We study in this paper the maximal version of the coarse BaumConnes assembly map for families of expanding graphs arising from residually finite groups. Unlike for the usual Roe algebra, we show that this assembly map is closely related to the (maximal) Baum-Connes assembly map for the group and is an isomorphism for a class of expanders. We also introduce a quantitative Baum-Connes assembly ma...

In a graph or network G = (V;E), a set S V is k-dependent if no node in S has more than k neighbors in S. We show that for each k 0 there is a self-stabilizing algorithm that identi es a maximal k-dependent set, and stabilizes in O(m + n) time. An interesting by-product of our paper is the new result that in any graph there exists a set that is both maximal k-dependent and minimal (k+1)-dominat...

A permutation σ ∈ Sn is a k-separator if all of its patterns of length k are distinct. Let F (k) denote the maximal length of a k-separator. Hegarty (2013) showed that k + ⌊√ 2k − 1 ⌋ − 1 6 F (k) 6 k + ⌊√ 2k − 3 ⌋ , and conjectured that F (k) = k + ⌊√ 2k − 1 ⌋ − 1. This paper will strengthen the upper bound to prove the conjecture for all sufficiently large k (in particular, for all k > 320801).

Let D = (V,E) be a minimally k-edge-connected simple directed graph. We prove that there is a function f(k) such that |V | ≥ f(k) implies |E| ≤ 2k(|V | − k). We also determine the extremal graphs whose size attains this upper bound.

A graph G = (V, E) is called a split graph if there exists a partition V = I ∪K such that the subgraphs G[I ] and G[K] of G induced by I and K are empty and complete graphs, respectively. In 1980, Burkard and Hammer gave a necessary condition for a split graph G with |I | < |K| to be hamiltonian. We will call a split graph G with |I | < |K| satisfying this condition a Burkard-Hammer graph. Furt...

TO THE EDITOR: Since the active reuptake of fiber K lost during action potentials in contracting muscles is limited by the maximal transport capacity of fiber Na -K pumps, substantial effort has been made to determine how this capacity is regulated during exercise. The methods employed includes ouabain-based determination of muscle Na -K pump content, 3-Omethylfluorescein phosphate-based determ...

Let G denote a connected reductive group over a nonarchimedean local field F . Let K denote a special maximal parahoric subgroup of G(F ). We establish a Satake isomorphism for the Hecke algebra HK of K-bi-invariant compactly supported functions on G(F ). The key ingredient is a Cartan decomposition describing the double coset space K\G(F )/K. We also describe how our results relate to the trea...

TO THE EDITOR: Since the active reuptake of fiber K lost during action potentials in contracting muscles is limited by the maximal transport capacity of fiber Na -K pumps, substantial effort has been made to determine how this capacity is regulated during exercise. The methods employed includes ouabain-based determination of muscle Na -K pump content, 3-Omethylfluorescein phosphate-based determ...

A graph is unfrozen with respect to k independent set if it has an independent set of size k after the addition of any edge. The problem of recognizing such graphs is known to be NP-complete. A graph is maximal if the addition of one edge means it is no longer unfrozen. We designate the problem of recognizing maximal unfrozen graphs as MAX(U(k-SET)) and show that this problem is CO-NP-complete....