On neighbourhood line graphs

On Diameter of Line Graphs

The diameter of a connected graph $G$, denoted by $diam(G)$, is the maximum distance between any pair of vertices of $G$. Let $L(G)$ be the line graph of $G$. We establish necessary and sufficient conditions under which for a given integer $k geq 2$, $diam(L(G)) leq k$.

Closed-neighbourhood anti-Sperner graphs

Configurations graphs of neighbourhood geometries

Configurations of type (κ +1)κ give rise to κ–regular simple graphs via configuration graphs. On the other hand, neighbourhood geometries of C4–free κ–regular simple graphs on κ 2 + 1 vertices turn out to be configurations of type (κ + 1)κ. We investigate which configurations of type (κ +1)κ are equal or isomorphic to the neighbourhood geometry of their configuration graph and conversely. We cl...

Studies on Neighbourhood Graphs for Communication in Multi Agent Systems

This paper addresses a special type of graph, the k-neighbourhood graph, for the usage in huge multi agent systems. It can be used to establish slim communication structures in extensive groups of agents as they are present e.g. in swarm applications. We will prove some properties of k-neighbourhood graphs in twoand three-dimensional Euclidean space, i.e. we show that the maximum number of inco...

On s-Hamiltonian Line Graphs

For an integer s ≥ 0, a graph G is s-hamiltonian if for any vertex subset S′ ⊆ V (G) with |S′| ≤ s, G − S′ is hamiltonian. It is well known that if a graph G is s-hamiltonian, then G must be (s + 2)-connected. The converse is not true, as there exist arbitrarily highly connected nonhamiltonian graphs. But for line graphs, we prove that when s ≥ 5, a line graph is s-hamiltonian if and only if it...