Experimental results show that in large complex networks such as internet or biological networks, there is a tendency to connect elements which have a common neighbor. This tendency in theoretical random graph models is depicted by the asymptotically constant clustering coefficient. Moreover complex networks have power law degree distribution and small diameter (small world phenomena), thus the...

In this paper, we continue the investigation of intersections of pairs of finitely generated subgroups Γ1 and Γ2 of a Kleinian group Γ, a question which has been examined by a number of authors. We give a brief survey of this work at the end of the introduction. We consider here intersections of topologically tame subgroups of Γ. We are interested primarily in determining the connection between...

A uniform random intersection graph G(n,m, k) is a random graph constructed as follows. Label each of n nodes by a randomly chosen set of k distinct colours taken from some finite set of possible colours of size m. Nodes are joined by an edge if and only if some colour appears in both their labels. These graphs arise in the study of the security of wireless sensor networks, in particular when m...

In 1966, Gallai asked whether every connected graph has a vertex that is common to all its longest paths. The answer to this question is negative. We prove that the answer is positive for outerplanar graphs. Another related question was raised in 1995 at the British Combinatorial Conference: Do any three longest paths in a connected graph have a vertex in common? We prove that, in a connected g...

J.H. Koolen and J. Park proved a lower bound for the intersection number c2 of a distance-regular graph Γ. Moreover, they showed that a graph Γ, for which equality is attained in this bound, is a Terwilliger graph. We prove that Γ is the icosahedron, the Doro graph or the Conway–Smith graph if equality is attained and c2 > 2.