A vertex partition $π= \{V_1, V_2, \ldots, V_k\}$ of $G$ is called a \emph{transitive partition} of size $k$ if $V_i$ dominates $V_j$ for all $1\leq i<j\leq k$. For two disjoint subsets $A$ and $B$ of $V$, we say $A$ \emph{strongly dominates} $B$ if for every vertex $y\in B$, there exists a vertex $x\in A$, such that $xy\in E$ and $deg_G(x)\geq deg_G(y)$. A vertex partition $π= \{V_1, V_2, \ldots, V_k\}$ of $G$ is called a \emph{strong transitive partition} of size $k$ if $V_i$ strongly dominates $V_j$ for all $1\leq i<j\leq k$. The \textsc{Maximum Strong Transitivity Problem} is to find a strong transitive partition of a given graph with the maximum number of parts. In this article, we initiate the study of this variation of transitive partition from algorithmic point of view. We show that the decision version of this problem is NP-complete for chordal graphs. On the positive side, we prove that this problem can be solved in linear time for trees and split graphs.
We study transitivity properties of graphs with more than one end. We completely classify the distance-transitive such graphs and, for all $k \geq 3$, the $k$-CS-transitive such graphs.
We compare three transitivity properties of finite graphs, namely, for a positive integer $s$, $s$-distance transitivity, $s$-geodesic transitivity and $s$-arc transitivity. It is known that if a finite graph is $s$-arc transitive but not $(s+1)$-arc transitive then $s\leq 7$ and $s\neq 6$. We show that there are infinitely many geodesic transitive graphs with this property for each of these values of $s$, and that these graphs can have arbitrarily large diameter if and only if $1\leq s\leq 3$. Moreover, for a prime $p$ we prove that there exists a graph of valency $p$ that is 2-geodesic trans…
The orbit graph of a k-orbit polytope is a graph on k nodes that shows how the flag orbits are related by flag adjacency. Using orbit graphs, we classify k-orbit polytopes and determine when a k-orbit polytope is i-transitive. We then provide an explicit classification of three-orbit polytopes, and we describe a generating set for their automorphism groups.
We give necessary and sufficient conditions for lobe-transitivity of locally finite and locally countable graphs whose connectivity equals 1. We show further that, given any biconnected graph $Λ$ and a "code" assigned to each orbit of Aut($Λ$), there exists a unique lobe-transitive graph $Γ$ of connectivity 1 whose lobes are copies of $Λ$ and is consistent with the given code at every vertex of $Γ$. These results lead to necessary and sufficient conditions for a graph of connectivity $1$ to be edge-transitive and to be arc-transitive. Countable graphs of connectivity 1 the action of whose auto…
Let $S$ be a set of transpositions generating the symmetric group $S_n$. The transposition graph of $S$ is defined to be the graph with vertex set $\{1,\ldots,n\}$, and with vertices $i$ and $j$ being adjacent in $T(S)$ whenever $(i,j) \in S$. In the present note, it is proved that two transposition graphs are isomorphic if and only if the corresponding two Cayley graphs are isomorphic. It is also proved that the transposition graph $T(S)$ is edge-transitive if and only if the Cayley graph $Cay(S_n,S)$ is edge-transitive.
We extend the notion of quasi-transitive orientations of graphs to 2-edge-coloured graphs. By relating quasi-transitive $2$-edge-colourings to an equivalence relation on the edge set of a graph, we classify those graphs that admit a quasi-transitive $2$-edge-colouring. As a contrast to Ghouilá-Houri's classification of quasi-transitively orientable graphs as comparability graphs, we find quasi-transitively $2$-edge-colourable graphs do not admit a forbiddden subgraph characterization. Restricting the problem to comparability graphs, we show that the family of uniquely quasi-transitively orient…
A directed graph is semi-transitive if and only if it is acyclic and for any directed path $u_1\rightarrow u_2\rightarrow \cdots \rightarrow u_t$, $t \geq 2$, either there is no edge from $u_1$ to $u_t$ or all edges $u_i\rightarrow u_j$ exist for $1 \leq i < j \leq t$. In this paper, we study semi-transitivity of families of directed split graphs obtained by iterations of morphisms applied to the adjacency matrices and giving in the limit infinite directed split graphs. A split graph is a graph in which the vertices can be partitioned into a clique and an independent set. We fully classify sem…
For a graph $Γ$, a positive integer $s$ and a subgroup $G\leq \Aut(Γ)$, we prove that $G$ is transitive on the set of $s$-arcs of $Γ$ if and only if $Γ$ has girth at least $2(s-1)$ and $G$ is transitive on the set of $(s-1)$-geodesics of its line graph. As applications, we first prove that the only non-complete locally cyclic $2$-geodesic transitive graphs are the complete multipartite graph $K_{3[2]}$ and the icosahedron. Secondly we classify 2-geodesic transitive graphs of valency 4 and girth 3, and determine which of them are geodesic transitive.
A criterion is established for the transitivity of connectedness in a transfinite graph. Its proof is much shorter than a prior argument published previously for that criterion.
Let $G=(V, E)$ be a graph, where $V$ and $E$ are the vertex and edge sets, respectively. For two disjoint subsets $A$ and $B$ of $V$, we say $A$ \textit{dominates} $B$ if every vertex of $B$ is adjacent to at least one vertex of $A$ in $G$. A vertex partition $π= \{V_1, V_2, \ldots, V_k\}$ of $G$ is called a \emph{transitive $k$-partition} if $V_i$ dominates $V_j$ for all $i,j$, where $1\leq i<j\leq k$. The maximum integer $k$ for which the above partition exists is called \emph{transitivity} of $G$ and it is denoted by $Tr(G)$. The \textsc{Maximum Transitivity Problem} is to find a transitive…
Let $G=(V, E)$ be a graph where $V$ and $E$ are the vertex and edge set, respectively. For two disjoint subsets $A$ and $B$, we say $A$ dominates $B$ if every vertex of $B$ is adjacent to at least one vertex of $A$. A vertex partition $π= \{V_1, V_2, \ldots, V_k\}$ of $G$ is called a \emph{transitive $k$-partition} if $V_i$ dominates $V_j$ for all $i,j$ where $1\leq i<j\leq k$. The maximum integer $k$ for which the above partition exists is called \emph{transitivity} of $G$ and it is denoted by $Tr(G)$. The \textsc{Maximum Transitivity Problem} is to find a transitive partition of a given grap…
We consider a \emph{code} to be a subset of the vertex set of a \emph{Hamming graph}. In this setting a \emph{neighbour} of the code is a vertex which differs in exactly one entry from some codeword. This paper examines codes with the property that some group of automorphisms acts transitively on the \emph{set of neighbours} of the code. We call these codes \emph{neighbour transitive}. We obtain sufficient conditions for a neighbour transitive group to fix the code setwise. Moreover, we construct an infinite family of neighbour transitive codes, with \emph{minimum distance} $δ=4$, where this i…
Distance-regular graphs have many beautiful combinatorial properties. Distance-transitive graphs have very strong symmetries, and they are distance-regular, i.e. distance-transitivity implies distance-regularity. In this paper, we give similar results, i.e. for special $s$ and graphs with other restrictions we show that $s$-distance-transitivity implies distance-regularity.
For every simplicial complex K there exists a vertex-transitive simplicial complex homotopy equivalent to a wedge of copies of K with some copies of the circle. It follows that every simplicial complex can occur as a homotopy wedge summand in some vertex-transitive complex. One can even demand that the vertex-transitive complex is the clique complex of a Cayley graph or that it is facet-transitive.
We introduce a new directed graph model for social networks, based on the transitivity of triads. In the Iterated Local Directed Transitivity (ILDT) model, new nodes are born over discrete time-steps, and inherit the link structure of their parent nodes. The ILDT model may be viewed as a directed analogue of the ILT model for undirected graphs introduced in \cite{ilt}. We investigate network science and graph theoretical properties of ILDT digraphs. We prove that the ILDT model exhibits a densification power law, so that the digraphs generated by the models densify over time. The number of dir…
We introduce uniformly vertex-transitive graphs as vertex-transitive graphs satisfying a stronger condition on their automorphism groups, motivated by a problem which arises from a Sinkhorn-type algorithm. We use the derangement graph $D(Γ)$ of a given graph $Γ$ to show that the uniform vertex-transitivity of $Γ$ is equivalent to the existence of cliques of sufficient size in $D(Γ)$. Using this method, we find examples of graphs that are vertex-transitive but not uniformly vertex-transitive, settling a previously open question. Furthermore, we develop sufficient criteria for uniform vertex-tra…
Recent work of Lazarovich provides necessary and sufficient conditions on a graph L for there to exist a unique simply-connected (k,L)-complex. The two conditions are symmetry properties of the graph, namely star-transitivity and st(edge)-transitivity. In this paper we investigate star-transitive and st(edge)-transitive graphs by studying the structure of the vertex and edge stabilisers of such graphs. We also provide new examples of graphs that are both star-transitive and st(edge)-transitive.
A graph is said to be edge-transitive if its automorphism group acts transitively on its edges. It is known that edge-transitive graphs are either vertex-transitive or bipartite. In this paper we present a complete classification of all connected edge-transitive graphs on less than or equal to $20$ vertices. We then present a construction for an infinite family of edge-transitive bipartite graphs, and use this construction to show that there exists a non-trivial bipartite subgraph of $K_{m,n}$ that is connected and edge-transitive whenever $gcd(m,n)>2$. Additionally, we investigate necessary a…
A directed graph $G=(V,E)$ is {\it strongly pseudo transitive} if there is a partition $\{A,E-A\}$ of $E$ so that graphs $G_1=(V,A)$ and $G_2=(V,E-A)$ are transitive, and additionally, if $ab\in A$ and $bc\in E $ implies that $ac\in E$. A strongly pseudo transitive graph $G=(V,E)$ is strongly pseudo transitive of the first type, if $ab\in A$ and $bc\in E$ implies $ac\in A$. An undirected graph is co-strongly pseudo transitive (co-strongly pseudo transitive of the first type) if its complement has an orientation which is strongly pseudo transitive (co-strongly pseudo transitive of the first typ…
One must add arrows which are forced by transitivity to form the transitive closure of a directed graph. We introduce a construction of a transitive directed graph which is formed by adding vertices instead of arrows and which preserves the transitive relationships formed by distinct vertices in the original directed graph. Our construction does not apply to all directed graphs.
The $k$-dimensional Weisfeiler-Leman algorithm ($k$-WL) is a very useful combinatorial tool in graph isomorphism testing. We address the applicability of $k$-WL to recognition of graph properties. Let $G$ be an input graph with $n$ vertices. We show that, if $n$ is prime, then vertex-transitivity of $G$ can be seen in a straightforward way from the output of 2-WL on $G$ and on the vertex-individualized copies of $G$. However, if $n$ is divisible by 16, then $k$-WL is unable to distinguish between vertex-transitive and non-vertex-transitive graphs with $n$ vertices as long as $k=o(\sqrt n)$. Si…
We introduce a notion of a girth-regular graph as a $k$-regular graph for which there exists a non-descending sequence $(a_1, a_2, \dots, a_k)$ (called the signature) giving, for every vertex $u$ of the graph, the number of girth cycles the edges with end-vertex $u$ lie on. Girth-regularity generalises two very different aspects of symmetry in graph theory: that of vertex transitivity and that of distance-regularity. For general girth-regular graphs, we give some results on the extremal cases of signatures. We then focus on the cubic case and provide a characterisation of cubic girth-regular g…
On-line social networks, such as in Facebook and Twitter, are often studied from the perspective of friendship ties between agents in the network. Adversarial ties, however, also play an important role in the structure and function of social networks, but are often hidden. Underlying generative mechanisms of social networks are predicted by structural balance theory, which postulates that triads of agents, prefer to be transitive, where friends of friends are more likely friends, or anti-transitive, where adversaries of adversaries become friends. The previously proposed Iterated Local Transit…
Given a connected graph G with p vertices and q edges, the G-graphicahedron is a vertex-transitive simple abstract polytope of rank q whose edge-graph is isomorphic to a Cayley graph of the symmetric group S_p associated with G. The paper explores combinatorial symmetry properties of G-graphicahedra, focussing in particular on transitivity properties of their automorphism groups. We present a detailed analysis of the graphicahedra for the q-star graphs K_{1,q} and the q-cycles C_q. The C_q-graphicahedron is intimately related to the geometry of the infinite Euclidean Coxeter group \tilde{A}_{q…
Given a finite undirected graph $X$, a vertex is $0$-dismantlable if its open neighbourhood is a cone and $X$ is $0$-dismantlable if it is reducible to a single vertex by successive deletions of $0$-dismantlable vertices. By an iterative process, a vertex is $(k+1)$-dismantlable if its open neighbourhood is $k$-dismantlable and a graph is $k$-dismantlable if it is reducible to a single vertex by successive deletions of $k$-dismantlable vertices. We introduce a graph family, the cubion graphs, in order to prove that $k$-dismantlabilities give a strict hierarchy in the class of graphs whose cliq…
The subdivision graph $S(Σ)$ of a graph $Σ$ is obtained from $Σ$ by `adding a vertex' in the middle of every edge of $\Si$. Various symmetry properties of $§(Σ)$ are studied. We prove that, for a connected graph $Σ$, $S(Σ)$ is locally $s$-arc transitive if and only if $Σ$ is $\lceil\frac{s+1}{2}\rceil$-arc transitive. The diameter of $S(Σ)$ is $2d+δ$, where $Σ$ has diameter $d$ and $0\leqslant δ\leqslant 2$, and local $s$-distance transitivity of $§(Σ)$ is defined for $1\leqslant s\leqslant 2d+δ$. In the general case where $s\leqslant 2d-1$ we prove that $S(Σ)$ is locally $s$-distance transiti…
