Let G be a graph and KG be the set of all cliques of G, then the clique graph of G denoted by K(G) is the graph with vertex set KG and two elements Qi, Qj ∈ KG form an edge if and only if Qi ∩ Qj 6= ∅. Iterated clique graphs are defined by K0(G) = G, and K(G) = K(Kn−1(G)) for n > 0. In this paper we determine the number of cliques in K(G) when G = G1+G2, prove a necessary and sufficient conditi...

We study amenability of algebras and modules (based on the notion of almost-invariant finite-dimensional subspace), and apply it to algebras associated with finitely generated groups. We show that a group G is amenable if and only if its group ring KG is amenable for some (and therefore for any) field K. Similarly, a G-set X is amenable if and only if its span KX is amenable as a KG-module for ...

These are notes from introductory survey lectures given at the Institute for Studies in Theoretical Physics and Mathematics (IPM), Teheran, in 2008 and 2010. We present the definition and the fundamental properties of Fomin-Zelevinsky’s cluster algebras. Then, we introduce quiver representations and show how they can be used to construct cluster variables, which are the canonical generator...

Generalising Nachbin's theory of ``topology and order'', in this paper we   continue the study of quantale-enriched categories equipped with a compact   Hausdorff topology. We compare these $V$-categorical compact Hausdorff spaces   with ultrafilter-quantale-enriched categories, and show that the presence of a   compact Hausdorff topology guarantees Cauchy completeness and (suitably   defined) ...