Related papers: The Smith group of the hypercube

We associate a graph to a possible non-zero zero-divisor in the group algebra of a torsion-free group.

A unit cube in $k$ dimensions ($k$-cube) is defined as the the Cartesian product $R_1\times R_2\times...\times R_k$ where $R_i$(for $1\leq i\leq k$) is a closed interval of the form $[a_i,a_i+1]$ on the real line. A graph $G$ on $n$ nodes…

Recently there has been growing interest in discrete homotopies and homotopies of graphs beyond treating graphs as 1-dimensional simplicial spaces. One such type of homotopy is $\times$-homotopy. Recent work by Chih-Scull has developed a…

An orbit of $G$ is a subset $S$ of $V(G)$ such that $\phi(u)=v$ for any two vertices $u,v\in S$, where $\phi$ is an isomorphism of $G$. The orbit number of a graph $G$, denoted by $\text{Orb}(G)$, is the number of orbits of $G$. In [A Note…

A matchstick graph is a plane graph with edges drawn as unit distance line segments. This class of graphs was introduced by Harborth who conjectured that a matchstick graph on $n$ vertices can have at most $\lfloor 3n - \sqrt{12n -…

The line graph $\Gamma$ of a multi-graph $\Delta$ is the graph whose vertices are the edges of $\Delta$, where two such edges are adjacent if and only if they meet in a single vertex of $\Delta$. We provide several characterizations of such…

The degree matrix of a graph is the diagonal matrix with diagonal entries equal to the degrees of the vertices of $X$. If $X_1$ and $X_2$ are graphs with respective adjacency matrices $A_1$ and $A_2$ and degree matrices $D_1$ and $D_2$, we…

Let $\Omega$ be a $m$-set, where $m>1$, is an integer. The Hamming graph $H(n,m)$, has $\Omega ^{n}$ as its vertex-set, with two vertices are adjacent if and only if they differ in exactly one coordinate. In this paper, we provide a proof…

A finite graph is called a tricirculant if admits a cyclic group of automorphism which has precisely three orbits on the vertex-set of the graph, all of equal size. We classify all finite connected cubic vertex-transitive tricirculants. We…

In this paper we introduce the graph $\Gamma_{sc}(G)$ associated with a group $G$, called the solvable conjugacy class graph (abbreviated as SCC-graph), whose vertices are the nontrivial conjugacy classes of $G$ and two distinct conjugacy…

A string graph is an intersection graph of curves in the plane. A $k$-string graph is a graph with a string representation in which every pair of curves intersects in at most $k$ points. We introduce the class of $(=k)$-string graphs as a…

In this note, we define a new graph $\Gamma_d(G)$ on a finite group $G$, where $d$ is a divisor of $|G|$. The vertices of $\Gamma_d(G)$ are the subgroups of $G$ of order $d$ and two subgroups $H_1$ and $H_2$ of $G$ are said to be adjacent…

For a positive integer $n$, a graph with at least $n$ vertices is $n$-existentially closed or simply $n$-e.c. if for any set of vertices $S$ of size $n$ and any set $T\subseteq S$, there is a vertex $x\not\in S$ adjacent to each vertex of…

We study random subcube intersection graphs, that is, graphs obtained by selecting a random collection of subcubes of a fixed hypercube $Q_d$ to serve as the vertices of the graph, and setting an edge between a pair of subcubes if their…

Finite connected cubic symmetric graphs of girth 6 have been classified by K. Kutnar and D. Maru\v{s}i\v{c}, in particular, each of these graphs has an abelian automorphism group with two orbits on the vertex set. In this paper all cubic…

A coalition in a graph $G$ with vertex set $V$ consists of two disjoint sets $V_1, V_2\subset V$ such that neither $V_1$ nor $V_2$ is a dominating set, but the union $V_1\cup V_2$ is a dominating set in $G$. A partition of graph vertices is…

A graph product kernel means the kernel of the natural surjection from a graph product to the corresponding direct product. We prove that a graph product kernel of countable groups is special, and a graph product of finite or cyclic groups…

The dissociation number ${\rm diss}(G)$ of a graph $G$ is the maximum order of a set of vertices of $G$ inducing a subgraph that is of maximum degree at most $1$. Computing the dissociation number of a given graph is algorithmically hard…

In 1985, Golumbic and Scheinerman established an equivalence between comparability graphs and containment graphs, graphs whose vertices represent sets, with edges indicating set containment. A few years earlier, McMorris and Zaslavsky…

In this note, we connect the $n$-torsions of the Picard group of an elliptic surface to the $n$-divisibility of the class group of torsion fields for a given integer $n>1$. We also connect the $n$-divisibility of the Selmer group to that of…