Related papers: Distance Sequences of Locally Infinite Primitive G…

A local subgraph of a graph is the subgraph induced by the neighborhood of a vertex. Thus a graph of order $n$ has $n$ local subgraphs. A graph $G$ is called locally nonforesty if every local subgraph of $G$ contains a cycle. Clearly, a…

We generalize the notion of quasirandom which concerns a class of equivalent properties that random graphs satisfy. We show that the convergence of a graph sequence under the spectral distance is equivalent to the convergence using the…

A finite graph $\Gamma$ is called $G$-symmetric if $G$ is a group of automorphisms of $\Gamma$ which is transitive on the set of ordered pairs of adjacent vertices of $\Gamma$. We study a family of symmetric graphs, called the unitary…

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 $u_i\rightarrow u_j$…

A graph is $1$-planar if it has a drawing in the plane such that each edge is crossed at most once by another edge. Moreover, if this drawing has the additional property that for each crossing of two edges the end vertices of these edges…

A graph $\Gamma$ is $k$-connected-homogeneous ($k$-CH) if $k$ is a positive integer and any isomorphism between connected induced subgraphs of order at most $k$ extends to an automorphism of $\Gamma$, and connected-homogeneous (CH) if this…

Distance-hereditary graphs are known to be the graphs that are totally decomposable for the split decomposition. We characterise distance-hereditary graphs whose complement is also distance-hereditary by their split decomposition and by…

The classical Hausdorff dimension of finite or countable metric spaces is zero. Recently, we defined a variant, called \emph{finite Hausdorff dimension}, which is not necessarily trivial on finite metric spaces. In this paper we apply this…

We develop a new framework for analysing finite connected, oriented graphs of valency 4, which admit a vertex-transitive and edge-transitive group of automorphisms preserving the edge orientation. We identify a sub-family of "basic" graphs…

Given a graph $G$ then a subgraph $H$ is $isometric$ if, for every pair of vertices $u,v$ of $H$, we have $d_H(u,v) = d_G(u,v)$. We say a graph $G$ is $distance\ preserving\ (dp)$ if it has an isometric subgraph of every possible order up…

Let $G$ be a finite, simple connected graph. The average distance of a vertex $v$ of $G$ is the arithmetic mean of the distances from $v$ to all other vertices of $G$. The remoteness $\rho(G)$ of $G$ is the maximum of the average distances…

Let $\Gamma$ be a locally finite graph, $L$ the normalized Laplacian of $\Gamma$. If $\Gamma$ is uniformy locally finite, i.e. if each vertex has no more than $d$ adjacent vertices, then the matrix of $L$ (with respect to the standard…

We show that a locally finite Borel graph is nonsmooth if and only if it admits marker sequences which are "far" from every point. Our proof uses the Galvin-Prikry theorem and the Glimm-Effros dichotomy.

Let $\Gamma$ be a finite connected $G$-vertex-transitive graph and let $v$ be a vertex of $\Gamma$. If the permutation group induced by the action of the vertex-stabiliser $G_v$ on the neighbourhood $\Gamma(v)$ is permutation isomorphic to…

For a simple graph $G$, the $3$-distance graph, $D_3(G)$, is a graph with the vertex set $V(G)$ and two vertices are adjacent if and only if their distance is $3$ in the graph $G$. For a connected graph $G$, we provide some conditions for…

Let $\Gamma$ be a simple undirected graph on a finite vertex set and let $A$ be its adjacency matrix. Then $\Gamma$ is {\it singular} if $A$ is singular. The problem of characterising singular graphs is easy to state but very difficult to…

The motion of a graph is the minimal degree of its full automorphism group. Babai conjectured that the motion of a primitive distance-regular graph on $n$ vertices of diameter greater than two is at least $n/C$ for some universal constant…

A connected graph is called \emph{geodetic} if there is a unique geodesic between each pair of vertices. In this paper we prove that if a finitely generated group admits a Cayley graph which is geodetic, then the group must be virtually…

A transitive group $G$ of permutations of a set $\Omega$ is primitive if the only $G$-invariant equivalence relations on $\Omega$ are the trivial and universal relations. If $\alpha \in \Omega$, then the orbits of the stabiliser $G_\alpha$…

A finite graph $\G$ is said to be {\em $(G,3)$-$($connected$)$ homogeneous} if every isomorphism between any two isomorphic (connected) subgraphs of order at most $3$ extends to an automorphism $g\in G$ of the graph, where $G$ is a group of…