energy ◽  
Skew Adjacency Matrix

An oriented graph ${G^{\sigma}}$ is a simple undirected graph $G$ with an orientation, which assigns to each edge of $G$ a direction so that ${G^{\sigma}}$ becomes a directed graph. $G$ is called the underlying graph of ${G^{\sigma}}$ and we denote by $S({G^{\sigma}})$ the skew-adjacency matrix of ${G^{\sigma}}$ and its spectrum $Sp({G^{\sigma}})$ is called the skew-spectrum of ${G^{\sigma}}$. In this paper, the skew spectra of two orientations of the Cartesian products are discussed, as applications, new families of oriented bipartite graphs ${G^{\sigma}}$ with $Sp({G^{\sigma}})={\bf i} Sp(G)$ are given and the orientation of a product graph with maximum skew energy is obtained.

scholarly journals Peg Solitaire on Cartesian Products of Graphs

2021 ◽  
Vol 37 (3) ◽  
pp. 907-917
Author(s):  
Martin Kreh ◽  
Jan-Hendrik de Wiljes
Keyword(s):  
Cartesian Product ◽  
Cartesian Products ◽  
Connected Graphs ◽  
Grid Graphs ◽  
Cartesian Products Of Graphs

AbstractIn 2011, Beeler and Hoilman generalized the game of peg solitaire to arbitrary connected graphs. In the same article, the authors proved some results on the solvability of Cartesian products, given solvable or distance 2-solvable graphs. We extend these results to Cartesian products of certain unsolvable graphs. In particular, we prove that ladders and grid graphs are solvable and, further, even the Cartesian product of two stars, which in a sense are the “most” unsolvable graphs.

scholarly journals On Cartesian Products of Orthogonal Double Covers

2013 ◽  
Vol 2013 ◽  
pp. 1-4 ◽  
Author(s):  
R. El Shanawany ◽  
M. Higazy ◽  
A. El Mesady
Keyword(s):  
Cartesian Product ◽  
Bipartite Graphs ◽  
Double Cover ◽  
Cartesian Products ◽  
Double Covers ◽  
Complete Bipartite Graphs ◽  
Orthogonal Double Cover

LetHbe a graph onnvertices and𝒢a collection ofnsubgraphs ofH, one for each vertex, where𝒢is an orthogonal double cover (ODC) ofHif every edge ofHoccurs in exactly two members of𝒢and any two members share an edge whenever the corresponding vertices are adjacent inHand share no edges whenever the corresponding vertices are nonadjacent inH. In this paper, we are concerned with the Cartesian product of symmetric starter vectors of orthogonal double covers of the complete bipartite graphs and using this method to construct ODCs by new disjoint unions of complete bipartite graphs.

scholarly journals A NOTE ON EDGE-CONNECTIVITY OF THE CARTESIAN PRODUCT OF GRAPHS

2011 ◽  
Vol 84 (1) ◽  
pp. 171-176
Author(s):  
LAKOA FITINA ◽  
C. T. LENARD ◽  
T. M. MILLS
Keyword(s):  
Cartesian Product ◽  
Edge Connectivity ◽  
Cartesian Products ◽  
Cartesian Product Of Graphs ◽  
Necessary And Sufficient ◽  
Product Of Graphs ◽  
Cartesian Products Of Graphs

AbstractThe main aim of this paper is to establish conditions that are necessary and sufficient for the edge-connectivity of the Cartesian product of two graphs to equal the sum of the edge-connectivities of the factors. The paper also clarifies an issue that has arisen in the literature on Cartesian products of graphs.

scholarly journals Separation of Cartesian products of graphs into several connected components by the removal of edges

2021 ◽  
pp. 18-18
Author(s):  
Simon Spacapan
Keyword(s):  
Cartesian Product ◽  
Connected Components ◽  
Edge Connectivity ◽  
Cartesian Products ◽  
Minimum Cardinality ◽  
Connected Graphs ◽  
Cartesian Products Of Graphs

Let G = (V (G),E(G)) be a graph. A set S ? E(G) is an edge k-cut in G if the graph G-S = (V (G), E(G) \ S) has at least k connected components. The generalized k-edge connectivity of a graph G, denoted as ?k(G), is the minimum cardinality of an edge k-cut in G. In this article we determine generalized 3-edge connectivity of Cartesian product of connected graphs G and H and describe the structure of any minimum edge 3-cut in G2H. The generalized 3-edge connectivity ?3(G2H) is given in terms of ?3(G) and ?3(H) and in terms of other invariants of factors G and H.

On Turán exponents of bipartite graphs

2021 ◽  
pp. 1-12
Author(s):  
Tao Jiang ◽  
Jie Ma ◽  
Liana Yepremyan
Keyword(s):  
Bipartite Graph ◽  
Rational Number ◽  
Upper Bound ◽  
Bipartite Graphs ◽  
Affirmative Answer ◽  
Finite Family ◽  
Turán Number ◽  
New Form ◽  
Single Graph

Abstract A long-standing conjecture of Erdős and Simonovits asserts that for every rational number $r\in (1,2)$ there exists a bipartite graph H such that $\mathrm{ex}(n,H)=\Theta(n^r)$ . So far this conjecture is known to be true only for rationals of form $1+1/k$ and $2-1/k$ , for integers $k\geq 2$ . In this paper, we add a new form of rationals for which the conjecture is true: $2-2/(2k+1)$ , for $k\geq 2$ . This in turn also gives an affirmative answer to a question of Pinchasi and Sharir on cube-like graphs. Recently, a version of Erdős and Simonovits $^{\prime}$ s conjecture, where one replaces a single graph by a finite family, was confirmed by Bukh and Conlon. They proposed a construction of bipartite graphs which should satisfy Erdős and Simonovits $^{\prime}$ s conjecture. Our result can also be viewed as a first step towards verifying Bukh and Conlon $^{\prime}$ s conjecture. We also prove an upper bound on the Turán number of theta graphs in an asymmetric setting and employ this result to obtain another new rational exponent for Turán exponents: $r=7/5$ .

scholarly journals A graph-theoretical approach to DNA similarity analysis

2021 ◽  
Author(s):  
Dong Quan Ngoc Nguyen ◽  
Lin Xing ◽  
Phuong Dong Tan Le ◽  
Lizhen Lin
Keyword(s):  
Dna Sequence ◽  
Dna Sequences ◽  
Influenza A ◽  
Cartesian Product ◽  
Space Structure ◽  
Similarity Analysis ◽  
Cartesian Products ◽  
Research Areas ◽  
Active Research ◽  
Cartesian Products Of Graphs

One of the very active research areas in bioinformatics is DNA similarity analysis. There are several approaches using alignment-based or alignment-free methods to analyze similarities/dissimilarities between DNA sequences. In this work, we introduce a novel representation of DNA sequences, using n-ary Cartesian products of graphs for arbitrary positive integers n. Each of the component graphs in the representing Cartesian product of each DNA sequence contain combinatorial information of certain tuples of nucleotides appearing in the DNA sequence. We further introduce a metric space structure to the set of all Cartesian products of graphs that represent a given collection of DNA sequences in order to be able to compare different Cartesian products of graphs, which in turn signifies similarities/dissimilarities between DNA sequences. We test our proposed method on several datasets including Human Papillomavirus, Human rhinovirus, Influenza A virus, and Mammals. We compare our method to other methods in literature, which indicates that our analysis results are comparable in terms of time complexity and high accuracy, and in one dataset, our method performs the best in comparison with other methods.

scholarly journals TWO RESULTS ON THE PALETTE INDEX OF GRAPHS

2021 ◽  
Vol 55 (1 (254)) ◽  
pp. 36-43
Author(s):  
Khachik S. Smbatyan
Keyword(s):  
Bipartite Graph ◽  
Upper Bound ◽  
Edge Coloring ◽  
Cartesian Products ◽  
Proper Edge Coloring ◽  
Minimum Number ◽  
Nearly Bipartite Graph ◽  
Cartesian Products Of Graphs

Given a proper edge coloring $\alpha$ of a graph $G$, we define the palette $S_G(v,\alpha)$ of a vertex $v\in V(G)$ as the set of all colors appearing on edges incident with $v$. The palette index $\check{s}(G)$ of $G$ is the minimum number of distinct palettes occurring in a proper edge coloring of $G$. A graph $G$ is called nearly bipartite if there exists $ v\in V(G)$ so that $G-v$ is a bipartite graph. In this paper, we give an upper bound on the palette index of a nearly bipartite graph $G$ by using the decomposition of $G$ into cycles. We also provide an upper bound on the palette index of Cartesian products of graphs. In particular, we show that for any graphs $G$ and $H$, $\check{s}(G\square H)\leq \check{s}(G)\check{s}(H)$.

scholarly journals On the Gonality of Cartesian Products of Graphs

10.37236/9307 ◽  
2020 ◽  
Vol 27 (4) ◽  
Author(s):  
Ivan Aidun ◽  
Ralph Morrison
Keyword(s):  
Upper Bound ◽  
Algebraic Curve ◽  
Cartesian Product ◽  
Metric Graphs ◽  
Systematic Treatment ◽  
Cartesian Products ◽  
Product Graphs ◽  
Chip Firing ◽  
Cartesian Products Of Graphs

In this paper we provide the first systematic treatment of Cartesian products of graphs and their divisorial gonality, which is a tropical version of the gonality of an algebraic curve defined in terms of chip-firing.  We prove an upper bound on the gonality of the Cartesian product of any two graphs, and determine instances where this bound holds with equality, including for the $m\times n$ rook's graph with $\min\{m,n\}\leq 5$.  We use our upper bound to prove that Baker's gonality conjecture holds for the Cartesian product of any two graphs with two or more vertices each, and we determine precisely which nontrivial product graphs have gonality equal to Baker's conjectural upper bound.  We also extend some of our results to metric graphs.

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