Related papers: The graphs of non-degenerate linear codes

Let $V$ be an $n$-dimensional left vector space over a division ring $R$. We write ${\mathcal G}_{k}(V)$ for the Grassmannian formed by $k$-dimensional subspaces of $V$ and denote by $\Gamma_{k}(V)$ the associated Grassmann graph. Let also…

Denote by $\mathbb G(k,n)$ the Grassmannian of linear subspaces of dimension $k$ in $\mathbb P^n$. We show that, if $\varphi:\mathbb G(l,n) \to \mathbb G(k,n)$ is a non constant morphism and $l \not=0,n-1$ then $l=k$ or $l=n-k-1$ and…

We discuss the Grassmann graph $J_q(N,D)$ with $N \geq 2D$, having as vertices the $D$-dimensional subspaces of an $N$-dimensional vector space over the finite field $\mathbb{F}_q$. This graph is distance-regular with diameter $D$; to avoid…

Let $V$ be an $n$-dimensional vector space over a finite field $\mathbb{F}_q$, where $q$ is a prime power. Define the \emph{generalized $q$-Kneser graph} $K_q(n,k,t)$ to be the graph whose vertices are the $k$-dimensional subspaces of $V$…

Let $S$ be an $n$-dimensional vector space over the finite field $\mathbb{F}_q$, where $q$ is necessarily a prime power. Denote $K_q(n,k)$ (resp. $J_q(n,k)$) to be the \emph{$q$-Kneser graph} (resp. \emph{Grassmann graph}) for $k\geq 1$…

Let $\Gamma_k(V)$ be the Grassmann graph whose vertex set is formed by all $k$-dimensional subspaces of an $n$-dimensional vector space $V$ over the finite field $F_q$ consisting of $q$ elements. We discuss its subgraph $\Pi(n,k)_q$ formed…

The family of generalized Paley graphs of prime power order $q$ and degree $(q-1)/k$ is studied. It is shown that the automorphism group of a graph in this family is a subgroup of ${\mathrm{A\Gamma L}}(1,q)$ whenever $q$ is sufficiently…

Let $k \subset K$ be an extension of fields, and let $A \subset M_{n}(K)$ be a $k$-algebra. We study parameter spaces of $m$-dimensional subspaces of $K^{n}$ which are invariant under $A$. The space $\mathbb{F}_{A}(m,n)$, whose $R$-rational…

We consider the graph whose vertex set is a conjugacy class ${\mathcal C}$ consisting of finite-rank self-adjoint operators on a complex Hilbert space $H$. The dimension of $H$ is assumed to be not less than $3$. In the case when operators…

The {\em Fibonacci cube} of dimension $n$, denoted as $\Gamma\_n$, is the subgraph of $n$-cube $Q\_n$ induced by vertices with no consecutive 1's. We study the maximum number of disjoint subgraphs in $\Gamma\_n$ isomorphic to $Q\_k$, and…

Grassmannian $\mathcal{G}_q(n,k)$ is the set of all $k$-dimensional subspaces of the vector space $\mathbb{F}_q^n.$ Recently, Etzion and Zhang introduced a new notion called covering Grassmannian code which can be used in network coding…

Let $n >3$ and $ 0< k < \frac{n}{2} $ be integers. In this paper, we investigate some algebraic properties of the line graph of the graph $ {Q_n}(k,k+1) $ where $ {Q_n}(k,k+1) $ is the subgraph of the hypercube $Q_n$ which is induced by the…

Let $H(n, q^2)$ be a non-degenerate Hermitian variety of $PG(n,q^2)$, $n \geq 2$. Let $NU(n+1,q^2)$ be the graph whose vertices are the points of $PG(n,q^2) \setminus H(n,q^2)$ and two vertices $u,~v$ are adjacent if the line joining $u$…

We investigate the existence of Boolean degree $d$ functions on the Grassmann graph of $k$-spaces in the vector space $\mathbb{F}_q^n$. For $d=1$ several non-existence and classification results are known, and no non-trivial examples are…

A graph is $k$-degenerate if every subgraph has minimum degree at most $k$. We provide lower bounds on the size of a maximum induced 2-degenerate subgraph in a triangle-free planar graph. We denote the size of a maximum induced 2-degenerate…

Consider the point line-geometry ${\mathcal P}_t(n,k)$ having as points all the $[n,k]$-linear codes having minimum dual distance at least $t+1$ and where two points $X$ and $Y$ are collinear whenever $X\cap Y$ is a $[n,k-1]$-linear code…

In this paper we investigate the chromatic number of the Grassmann graphs and of their powers, denoted $J_q(n,m,t)$. In this graph, the vertices correspond to the $m$-dimensional subspaces in $\mathbb{F}_q^n$ and two vertices are adjacent…

Let $\mathbb{F}_q$ denote a finite field with $q$ elements. Let $N$ and $D$ denote integers with $N>D \ge 1$. Let $\mathcal{V}$ denote an $N$-dimensional vector space over $\mathbb{F}_q$. The Grassmann graph $J_q(N,D)$ is the graph with…

An automorphism of a graph $G=(V,E)$ is a bijective map $\phi$ from $V$ to itself such that $\phi(v_i)\phi(v_j)\in E$ $\Leftrightarrow$ $v_i v_j\in E$ for any two vertices $v_i$ and $v_j$. Denote by $\mathfrak{G}$ the group consisting of…

Let $G$ be a connected graph on $n$ vertices and $1 \le k \le n-1$ an integer. The $k$-token graph of $G$ is the graph $F_k(G)$ whose vertices are all the $k$-subsets of vertices of $G$, two of which are adjacent whenever their symmetric…