Motivated by the Watts-Strogatz model for a complex network, we introduce a generalization of the Erdős-Rényi random graph. We derive a combinatorial formula for the moment sequence of its spectral distribution in the sparse limit.
Two new examples are given for illustrating the method of quantum decomposition in the asymptotic spectral analysis for a growing family of graphs. The odd graphs form a growing family of distance-regular graphs and the two-sided Rayleigh distribution appears in the limit of vacuum spectral distribution of the adjacency matrix. For a spidernet as well as for a growing family of spidernets the vacuum distribution of the adjacency matrix is the free Meixner law. These distributions are calculated...
We derive the asymptotic spectral distribution of the distance k-graph of N-dimensional hypercube as N → ∞.
We prove that every biorthogonality preserving linear surjection from a weakly compact JB*-triple containing no infinite-dimensional rank-one summands onto another JB*-triple is automatically continuous. We also show that every biorthogonality preserving linear surjection between atomic JBW*-triples containing no infinite-dimensional rank-one summands is automatically continuous. Consequently, two atomic JBW*-triples containing no rank-one summands are isomorphic if and only if there exists a (not...
Given a von Neumann algebra M we consider its central extension E(M). For type I von Neumann algebras, E(M) coincides with the algebra LS(M) of all locally measurable operators affiliated with M. In this case we show that an arbitrary automorphism T of E(M) can be decomposed as , where is an inner automorphism implemented by an element a ∈ E(M), and is a special automorphism generated by an automorphism ϕ of the center of E(M). In particular if M is of type then every band preserving automorphism...