scholarly journals On the Crossing Numbers of the Join Products of Six Graphs of Order Six with Paths and Cycles

Symmetry ◽  
2021 ◽  
Vol 13 (12) ◽  
pp. 2441
Author(s):  
Michal Staš
Keyword(s):  
Lower Bounds ◽  
Crossing Number ◽  
Crossing Numbers ◽  
Symmetric Graphs ◽  
The Family ◽  
Minimum Number ◽  
The Common ◽  
First Time ◽  
Edge Crossings

The crossing number of a graph G is the minimum number of edge crossings over all drawings of G in the plane. The main purpose of this paper is to determine the crossing numbers of the join products of six symmetric graphs on six vertices with paths and cycles on n vertices. The idea of configurations is generalized for the first time onto the family of subgraphs whose edges cross the edges of the considered graph at most once, and their lower bounds of necessary numbers of crossings are presented in the common symmetric table. Some proofs of the join products with cycles are done with the help of several well-known auxiliary statements, the idea of which is extended by a suitable classification of subgraphs that do not cross the edges of the examined graphs.

scholarly journals On the crossing numbers of join products of W_{4}+P_{n} and W_{4}+C_{n}

2021 ◽  
Vol 41 (1) ◽  
pp. 95-112
Author(s):  
Michal Staš ◽  
Juraj Valiska
Keyword(s):  
Crossing Number ◽  
Crossing Numbers ◽  
Minimum Number ◽  
Edge Crossings

The crossing number \(\mathrm{cr}(G)\) of a graph \(G\) is the minimum number of edge crossings over all drawings of \(G\) in the plane. The main aim of the paper is to give the crossing number of the join product \(W_4+P_n\) and \(W_4+C_n\) for the wheel \(W_4\) on five vertices, where \(P_n\) and \(C_n\) are the path and the cycle on \(n\) vertices, respectively. Yue et al. conjectured that the crossing number of \(W_m+C_n\) is equal to \(Z(m+1)Z(n)+(Z(m)-1) \big \lfloor \frac{n}{2} \big \rfloor + n+ \big\lceil\frac{m}{2}\big\rceil +2\), for all \(m,n \geq 3\), and where the Zarankiewicz's number \(Z(n)=\big \lfloor \frac{n}{2} \big \rfloor \big \lfloor \frac{n-1}{2} \big \rfloor\) is defined for \(n\geq 1\). Recently, this conjecture was proved for \(W_3+C_n\) by Klešč. We establish the validity of this conjecture for \(W_4+C_n\) and we also offer a new conjecture for the crossing number of the join product \(W_m+P_n\) for \(m\geq 3\) and \(n\geq 2\).

scholarly journals Join Products K2,3 + Cn

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 925
Author(s):  
Michal Staš
Keyword(s):  
Bipartite Graph ◽  
Complete Bipartite Graph ◽  
Crossing Number ◽  
Arithmetic Means ◽  
Minimum Number ◽  
Edge Crossings

The crossing number cr ( G ) of a graph G is the minimum number of edge crossings over all drawings of G in the plane. The main goal of the paper is to state the crossing number of the join product K 2 , 3 + C n for the complete bipartite graph K 2 , 3 , where C n is the cycle on n vertices. In the proofs, the idea of a minimum number of crossings between two distinct configurations in the various forms of arithmetic means will be extended. Finally, adding one more edge to the graph K 2 , 3 , we also offer the crossing number of the join product of one other graph with the cycle C n .

scholarly journals Improvement on the Crossing Number of Crossing-Critical Graphs

2020 ◽  
Author(s):  
János Barát ◽  
Géza Tóth
Keyword(s):  
Crossing Number ◽  
Critical Graphs ◽  
Minimum Number ◽  
Edge Crossings

AbstractThe crossing number of a graph G is the minimum number of edge crossings over all drawings of G in the plane. A graph G is k-crossing-critical if its crossing number is at least k, but if we remove any edge of G, its crossing number drops below k. There are examples of k-crossing-critical graphs that do not have drawings with exactly k crossings. Richter and Thomassen proved in 1993 that if G is k-crossing-critical, then its crossing number is at most $$2.5\, k+16$$ 2.5 k + 16 . We improve this bound to $$2k+8\sqrt{k}+47$$ 2 k + 8 k + 47 .

A comparison of methodological approaches to the subfamilial classification of the Naididae (Oligochaeta)

1987 ◽  
Vol 65 (3) ◽  
pp. 691-707 ◽  
Author(s):  
A. F. L. Nemec ◽  
R. O. Brinkhurst
Keyword(s):  
Data Matrix ◽  
Jaccard Coefficient ◽  
Data Set ◽  
Parsimony Method ◽  
Fit Statistics ◽  
Reconstruction Methods ◽  
Linkage Methods ◽  
The Family ◽  
First Time

A data matrix of 23 generic or subgeneric taxa versus 24 characters and a shorter matrix of 15 characters were analyzed by means of ordination, cluster analyses, parsimony, and compatibility methods (the last two of which are phylogenetic tree reconstruction methods) and the results were compared inter alia and with traditional methods. Various measures of fit for evaluating the parsimony methods were employed. There were few compatible characters in the data set, and much homoplasy, but most analyses separated a group based on Stylaria from the rest of the family, which could then be separated into four groups, recognized here for the first time as tribes (Naidini, Derini, Pristinini, and Chaetogastrini). There was less consistency of results within these groups. Modern methods produced results that do not conflict with traditional groupings. The Jaccard coefficient minimizes the significance of symplesiomorphy and complete linkage avoids chaining effects and corresponds to actual similarities, unlike single or average linkage methods, respectively. Ordination complements cluster analysis. The Wagner parsimony method was superior to the less flexible Camin–Sokal approach and produced better measure of fit statistics. All of the aforementioned methods contain areas susceptible to subjective decisions but, nevertheless, they lead to a complete disclosure of both the methods used and the assumptions made, and facilitate objective hypothesis testing rather than the presentation of conflicting phylogenies based on the different, undisclosed premises of manual approaches.

Cyclic permutations and crossing numbers of join products of two symmetric graphs of order six

2019 ◽  
Vol 35 (2) ◽  
pp. 137-146
Author(s):  
STEFAN BEREZNY ◽  
MICHAL STAS ◽  
◽  
Keyword(s):  
Crossing Number ◽  
Crossing Numbers ◽  
Symmetric Graphs ◽  
Isolated Vertex ◽  
Join Of Graphs ◽  
Discrete Graph

The main purpose of this article is broaden known results concerning crossing numbers for join of graphs of order six. We give the crossing number of the join product G + Dn, where the graph G consists of one 5-cycle and of one isolated vertex, and Dn consists on n isolated vertices. The proof is done with the help of software that generates all cyclic permutations for a given number k, and creates a new graph COG for calculating the distances between all vertices of the graph. Finally, by adding some edges to the graph G, we are able to obtain the crossing numbers of the join product with the discrete graph Dn and with the path Pn on n vertices for other two graphs.

A Bayesian approach to the classification of the Turkic languages

2020 ◽  
pp. 114-124
Author(s):  
Alexander Savelyev
Keyword(s):  
Bayesian Approach ◽  
Controversial Issue ◽  
Language Family ◽  
Independent Verification ◽  
Linguistic History ◽  
The Family ◽  
Turkic Languages ◽  
The Common ◽  
Widespread View

Despite more than 150 years of research, the internal structure of the Turkic language family remains a controversial issue. In this study, the Bayesian phylogenetic approach is employed in order to provide an independent verification of the contemporary views on Turkic linguistic history. The data underlying the study are Turkic basic vocabularies, which are resistant to replacement and likely to reflect the genealogical relationships among the Turkic languages. The method tested in the chapter is based on the strict clock model of evolution, which assumes that relevant changes occur at the same rate at every branch of the family. This study supports the widespread view that the binary split between Bulgharic and Common Turkic was the earliest split in the Turkic family. The model further replicates most of the conventional subgroups within the Common Turkic branch. Based on a Bayesian analysis, the time depth of Proto-Turkic is estimated to be around 2,119 years BP, which is in accordance with the traditional estimates of 2,000–2,500 years BP.

scholarly journals Crossing Numbers and Hard Erdős Problems in Discrete Geometry

1997 ◽  
Vol 6 (3) ◽  
pp. 353-358 ◽  
Author(s):  
LÁSZLÓ A. SZÉKELY
Keyword(s):  
Lower Bound ◽  
Discrete Geometry ◽  
Crossing Number ◽  
Plane Geometry ◽  
Crossing Numbers ◽  
Minimum Number

We show that an old but not well-known lower bound for the crossing number of a graph yields short proofs for a number of bounds in discrete plane geometry which were considered hard before: the number of incidences among points and lines, the maximum number of unit distances among n points, the minimum number of distinct distances among n points.

scholarly journals Coherent band pathways between knots and links

2015 ◽  
Vol 24 (02) ◽  
pp. 1550006 ◽  
Author(s):  
Dorothy Buck ◽  
Kai Ishihara
Keyword(s):  
Lower Bounds ◽  
Recombinant Dna ◽  
Crossing Number ◽  
Dna Molecules ◽  
Minimum Number ◽  
Knots And Links

We categorize coherent band (aka nullification) pathways between knots and 2-component links. Additionally, we characterize the minimal coherent band pathways (with intermediates) between any two knots or 2-component links with small crossing number. We demonstrate these band surgeries for knots and links with small crossing number. We apply these results to place lower bounds on the minimum number of recombinant events separating DNA configurations, restrict the recombination pathways and determine chirality and/or orientation of the resulting recombinant DNA molecules.

The comparative morphology of the male and female postabdomen of the Australian Tachinidae (Diptera), with descriptions of some first-instar larvae and pupae

1988 ◽  
Vol 2 (1) ◽  
pp. 81 ◽  
Author(s):  
BK Cantrell
Keyword(s):  
Comparative Morphology ◽  
Instar Larva ◽  
Taxonomic Position ◽  
Specific Level ◽  
Male And Female ◽  
First Instar ◽  
New Information ◽  
The Family ◽  
First Time

The comparactive morphology of the male and female postabdomen of the Australian Tachinidae was studied in a survey which included 152 species of representative genera from all subfamilies except Dufouriinae. The value of the structure of the postabdomen for the higher classification of the family was ascertained, and new information gained which has allowed a better understanding of the correct taxonomic position of some problem groups of tachinids. It was possible to recognise suites of characters for each sex to define each subfamily, but this was not possible at tribal or generic levels in most groups. The male terininalia are diagnostic at specific level. The survey also allowed the discovery of, or confirmed, the reproductive habit of the included genera and provided information on the first-instar larva of 52 species, many of which larvae are described below for the first time. Characters of puparia which may have taxonomic value are discussed and illustrated.

scholarly journals ON THE CROSSING NUMBER OF THE CARTESIAN PRODUCT OF A SUNLET GRAPH AND A STAR GRAPH

2019 ◽  
Vol 100 (1) ◽  
pp. 5-12
Author(s):  
MICHAEL HAYTHORPE ◽  
ALEX NEWCOMBE
Keyword(s):  
Upper Bound ◽  
Cartesian Product ◽  
Crossing Number ◽  
Star Graph ◽  
Cartesian Products ◽  
Crossing Numbers ◽  
First Time

The exact crossing number is only known for a small number of families of graphs. Many of the families for which crossing numbers have been determined correspond to cartesian products of two graphs. Here, the cartesian product of the sunlet graph, denoted ${\mathcal{S}}_{n}$, and the star graph, denoted $K_{1,m}$, is considered for the first time. It is proved that the crossing number of ${\mathcal{S}}_{n}\Box K_{1,2}$ is $n$, and the crossing number of ${\mathcal{S}}_{n}\Box K_{1,3}$ is $3n$. An upper bound for the crossing number of ${\mathcal{S}}_{n}\Box K_{1,m}$ is also given.

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