Related papers: On region crossing change and incidence matrix
A switching method is a graph operation that results in cospectral graphs (graphs with the same spectrum). Work by Wang and Xu [Discrete Math. 310 (2010)] suggests that most cospectral graphs with cospectral complements can be constructed…
We present evidence in support of a conjecture that a bipartite graph with at least five vertices in each part and |E(G)| \geq 4 |V(G)| - 17 is intrinsically knotted. We prove the conjecture for graphs that have exactly five or exactly six…
Graph matching consists of aligning the vertices of two unlabeled graphs in order to maximize the shared structure across networks; when the graphs are unipartite, this is commonly formulated as minimizing their edge disagreements. In this…
An incidence in a graph $G$ is a pair $(v,e)$ where $v$ is a vertex of $G$ and $e$ is an edge of $G$ incident to $v$. Two incidences $(v,e)$ and $(u,f)$ are adjacent if at least one of the following holds: $(a)$ $v = u$, $(b)$ $e = f$, or…
Using Boolean algebra, we discuss the region unknotting number of a knot, and show that the region unknotting number is less than or equal to (c+1)/2 for any knot with crossing number c. This is a progress from (c+2)/2.
We introduce natural language processing into the study of knot theory, as made natural by the braid word representation of knots. We study the UNKNOT problem of determining whether or not a given knot is the unknot. After describing an…
In the United States, regions are frequently divided into districts for the purpose of electing representatives. How the districts are drawn can affect who's elected, and drawing districts to give an advantage to a certain group is known as…
Charts are oriented labeled graphs in a disk. Any simple surface braid (2-dimensional braid) can be described by using a chart. Also, a chart represents an oriented closed surface embedded in 4-space. In this paper, we investigate embedded…
Let $G$ be a signed graph. Let $\hat{G}$ be the graph obtained from $G$ by replacing each edge $e$ by a chain or a sheaf. We first establish a relation between the $Q$-polynomial of $\hat{G}$[6] and the $W$-polynomial of $G$ [9]. Two…
There is a well-known way to describe a link diagram as a (signed) plane graph, called its Tait graph. This concept was recently extended, providing a way to associate a set of embedded graphs (or ribbon graphs) to a link diagram. While…
This study delves into the incidence matrices of hypergraphs, with a focus on two types: the edge-vertex incidence matrix and the vertex-edge incidence matrix. The edge-vertex incidence matrix is a matrix in which the rows represent…
In this paper, we discuss the region unknotting number of different classes of 2-bridge knots. In particular, we provide region unknotting number for the classes of $2$-bridge knots whose Conway notation is $C(m,\ n), C(m,\ 2,\ m),$ $ C(m,\…
We study SIR type epidemics on graphs in two scenarios: (i) when the initial infections start from a well connected central region, (ii) when initial infections are distributed uniformly. Previously, \'Odor et al. demonstrated on a few…
Fleming and Foisy recently proved the existence of a digraph whose every embedding contains a $4$-component link, and left open the possibility that a directed graph with an intrinsic $n$-component link might exist. We show that, indeed,…
We study relations between unknotting number and crossing number of a spatial embedding of a handcuff-graph and a theta curve. It is well known that for any non-trivial knot $K$ twice the unknotting number of $K$ is less than or equal to…
Structural changes occur in dynamic networks quite frequently and its detection is an important question in many situations such as fraud detection or cybersecurity. Real-life networks are often incompletely observed due to individual…
We obtain a localized version of the configuration space integral for the Casson knot invariant, where the standard symmetric Gauss form is replaced with a locally supported form. An interesting technical difference between the arguments…
Rectangular diagrams of links are link diagrams in the plane ${\mathbb R}^2$ such that they are composed of vertical line segments and horizontal line segments and vertical segments go over horizontal segments at all crossings. P. R.…
Associated to every state surface for a knot or link is a state graph, which embeds as a spine of the state surface. A state graph can be decomposed along cut-vertices into graphs with induced planar embeddings. Associated with each such…
Given a knot diagram $D$, we construct a semi-threading circle for it which can be an axis of $D$ as a closed braid depending on knot diagrams. In particular, we consider semi-threading circles for minimal diagrams of a knot with respect to…