Strongly Regular Graphs with Maximal Energy

The energy of a graph is the sum of the absolute values of the eigenvalues of its adjacency matrix. Koolen and Moulton have proved that the energy of a graph on n vertices is at most n(1 + √n)/2, and that equality holds if and only if the graph is strongly regular with parameters (n, (n+√n)/2, (n+2√n)/4, (n+2√n)/4). Such graphs are equivalent to a certain type of Hadamard matrices. Here we survey constructions of these Hadamard matrices and the related strongly regular graphs.Graph energy;Strongly regular graph;Hadamard matrix.

Conditions for Singular Incidence Matrices

Suppose one looks for a square integral matrixN, for which NN has a prescribed form.Then the Hasse-Minkowski invariants and the determinant of NN lead to necessary conditions for existence.The Bruck-Ryser-Chowla theorem gives a famous example of such conditions in case N is the incidence matrix of a square block design.This approach fails when N is singular.In this paper it is shown that in some cases conditions can still be obtained if the kernels of N and N are known, or known to be rationally equivalent.This leads for example to non-existence conditions for selfdual generalised polygons, semi-regular square divisible designs and distance-regular graphs.singularities;matrices;graphs

A targeted public: public services in fifteenth-century Ghent and Bruges

Though the phrase 'public services' is a nineteenth-century invention, which was supported by a developed rhetoric of political economy, this article shows that the concept, practice and supply of such services could also be found in the medieval city. It specifically analyses three areas of urban service provision: jurisprudence and legal security, infrastructure and finally health care and poor relief. Although the available sources tend to stress the involvement of municipal authorities in providing public services, it turns out that in fact the furnishing of services was highly multi-layered. In all three areas studied, a wide range of public and private institutions offered services to specific groups within late medieval urban society. In contrast to what the notion of 'public services' lets us presume, however, public services in the medieval city were not available to all inhabitants. Instead, the provision of services was usually quite restrictive, and targeted particular groups in society

More about Divisible Design Graphs

Abstract: Divisible design graphs (DDG for short) have been recently defined by Kharaghani, Meulenberg and the second author as a generalization of (v, k, λ)-graphs. In this paper we give some new constructions of DDGs, most of them using Hadamard matrices and (v, k, λ)-graphs. For three parameter sets we give a nonexistence proof. Furthermore, we find conditions for a DDG to be walk-regular. It follows that most of the known examples are walk-regular, but some are not. In case walk-regularity of a DDG is forced by the parameters, necessary conditions for walk-regularity lead to new nonexistence results for DDGs. We examine all feasible parameter sets for DDGs on at most 27 vertices, establish existence in all but one cases, and decide on existence of a walk-regular DDG in all cases.divisible design graph;divisible design;walk-regular graph;(v;k;λ)-graph;Hadamard matrix

Eigenvalues and Perfect Matchings

AMS classification: 05C50, 05C70, 05E30.graph;perfect matching;Laplacian matrix;eigenvalues.

The Integral Trees with Spectral Radius 3

There are eleven integral trees with largest eigenvalue 3.Integral graphs;graph spectra;trees

5-Chromatic Strongly Regular Graphs

In this paper, we begin the determination of all primitive strongly regular graphs with chromatic number equal to 5.Using eigenvalue techniques, we show that there are at most 43 possible parameter sets for such a graph.For each parameter set, we must decide which strongly regular graphs, if any, possessing the set are 5-chromatic.In this way, we deal completely with 34 of these parameter sets using eigenvalue techniques and computer enumerations.graphs;eigenvalues

Universal Adjacency Matrices with Two Eigenvalues

AMS Mathematics Subject Classification: 05C50.Adjacency matrix;Universal adjacency matrix;Laplacian matrix;signless Laplacian;Graph spectra;Eigenvalues;Strongly regular graphs