A code $C$ is a subset of the vertex set of a graph and $C$ is $s$-neighbour-transitive if its automorphism group ${\rm Aut}(C)$ acts transitively on each of the first $s+1$ parts $C_0,C_1,\ldots,C_s$ of the distance partition $\{C=C_0,C_1,\ldots,C_ρ\}$, where $ρ$ is the covering radius of $C$. While codes have traditionally been studied in the Hamming and Johnson graphs, we consider here codes in the Kneser graphs. Let $Ω$ be the underlying set on which the Kneser graph $K(n,k)$ is defined. Our first main result says that if $C$ is a $2$-neighbour-transitive code in $K(n,k)$ such that $C$ has minimum distance at least $5$, then $n=2k+1$ (i.e., $C$ is a code in an odd graph) and $C$ lies in a particular infinite family or is one particular sporadic example. We then prove several results when $C$ is a neighbour-transitive code in the Kneser graph $K(n,k)$. First, if ${\rm Aut}(C)$ acts intransitively on $Ω$ we characterise $C$ in terms of certain parameters. We then assume that ${\rm Aut}(C)$ acts transitively on $Ω$, first proving that if $C$ has minimum distance at least $3$ then either $K(n,k)$ is an odd graph or ${\rm Aut}(C)$ has a $2$-homogeneous (and hence primitive) action on $Ω$. We then assume that $C$ is a code in an odd graph and ${\rm Aut}(C)$ acts imprimitively on $Ω$ and characterise $C$ in terms of certain parameters. We give examples in each of these cases and pose several open problems.
An orientation of a graph is semi-transitive if it is acyclic, and for any directed path $v_0\rightarrow v_1\rightarrow \cdots\rightarrow v_k$ either there is no edge between $v_0$ and $v_k$, or $v_i\rightarrow v_j$ is an edge for all $0\leq i<j\leq k$. An undirected graph is semi-transitive if it admits a semi-transitive orientation. Semi-transitive graphs include several important classes of graphs such as 3-colorable graphs, comparability graphs, and circle graphs, and they are precisely the class of word-representable graphs studied extensively in the literature. In this paper, we study se…
A code $C$ in the Hamming graph $\varGamma=H(m,q)$ is $2\it{\text{-neighbour-transitive}}$ if ${\rm Aut}(C)$ acts transitively on each of $C=C_0$, $C_1$ and $C_2$, the first three parts of the distance partition of $V\varGamma$ with respect to $C$. Previous classifications of families of $2$-neighbour-transitive codes leave only those with an affine action on the alphabet to be investigated. Here, $2$-neighbour-transitive codes with minimum distance at least $5$ and that contain "small" subcodes as blocks of imprimitivity are classified. When considering codes with minimum distance at least $5…
The main result here is a characterisation of binary $2$-neighbour-transitive codes with minimum distance at least $5$ via their minimal subcodes, which are found to be generated by certain designs. The motivation for studying this class of codes comes primarily from their relationship to the class of completely regular codes. The results contained here yield many more examples of $2$-neighbour-transitive codes than previous classification results of families of $2$-neighbour-transitive codes. In the process, new lower bounds on the minimum distance of particular sub-families are produced. Sev…
A code $C$ in a generalised quadrangle ${\mathcal Q}$ is defined to be a subset of the vertex set of the point-line incidence graph $\varGamma$ of ${\mathcal Q}$. The minimum distance $δ$ of $C$ is the smallest distance between a pair of distinct elements of $C$. The graph metric gives rise to the distance partition $\{C,C_1,\ldots,C_ρ\}$, where $ρ$ is the maximum distance between any vertex of $\varGamma$ and its nearest element of $C$. Since the diameter of $\varGamma$ is $4$, both $ρ$ and $δ$ are at most $4$. If $δ=4$ then $C$ is a partial ovoid or partial spread of ${\mathcal Q}$, and if, …
Seymour's second neighbourhood conjecture asserts that every oriented graph has a vertex whose second out-neighbourhood is at least as large as its out-neighbourhood. In this paper, we prove that the conjecture holds for quasi-transitive oriented graphs, which is a superclass of tournaments and transitive acyclic digraphs. A digraph $D$ is called quasi-transitive is for every pair $xy,yz$ of arcs between distinct vertices $x,y,z$, $xz$ or $zx$ ("or" is inclusive here) is in $D$.
An orientation of a graph is semi-transitive if it is acyclic, and for any directed path $v_0\rightarrow v_1\rightarrow \cdots\rightarrow v_k$ either there is no arc between $v_0$ and $v_k$, or $v_i\rightarrow v_j$ is an arc for all $0\leq i<j\leq k$. An undirected graph is semi-transitive if it admits a semi-transitive orientation. Semi-transitive graphs generalize several important classes of graphs and they are precisely the class of word-representable graphs studied extensively in the literature. Determining if a triangle-free graph is semi-transitive is an NP-hard problem. The existence o…
A code $C$ in the Hamming metric, that is, is a subset of the vertex set $V\varGamma$ of the Hamming graph $\varGamma=H(m,q)$, gives rise to a natural distance partition $\{C,C_1,\ldots,C_ρ\}$, where $ρ$ is the covering radius of $C$. Such a code $C$ is called completely transitive if the automorphism group $\rm{Aut}(C)$ acts transitively on each of the sets $C$, $C_1$, \ldots, $C_ρ$. A code $C$ is called $2$-neighbour-transitive if $ρ\geq 2$ and $\rm{Aut}(C)$ acts transitively on each of $C$, $C_1$ and $C_2$. Let $C$ be a completely transitive code in a binary ($q=2$) Hamming graph having ful…
In this article, we study quantum automorphism groups of distance-transitive graphs. We show that the odd graphs, the Hamming graphs $H(n,3)$, the Johnson graphs $J(n,2)$ and the Kneser graphs $K(n,2)$ do not have quantum symmetry. We also give a table with the quantum automorphism groups of all cubic distance-transitive graphs. Furthermore, with one graph missing, we can now decide whether or not a distance-regular graph of order $\leq 20$ has quantum symmetry. Moreover, we prove that the Hoffman-Singleton graph has no quantum symmetry. On a final note, we present an example of a pair of grap…
This paper begins the classification of all edge-primitive 3-arc-transitive graphs by classifying all such graphs where the automorphism group is an almost simple group with socle an alternating or sporadic group, and all such graphs where the automorphism group is an almost simple classical group with a vertex-stabiliser acting faithfully on the set of neighbours.
A graph ${\mathcal G}$ is locally X if the graphs induced on the neighbours of every vertex of ${\mathcal G}$ are isomorphic to the graph $X$. We prove that the infinite family of incidence graphs of the $r$-rank incidence geometries, $Γ(KG(n,k),r)$, constructed using the Kneser graphs $KG(n,k)$, are locally $X$ with $X$ being the incidence graphs of the rank $r-1$ residues of $Γ(KG(n,k),r)$.
It is known that there are precisely three transitive permutation groups of degree $6$ that admit an invariant partition with three parts of size $2$ such that the kernel of the action on the parts has order $4$; these groups are called $A_4(6)$, $S_4(6d)$ and $S_4(6c)$. For each $L\in \{A_4(6), S_4(6d), S_4(6c)\}$, we construct an infinite family of finite connected $6$-valent graphs $\{Γ_n\}_{n\in \mathbb{N}}$ and arc-transitive groups $G_n \le \rm{Aut}(Γ_n)$ such that the permutation group induced by the action of the vertex-stabiliser $(G_n)_v$ on the neighbourhood of a vertex $v$ is permu…
A code is a subset of the vertex set of a Hamming graph. The set of $s$-neighbours of a code is the set of all vertices at Hamming distance $s$ from their nearest codeword. A code $C$ is $s$-elusive if there exists a distinct code $C'$ that is equivalent to $C$ under the full automorphism group of the Hamming graph such that $C$ and $C'$ have the same set of $s$-neighbours. It is proved here that the minimum distance of an $s$-elusive code is at most $2s+2$, and that an $s$-elusive code with minimum distance at least $2s+1$ gives rise to a $q$-ary $t$-design with certain parameters. This leads…
A graph $Γ$ is $G$-symmetric if $G$ is a group of automorphisms of $Γ$ which is transitive on the set of ordered pairs of adjacent vertices of $Γ$. If $V(Γ)$ admits a nontrivial $G$-invariant partition ${\cal B}$ such that for blocks $B, C \in {\cal B}$ adjacent in the quotient graph $Γ_{\cal B}$ of $Γ$ relative to ${\cal B}$, exactly one vertex of $B$ has no neighbour in $C$, then $Γ$ is called an almost multicover of $Γ_{\cal B}$. In this case an incidence structure with point set ${\cal B}$ arises naturally, and it is a $(G, 2)$-point-transitive and $G$-block-transitive 2-design if in addit…
We start up the study of the stability of general graph pairs. This notion is a generalization of the concept of the stability of graphs. We say that a pair of graphs $(Γ,Σ)$ is stable if $Aut(Γ\timesΣ) \cong Aut(Γ)\times Aut(Σ)$ and unstable otherwise, where $Γ\timesΣ$ is the direct product of $Γ$ and $Σ$. An unstable graph pair $(Γ,Σ)$ is said to be a nontrivially unstable graph pair if $Γ$ and $Σ$ are connected coprime graphs, at least one of them is non-bipartite, and each of them has the property that different vertices have distinct neighbourhoods. We obtain necessary conditions for a pa…
Exact certification of the coordinate fields of the triangle-free Exoo-Ismailescu unit-distance graphs EI17 and EI19 (HoG 51375, 51376): a solvable-non-solvable dichotomy (origami vs. S20) and the Laman-number conjecture
CEFET-RJ（里约热内卢联邦技术教育中心）；Universidade do Estado do Rio de Janeiro (UERJ)（里约热内卢州立大学）
