Related papers: Replacing a graph clasper by tree claspers
One of the most basic results in graph theory states that every graph with at least two vertices has two vertices with the same degree. Since there are graphs without $3$ vertices of the same degree, it is natural to ask if for any fixed…
Grouping the nodes of a graph into clusters is a standard technique for studying networks. We study a problem where we are given a directed network and are asked to partition the graph into a sequence of coherent groups. We assume that…
We describe a correspondence between GL_n-invariant tensors and graphs, and show how this correspondence accomodates various types of symmetries and orientations.
We examine the adjacency spectrum of trees with diameter three, also referred to as double stars. Using $P_2(a,b)$ to denote a double star with $ a$ and $b$ leaves at its respective endpoints, we discuss graphs which are cospectral to…
A unicellular collection on a surface is a collection of curves whose complement is a single disk. There is a natural surgery operation on unicellular collections, endowing the set of such with a graph structure where the edge relation is…
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…
This paper studies the structure of graphs with given tree-width and excluding a fixed complete bipartite subgraph, which generalises the bounded degree setting. We give a new structural description of such graphs in terms of so-called…
Let $G$ be a graph with adjacency matrix $A(G)$ and degree matrix $D(G)$, and let $L_\mu(G):=A(G)-\mu D(G)$. Two graphs $G_1$ and $G_2$ are called \emph{degree-similar} if there exists an invertible matrix $M$ such that $M^{-1} A(G_1) M…
A contraction sequence of a graph consists of iteratively merging two of its vertices until only one vertex remains. The recently introduced twin-width graph invariant is based on contraction sequences. More precisely, if one puts red edges…
We present a streamlined exposition of a construction by R. Chen, A. Poulin, R. Tao, and A. Tserunyan, which proves the treeability of equivalence relations generated by any locally-finite Borel graph such that each component is a…
In this paper, we establish that the arc shift operation on a $n$-component virtual link diagram acts as an unknotting operation when the virtual link is $n$-homogeneous proper, aiding in the classification of \( n \)-component virtual…
We address a question posed by Fessler-Jensen-Kelsey-Owen regarding graphs whose second gonality is greater than the first by exactly 1. We answer the question affirmatively under a stronger condition, thereby characterising the entire…
In graph realization problems one is given a degree sequence and the task is to decide whether there is a graph whose vertex degrees match to the given sequence. This realization problem is known to be polynomial-time solvable when the…
We say that a graph $G$ is $(2,m)$-linked if, for any distinct vertices $a_1,\ldots, a_m, b_1,b_2$ in $G$, there exist vertex disjoint connected subgraphs $A,B$ of $G$ such that $\{a_1, \ldots, a_m\}$ is contained in $A$ and $\{b_1,b_2\}$…
We consider two CSP problems: the first CSP encodes 2D Sperner's lemma for the standard triangulation of the right triangle on $n^2$ small triangles; the second CSP encodes the fact that it is impossible to match cells of $n \times n$…
Cut-diagrams are diagrammatic objects, defined in dimensions 1 and 2, that generalize links in 3-space and surface-links in 4-space; in dimension 1, this coincides with the theory of welded links. Using cut-diagrams, we introduce an…
A C_n-move is a local move on links defined by Habiro and Goussarov, which can be regarded as a `higher order crossing change'. We use Milnor invariants with repeating indices to provide several classification results for links up to…
A graph G is c-closed if every two vertices with at least c common neighbors are adjacent to each other. Introduced by Fox, Roughgarden, Seshadhri, Wei and Wein [ICALP 2018, SICOMP 2020], this definition is an abstraction of the triadic…
Given a graph, we associate each edge with the transposition which exchanges the endvertices. Fixing a linear order on the edge set, we obtain a permutation of the vertices. D\'enes proved that the permutation is a full cyclic permutation…
The $k$-deck of a graph is the multiset of its subgraphs induced by $k$ vertices. A graph or graph property is $l$-reconstructible if it is determined by the deck of subgraphs obtained by deleting $l$ vertices. We show that the degree list…