Laplacian Distribution and Domination

Let $m_G(I)$ denote the number of Laplacian eigenvalues of a graph $G$ in an interval $I$, and let $\gamma(G)$ denote its domination number. We extend the recent result $m_G[0,1) \leq \gamma(G)$, and show that isolate-free graphs also satisfy $\gamma(G) \leq m_G[2,n]$. In pursuit of better understanding Laplacian eigenvalue distribution, we find applications for these inequalities. We relate these spectral parameters with the approximability of $\gamma(G)$, showing that $\frac{\gamma(G)}{m_G[0,1)} \not\in O(\log n)$. However, $\gamma(G) \leq m_G[2, n] \leq (c + 1) \gamma(G)$ for $c$-cyclic graphs, $c \geq 1$. For trees $T$, $\gamma(T) \leq m_T[2, n] \leq 2 \gamma(G)$

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

Characterization of some Kunitz domain containing genes - possible link to Salivary gland

As a hematophagous parasite, anticoagulants are crucial for L. salmonis. In hematophagous animals specific anticoagulants are produced by salivary gland in order to keep the blood liquid and to allow the parasite to process it properly. Such proteins are unknown in L. salmonis as well as its site of expression. At the same time the function of the salivary gland as production of anticoagulant factors has not been confirmed in L. salmonis. Genes with Kunitz domain are typically proteinase inhibitors and some are involved in anticoagulation. They are present in L. salmonis but with unknown function and site of expression. This studied demonstrated the presence of two salivary gland specific genes belonging to the Kunitz family and other highly expressed in the intestine. The silencing of these genes did not give any distinct phenotypes in adults or larvae stages. The present study could not conclude if the three investigated genes are involved in anticoagulation in the salmon louse. However, the lack of detectable phenotypes in the RNAi experiments indicates that could be other compensating molecules in the lice for the processes that LsKunitz1-3 are involved in.MAMN-MARMAR39

Recent results on graphs with convex quadratic stability number

The main results about graphs with convex quadratic stability number (that is, graphs for which the stability number can be determined by convex quadratic programming) are surveyed including the most recently obtained. Furthermore, a few algorithmic techniques for the recognition of this type of graphs in particular families are presented

Reconhecimento de grafos com número de estabilidade quadrático convexo

São apresentados os principais e mais recentes resultados sobre grafos com número de estabilidade quadrático convexo (que são grafos cujo número de estabilidade pode ser determinado através de técnicas de programação quadrática convexa) e descritas algumas estratégias algorítmicas para o reconhecimento de grafos deste tipo em famílias particulares