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

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.

The Crossing Numbers of Join Products of Paths and Cycles with Four Graphs of Order Five

The main aim of the paper is to establish the crossing numbers of the join products of the paths and the cycles on n vertices with a connected graph on five vertices isomorphic to the graph K1,1,3\e obtained by removing one edge e incident with some vertex of order two from the complete tripartite graph K1,1,3. The proofs are done with the help of well-known exact values for the crossing numbers of the join products of subgraphs of the considered graph with paths and cycles. Finally, by adding some edges to the graph under consideration, we obtain the crossing numbers of the join products of other graphs with the paths and the cycles on n vertices.

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

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 \(n\geq 2\).

Enhanced of Attendance Records Technology used Geospatial Retrieval based on Crossing Number

<span lang="EN-US">Nowadays, the fingerprint scanner widely used to records attendance. However, this technology has a weakness. Much research has done to improve the attendance system by utilizing mobile technology, like usage a fingerprint smartphone and location by GPS sensor to validate user location manually. In this research, we developed an application to enhance the records attendance system with a smartphone by crossing numbers to verify user position automatically, which implemented in a mobile app. This application using the PNPOLY method for detecting the location of the user inside of the polygon area predetermined. This method is part of the crossing number algorithm for increasing x and fixed y from point P, which x is latitude, and y is a longitude. The result of the experiment demonstrated that the percentage of successful validate user coordinate inside edges of the polygon boundary is 87%, depending on the GPS sensor embedded into a mobile device.</span>

Untangle problems, with knot theory

“Untangle problems, with knot theory” offers a basic introduction to the mathematical subfield of knot theory, including the classification of knots by crossing numbers. A mathematical knot is a closed loop that may or may not be tangled. Two knots are considered the same if one may be manipulated into the other using easy-to-understand techniques. Readers learn to identify knots by crossing numbers and encounter numerous hand-drawn sketches of knots, including the trivial knot, trefoil knot, figure-eight knot, and more. Mathematics students and enthusiasts are encouraged to employ knot theory methods for untangling problems in mathematics or life by asking whether they have encountered the problem before. At the chapter’s end, readers may check their understanding by working on a problem. A solution is provided.

Braid index bounds ropelength from below

For an unoriented link [Formula: see text], let [Formula: see text] be the ropelength of [Formula: see text]. It is known that in general [Formula: see text] is at least of the order [Formula: see text], and at most of the order [Formula: see text] where [Formula: see text] is the minimum crossing number of [Formula: see text]. Furthermore, it is known that there exist families of (infinitely many) links with the property [Formula: see text]. A long standing open conjecture states that if [Formula: see text] is alternating, then [Formula: see text] is at least of the order [Formula: see text]. In this paper, we show that the braid index of a link also gives a lower bound of its ropelength. More specifically, we show that there exists a constant [Formula: see text] such that [Formula: see text] for any [Formula: see text], where [Formula: see text] is the largest braid index among all braid indices corresponding to all possible orientation assignments of the components of [Formula: see text] (called the maximum braid index of [Formula: see text]). Consequently, [Formula: see text] for any link [Formula: see text] whose maximum braid index is proportional to its crossing number. In the case of alternating links, the maximum braid indices for many of them are proportional to their crossing numbers hence the above conjecture holds for these alternating links.