On the girth of the bipartite graph D(k,q)

Abstract

In this paper, we deal with a generalization Γ(Ω, q) of the bipartite graphs D(k, q) proposed by Lazebnik and Ustimenko, where Ω is a set of binary sequences that are adopted to index the entries of the vertices. A few sufficient conditions on Ω for Γ(Ω, q) to admit a variety of automorphisms are proposed. A sufficient condition for Γ(Ω, q) to be edge-transitive is proposed further. A lower bound of the number of the connected components of Γ(Ω, q) is given by showing some invariants for the components. For Γ(Ω, q), paths and cycles which contain vertices of some specified form are investigated in details. Some lower bounds for the girth of Γ(Ω, q) are then shown. In particular, one can give very simple conditions on the index set Ω so as to assure the generalized graphs Γ(Ω, q) to be a family of graphs with large girth.

Vitamin D impacts on the intestinal health, immune status and metabolism in turbot (Scophthalmus maximus L.)

“…The study on European perch (Dicentrarchus labrax L.) also showed that VD 3 increased the phagocytosis of white blood cells in the head kidney, while inhibited the expression of il-1β in the head kidney and intestinal tract (49) . Furthermore, the addition of dietary VD 3 inhibited the upregulation of pro-inflammatory cytokines induced by bacterial infection in yellow catfish (25) and Jian carp (Cyprinus carpio var.jian) (47) . Consistently, our results demonstrated that the gene expression of the pro-inflammatory cytokines, including il-1β, il-8, il-6 and tnf-α, was significantly lower in the liver and gut of the turbots in 400 IU group (Figure 4A-B),…”