Spectral conditions for the reconstructibility of a graph

Various conditions on the eigenvalues and eigenvectors of a graph are found to be sufficient for its reconstructibility. In particular, a graph is reconstructible if all but at most one of its eigenvalues are simple and have eigenvectors not orthogonal to c, where c is the vector with each entry equal to one.     

Polynomials with PSL(2, 7) as Galois group

We describe a technique for determining the set-transitivity of the Galois group of a polynomial over the rationals. As an application we give a short proof that the polynomial P₇(x) = x⁷ ? 154x + 99 has the simple group PSL(2, 7) of order 168 as its Galois group over the rationals. A similar method is used to prove that the associated splitting field is not that of the polynomial x⁷ ? 7x + 3 given by Trinks [9].     

Computational insight into the reductive oligomerisation of CO at uranium(III) mixed-sandwich complexes

Calculations reveal a multistep pathway towards formation of linear [U](2)-(μ-η(1):η(1)-C(2)O(2)); [U] = U(η-C(8)H(6){SiH(3)-1,4}(2))(η-Cp). However formation of deltate-bridged [U](2)-(μ-η(1):η(2)-C(3)O(3)) requires an alternative mechanism, involving a side-on [U](2)-(μ-η(2):η(2)-CO) complex and whereby the bridging units of [U](2)-(μ-η(2):η(2)-C(n)O(n)) intermediates (n = 1, 2) react directly with free CO.     

Switching Reconstruction of Digraphs

Switching about a vertex in a digraph means to reverse the direction of every edge incident with that vertex. Bondy and Mercier introduced the problem of whether a digraph can be reconstructed up to isomorphism from the multiset of isomorphism types of digraphs obtained by switching about each vertex. Since the largest known nonreconstructible oriented graphs have eight vertices, it is natural to ask whether there are any larger nonreconstructible graphs. In this article, we continue the investigation of this question. We find that there are exactly 44 nonreconstructible oriented graphs whose underlying undirected graphs have maximum degree at most 2. We also determine the full set of switching‐stable oriented graphs, which are those graphs for which all switchings return a digraph isomorphic to the original.     

Cellular consequences of copper complexes used to catalyze bioorthogonal click reactions

Copper toxicity is a critical issue in the development of copper-based catalysts for copper(I)-catalyzed azide-alkyne cycloaddition (CuAAC) reactions for applications in living systems. The effects and related toxicity of copper on mammalian cells are dependent on the ligand environment. Copper complexes can be highly toxic, can induce changes in cellular metabolism, and can be rapidly taken up by cells, all of which can affect their ability to function as catalysts for CuAAC in living systems. Herein, we have evaluated the effects of a number of copper complexes that are typically used to catalyze CuAAC reactions on four human cell lines by measuring mitochondrial activity based on the metabolism of 3-(4,5-dimethylthiazol-2-yl)-2,5-diphenyltetrazolium bromide (MTT) to study toxicity, inductively coupled plasma mass spectrometry to study cellular uptake, and coherent anti-Stokes Raman scattering (CARS) microscopy to study effects on lipid metabolism. We find that ligand environment around copper influences all three parameters. Interestingly, for the Cu(II)-bis-L-histidine complex (Cu(his)(2)), cellular uptake and metabolic changes are observed with no toxicity after 72 h at micromolar concentrations. Furthermore, we show that under conditions where other copper complexes kill human hepatoma cells, Cu(I)-L-histidine is an effective catalyst for CuAAC labeling of live cells following metabolic incorporation of an alkyne-labeled sugar (Ac(4)ManNAl) into glycosylated proteins expressed on the cell surface. This result suggests that Cu(his)(2) or derivatives thereof have potential for in vivo applications where toxicity as well as catalytic activity are critical factors for successful bioconjugation reactions.     

Automorphisms of random graphs with specified vertices

Conditions are found under which the expected number of automorphisms of a large random labelled graph with a given degree sequence is close to 1. These conditions involve the probability that such a graph has a given subgraph. One implication is that the probability that a random unlabelledk-regular simple graph onn vertices has only the trivial group of automorphisms is asymptotic to 1 asn → ∞ with 3≦k=O(n ^1/2−c). In combination with previously known results, this produces an asymptotic formula for the number of unlabelledk-regular simple graphs onn vertices, as well as various asymptotic results on the probable connectivity and girth of such graphs. Corresponding results for graphs with more arbitrary degree sequences are obtained. The main results apply equally well to graphs in which multiple edges and loops are permitted, and also to bicoloured graphs. Research of the second author supported by U. S. National Science Foundation Grant MCS-8101555, and by the Australian Department of Science and Technology under the Queen Elizabeth II Fellowships Scheme. Current address: Mathematics Department, University of Auckland, Auckland, New Zealand.     

A chain rule for the resultant of two polynomials

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