The spectrum for 2-perfect 6-cycle systems

AbstractRecently, the spectrum problem for 2-perfectm-cycle systems has been studied by several authors. In this paper we find the spectrum for 2-perfect 6-cycle systems with two possible exceptions. The connection between these systems and quasigroups satisfying some 2 variable identities is discussed

Switching for Small Strongly Regular Graphs

We provide an abundance of strongly regular graphs (SRGs) for certain parameters $(n, k, \lambda, \mu)$ with $n < 100$. For this we use Godsil-McKay (GM) switching with a partition of type $4,n-4$ and Wang-Qiu-Hu (WQH) switching with a partition of type $3^2,n-6$. In most cases, we start with a highly symmetric graph which belongs to a finite geometry. To our knowledge, most of the obtained graphs are new. For all graphs, we provide statistics about the size of the automorphism group. We also find the recently discovered Kr\v{c}adinac partial geometry, therefore finding a third method of constructing it.Comment: 15 page

2-perfect m-cycle systems

AbstractThe spectrum for 2-perfect m-cycle systems of Kn has been considered by several authors in the case when m⩽7. In this paper we essentially solve the problem for 2-perfect m-cycle systems of Kn in the case where m is prime and 2m+1 is a prime power. In particular we settle the problem for m = 11 and 13 except for two or one possible exceptions respectively. The problem for m = 9 is also considered

On nesting of G-decompositions of λKv where G has four nonisolated vertices or less

AbstractThe complete multigraph λKv is said to have a G-decomposition if it is the union of edge disjoint subgraphs of Kv each of them isomorphic to a fixed graph G. The spectrum problem for G-decompositions of λKv that have a nesting was first considered in the case G=K3 by Colbourn and Colbourn (Ars Combin. 16 (1983) 27–34) and Stinson (Graphs and Combin. 1 (1985) 189–191). For λ=1 and G=Cm (the cycle of length m) this problem was studied in many papers, see Lindner and Rodger (in: J.H. Dinitz, D.R. Stinson (Eds.), Contemporary Design Theory: A Collection of Surveys, Wiley, New York, 1992, p. 325–369), Lindner et al. (Discrete Math. 77 (1989) 191–203), Lindner and Stinson (J. Combin. Math. Combin. Comput. 8 (1990) 147–157) for more details and references. For λ=1 and G=Pk (the path of length k−1) the analogous problem was considered in Milici and Quattrocchi (J. Combin. Math. Combin. Comput. 32 (2000) 115–127). In this paper we solve the spectrum problem of nested G-decompositions of λKv for all the graphs G having four nonisolated vertices or less, leaving eight possible exceptions

Peptidomics of a single identified neuron reveals diversity of multiple neuropeptides with convergent actions on cellular excitability

In contrast to classical transmitters, the detailed structures and cellular and synaptic actions of neuropeptides are less well described. Peptide mass profiling of single identified neurons of the mollusc Lymnaea stagnalis indicated the presence of 17 abundant neuropeptides in the cardiorespiratory neuron, visceral dorsal 1 (VD1), and a subset of 14 peptides in its electrically coupled counterpart, right parietal dorsal 2. Altogether, based on this and previous work, we showed that the high number of peptides arises from the expression and processing of four distinct peptide precursor proteins, including a novel one. Second, we established a variety of posttranslational modifications of the generated peptides, including phosphorylation, disulphide linkage, glycosylation, hydroxylation, N-terminal pyro-glutamylation, and C-terminal amidation. Specific synapses between VD1 and its muscle targets were formed, and their synaptic physiology was investigated. Whole-cell voltage-clamp analysis of dissociated heart muscle cells revealed, as tested for a selection of representative family members and their modifications, that the peptides of VD1 exhibit convergent activation of a high-voltage-activated Ca current. Moreover, the differentially glycosylated and hydroxylated α2 peptides were more potent than the unmodified α2 peptide in enhancing these currents. Together, this study is the first to demonstrate that single neurons exhibit such a complex pattern of peptide gene expression, precursor processing, and differential peptide modifications along with a remarkable degree of convergence of neuromodulatory actions. This study thus underscores the importance of a detailed mass spectrometric analysis of neuronal peptide content and peptide modifications related to neuromodulatory function. Copyright © 2006 Society for Neuroscience

Hypergraph matchings and designs

We survey some aspects of the perfect matching problem in hypergraphs, with particular emphasis on structural characterisation of the existence problem in dense hypergraphs and the existence of designs.Comment: 19 pages, for the 2018 IC