Towards a spectral theory of graphs based on the signless Laplacian, II

A spectral graph theory is a theory in which graphs are studied by means of eigenvalues of a matrix M which is in a prescribed way defined for any graph. This theory is called M-theory. We outline a spectral theory of graphs based on the signless Laplacians Q and compare it with other spectral theories, in particular to those based on the adjacency matrix A and the Laplacian L. As demonstrated in the first part, the Q-theory can be constructed in part using various connections to other theories: equivalency with A-theory and L-theory for regular graphs, common features with L-theory for bipartite graphs, general analogies with A-theory and analogies with A-theory via line graphs and subdivision graphs. In this part, we introduce notions of enriched and restricted spectral theories and present results on integral graphs, enumeration of spanning trees, characterizations by eigenvalues, cospectral graphs and graph angles.     

Graph spectra in Computer Science

In this paper, we shall give a survey of applications of the theory of graph spectra to Computer Science. Eigenvalues and eigenvectors of several graph matrices appear in numerous papers on various subjects relevant to information and communication technologies. In particular, we survey applications in modeling and searching Internet, in computer vision, data mining, multiprocessor systems, statistical databases, and in several other areas. Some related new mathematical results are included together with several comments on perspectives for future research. In particular, we claim that balanced subdivisions of cubic graphs are good models for virus resistent computer networks and point out some advantages in using integral graphs as multiprocessor interconnection networks.     

Relations between (κ, τ)-regular sets and star complements

Let G be a finite graph with an eigenvalue µ of multiplicity m. A set X of m vertices in G is called a star set for µ in G if µ is not an eigenvalue of the star complement G\X which is the subgraph of G induced by vertices not in X. A vertex subset of a graph is (κ, τ)-regular if it induces a κ-regular subgraph and every vertex not in the subset has τ neighbors in it. We investigate the graphs having a (κ, τ)-regular set which induces a star complement for some eigenvalue. A survey of known results is provided and new properties for these graphs are deduced. Several particular graphs where these properties stand out are presented as examples.     

A one parameter method for the matrix inverse square root

This paper is motivated by the paper [3], where an iterative method for the computation of a matrix inverse square root was considered. We suggest a generalization of the method in [3]. We give some sufficient conditions for the convergence of this method, and its numerical stabillity property is investigated. Numerical examples showing that sometimes our generalization converges faster than the methods in [3] are presented.     

Viktor Anatol’evich Akulichev (On his 70th birthday)

Acoustical Physics -     

über die Darstellung von kondensierten Azolo-[1,2,4]-triazinen aus heterocyclischen Diazoverbindungen

Diazotized 3-amino-1,2,4-triazole reacted with 1,3-cyclohexanedione or its 5,5-dimethyl derivative to give the tricyclic 1,2,4-triazine derivatives1 a and1 b. With 1,3-indanedione the tetracyclic compound8 was obtained. Compound1 a or the diacetyl derivative4 a are transformed in an acid catalyzed reaction into the fully aromatic system6 a with simultaneous rearrangement. Chemical reductions of systems1, 10 and12 have been investigated and the compounds9, 11 and13 were isolated and identified. Diazotized 2-aminothiazole readily formed the coupling products with reactive methylene compounds, but only14 could be prepared pure by cyclodehydration.     

Analysis of relative isotopologue abundances for quantitative profiling of complex protein mixtures labelled with the acrylamide/D3-acrylamide alkylation tag system

The new method of analysis of relative isotopologue abundances (ARIA) applied here is based on the evaluation of total isotope patterns of tryptic protein fragments measured by matrix-assisted laser desorption/ionisation time-of-flight mass spectrometry (MALDI-TOFMS) to calculate the mixing ratios of composites consisting of stable isotope labelled and isotopically natural (unlabelled) proteins, as described in an accompanying paper in this issue. Recently, Sechi (Rapid Commun. Mass Spectrom. 2002; 16: 1416-1424) and Gehanne et al. (Rapid Commun. Mass Spectrom. 2002; 16: 1692-1698) introduced the use of differential quantitative mass analysis by MALDI-TOFMS using mixtures of standard proteins alkylated prior to two-dimensional polyacrylamide gel electrophoresis (2D-PAGE) with either acrylamide (AA) or deuterium-labelled [2,3,3'-D(3)]-acrylamide (D3AA). In the present study we validate the AA/D3AA system, firstly by measuring the yield of proteins alkylated with AA, and secondly by using differential radioactive labels ((125)I and (131)I) to quantitatively establish that non-comigration in 2D-PAGE is negligible. ARIA is then applied to quantitatively estimate the relative proportions of peptides labelled with AA or D3AA in the validated system, using typical silver-stained 2D-PAGE protein spots from 2D gels loaded with 150 microg of total liver protein. The precision and limitations of ARIA quantification of peptides differentially alkylated with isotopomeric reagents are discussed.     

Electrochemical Oxidation of Donepezil and Its Voltammetric Determination at Gold Electrode

The oxidative ability of donepezil, a frequently prescribed drug for the treatment of Alzheimer’s disease is reported for the first time at a gold electrode. It was oxidized by cyclic voltammetry and determined by square wave voltammetry in phosphate buffer electrolyte. Electrochemical degradation of donepezil was carried out by long term potential cycling. The identification and characterization of the major product, isolated by preparative high performance liquid chromatography, was performed by high resolution mass spectrometry and 1D and 2D nuclear magnetic resonance spectroscopy. Donepezil hydroxy derivative was identified as the major electrochemical oxidation product from donepezil.