Related papers: Codes and Designs in Johnson Graphs From Symplecti…

The Johnson graph J(v,k) has, as vertices, the k-subsets of a v-set V, and as edges the pairs of k-subsets with intersection of size k-1. We introduce the notion of a neighbour-transitive code in J(v,k). This is a vertex subset \Gamma such…

Extending earlier results of Godsil and of Dobson and Malnic on Johnson graphs, we characterise those merged Johnson graphs $J=J(n,k)_I$ which are Cayley graphs, that is, which are connected and have a group of automorphisms acting…

This is a chapter in a forthcoming book on completely regular codes in distance regular graphs. The chapter provides an overview, and some original results, on codes in distance regular graphs which admit symmetries via a permutation group…

We classify the neighbour-transitive codes in Johnson graphs J(v, k) of minimum distance at least three which admit a neighbour-transitive group of automorphisms that is an almost simple two-transitive group of degree v and does not occur…

For positive integers $n,k$ and $t$, the uniform subset graph $G(n, k, t)$ has all $k$-subsets of $\{1,2,\ldots, n\}$ as vertices and two $k$-subsets are joined by an edge if they intersect at exactly $t$ elements. The Johnson graph…

The Johnson graph $J(n, i)$ is defined as the graph whose vertex set is the set of all $i$-element subsets of $\{1, . . ., n \}$, and two vertices are adjacent whenever the cardinality of their intersection is equal to $i$-1. In Ramras and…

The Johnson graph $J(n,i)$ is defined to the graph whose vertex set is the set of all $i$-element subsets of $\{1,\ldots,n\}$, and two vertices are joined whenever the cardinality of their intersection is equal to $i-1$. In Ramras and…

We determine all factorisations $X=AB$, where $X$ is a finite almost simple group and $A,B$ are core-free subgroups such that $A\cap B$ is cyclic or dihedral. As a main application, we classify the graphs $\Gamma$ admitting an almost simple…

We say that a Hamilton cycle $C=(x_1,\ldots,x_n)$ in a graph $G$ is $k$-symmetric, if the mapping $x_i\mapsto x_{i+n/k}$ for all $i=1,\ldots,n$, where indices are considered modulo $n$, is an automorphism of $G$. In other words, if we lay…

We define a Johnson graph code as a subspace of labelings of the vertices in a Johnson graph with the property that labelings are uniquely determined by their restriction to vertex neighborhoods specified by the parameters of the code. We…

The set $X$ of $k$-subsets of an $n$-set has a natural graph structure where two $k$-subsets are connected if and only if the size of their intersection is $k-1$. This is known as the Johnson graph. The symmetric group $S_n$ acts on the…

We classify trivalent vertex-transitive graphs whose edge sets have a partition into a 2-factor composed of two cycles and a 1-factor that is invariant under the action of the automorphism group.

The symplectic graph Sp(2d, q) is the collinearity graph of the symplectic space of dimension 2d over a finite field of order q. A k-regular graph on v vertices is a divisible design graph with parameters (v, k, lambda_1, lambda_2 ,m,n) if…

The Johnson graph J(n,N) is defined as the graph whose vertices are the n-subsets of the set {1,2,...,N}, where two vertices are adjacent if they share exactly n - 1 elements. Unlike Johnson graphs, induced subgraphs of Johnson graphs (JIS…

For a graph $H$ and an integer $k\ge 1$, the \emph{Token Sliding reconfiguration graph} $\mathsf{TS}_k(H)$ and the \emph{Token Jumping reconfiguration graph} $\mathsf{TJ}_k(H)$ have as vertices the $k$-cliques of $H$, with two vertices…

Applying a method of Godsil and McKay \cite{GM} to some graphs related to the symplectic graph, a series of new infinite families of strongly regular graphs with parameters…

The set $X$ of $k$-subsets of an $n$-set has a natural graph structure where two $k$-subsets are connected if and only if the size of their intersection is $k-1$. This is known as the Johnson graph. The symmetric group $S_n$ acts on the…

We initiate a study of the vertex clique covering numbers of Johnson graphs $J(N, k)$, the smallest numbers of cliques necessary to cover the vertices of those graphs. We prove identities for the values of these numbers when $k \leq 3$, and…

Let $V$ be an $n$-dimensional vector space ($4\le n <\infty$) and let ${\mathcal G}_{k}(V)$ be the Grassmannian formed by all $k$-dimensional subspaces of $V$. The corresponding Grassmann graph will be denoted by $\Gamma_{k}(V)$. We…

A Johnson type graph $J_{\pm}(n,k,t)$ is a graph whose vertex set consists of vectors from $\{-1,0,1\}^n$ of the length $\sqrt{k}$ and edges connect vertices with scalar product $t$. The paper determines the order of growth of the chromatic…