Main eigenvalues and (κ, τ)-regular sets

A (κ, τ)-regular set is a subset of the vertices of a graph G, inducing a κ-regular subgraph such that every vertex not in the subset has τ neighbors in it. A main eigenvalue of the adjacency matrix A of a graph G has an eigenvector not orthogonal to the all-one vector j. For graphs with a (κ, τ)-regular set a necessary and sufficient condition for an eigenvalue be non-main is deduced and the main eigenvalues are characterized. These results are applied to the construction of infinite families of bidegreed graphs with two main eigenvalues and the same spectral radius (index) and some relations with strongly regular graphs are obtained. Finally, the determination of (κ, τ)-regular sets is analyzed. © 2009 Elsevier Inc. All rights reserved.

Systematics and evolution of coastal and deepwater Hydrozoa from the NE Atlantic

The study of the Portuguese Hydrozoa fauna has been abandoned for more than half a century, except for the Azores archipelago. One of the main aims of this Ph.D. project was to contribute new hydrozoan records leading to a more accurate perception of the actual hydrozoan diversity found in Portuguese waters, including the archipelagos of Azores and Madeira, and neighbouring geographical areas, for habitats ranging from the deep sea to the intertidal. Shallow water hydroids from several Portuguese marine regions (including the Gorringe Bank) were sampled by scuba-diving. Deep-water hydroids, from the Azores, Madeira, Gulf of Cadiz and Alboran Sea, were collected by researchers of different institutions during several oceanographic campaigns. Occasional hydroid sampling by scuba-diving was performed in the UK, Malta and Spain. Over 300 hydroid species were identified and about 600 sequences of the hydrozoan ‘DNA barcode’ 16S mRNA were generated. The families Sertulariidae, Plumulariidae, Lafoeidae, Hebellidae, Aglaopheniidae, Campanulinidae, Halopterididae, Kirchenpaueriidae, Haleciidae and Eudendriidae, were studied in greater detail. About 350 16S sequences were generated for these taxa, allowing phylogenetic, phylogeographic and evolutionary inferences, and also more accurate taxonomic identifications. Phylogenetic analyses integrated molecular and morphological characters. Subsequent results revealed: particularly high levels of cryptic biodiversity, polyphyly in many taxonomic groups, pairs of species that were synonymous, the identity of several varieties as valid species, and highlighted phylogeographic associations of hydroids in deep and shallow-water areas of the NE Atlantic and W Mediterranean. It was proved that many (but not all) marine hydroid species with supposedly widespread vertical and/or horizontal geographical distributions, correspond in fact to complexes of cryptic taxa. This study further revealed that, in the NE Atlantic, shallow environments sustain higher hydrozoan diversity and abundance, but the importance of bathyal habitats as a source of phylogenetic diversity was also revealed. The Azorean seamounts were shown to be particularly important in the segregation of populations of hydroids with reduced dispersive potential. The bathyal habitats of the Gulf of Cadiz proved to harbour a considerably high number of cryptic species, which may mainly be a consequence of habitat heterogeneity and convergence of various water masses in the Gulf. The main causes proposed for speciation and population divergence of hydroids were: species population size, dispersal mechanisms and plasticity to inhabit different environmental conditions, but also the influence of oceanic currents (and its properties), habitat heterogeneity, climate change and continental drift. Higher phylogenetic resolution obtained for the family Plumulariidae revealed particularly that glacial cycles likely facilitated population divergence, ultimately speciation, and also faunal evolutionary transitions from deep to shallow waters.

Necessary and sufficient conditions for a Hamiltonian graph

A graph is singular if the zero eigenvalue is in the spectrum of its 0-1 adjacency matrix A. If an eigenvector belonging to the zero eigenspace of A has no zero entries, then the singular graph is said to be a core graph. A ( k,t)-regular set is a subset of the vertices inducing a k -regular subgraph such that every vertex not in the subset has t neighbours in it. We consider the case when k=t which relates to the eigenvalue zero under certain conditions. We show that if a regular graph has a ( k,k )-regular set, then it is a core graph. By considering the walk matrix we develop an algorithm to extract ( k,k )-regular sets and formulate a necessary and sufficient condition for a graph to be Hamiltonian.