Connectivity of Kronecker products by K2

Let $\kappa(G)$ be the connectivity of $G$. The Kronecker product $G_1\times G_2$ of graphs $G_1$ and $G_2$ has vertex set $V(G_1\times G_2)=V(G_1)\times V(G_2)$ and edge set $E(G_1\times G_2)=\{(u_1,v_1)(u_2,v_2):u_1u_2\in E(G_1),v_1v_2\in E(G_2)\}$. In this paper, we prove that $\kappa(G\times K_2)=\textup{min}\{2\kappa(G), \textup{min}\{|X|+2|Y|\}\}$, where the second minimum is taken over all disjoint sets $X,Y\subseteq V(G)$ satisfying (1)$G-(X\cup Y)$ has a bipartite component $C$, and (2) $G[V(C)\cup \{x\}]$ is also bipartite for each $x\in X$.Comment: 6 page

Zebrafish in the sea of mineral (iron, zinc, and copper) metabolism

Iron, copper, zinc, and eight other minerals are classified as essential trace elements because they present in minute in vivo quantities and are essential for life. Because either excess or insufficient levels of trace elements can be detrimental to life (causing human diseases such as iron-deficiency anemia, hemochromatosis, Menkes syndrome and Wilson's disease), the endogenous levels of trace minerals must be tightly regulated. Many studies have demonstrated the existence of systems that maintain trace element homeostasis, and these systems are highly conserved in multiple species ranging from yeast to mice. As a model for studying trace mineral metabolism, the zebrafish is indispensable to researchers. Several large-scale mutagenesis screens have been performed in zebrafish, and these screens led to the identification of a series of metal transporters and the generation of several mutagenesis lines, providing an in-depth functional analysis at the system level. Moreover, because of their developmental advantages, zebrafish have also been used in mineral metabolism-related chemical screens and toxicology studies. Here, we systematically review the major findings of trace element homeostasis studies using the zebrafish model, with a focus on iron, zinc, copper, selenium, manganese, and iodine. We also provide a homology analysis of trace mineral transporters in fish, mice and humans. Finally, we discuss the evidence that zebrafish is an ideal experimental tool for uncovering novel mechanisms of trace mineral metabolism and for improving approaches to treat mineral imbalance-related diseases

On the determinant of the QQ-walk matrix of rooted product with a path

Let $G$ be an $n$-vertex graph and $Q(G)$ be its signless Laplacian matrix. The $Q$-walk matrix of $G$, denoted by $W_Q(G)$, is $[e,Q(G)e,\ldots,Q^{n-1}(G)e]$, where $e$ is the all-one vector. Let $G\circ P_m$ be the graph obtained from $G$ and $n$ copies of the path $P_m$ by identifying the $i$-th vertex of $G$ with an endvertex of the $i$-th copy of $P_m$ for each $i$. We prove that, $$\det W_Q(G\circ P_m)=\pm (\det Q(G))^{m-1}(\det W_Q(G))^m$$ holds for any $m\ge 2$. This gives a signless Laplacian counterpart of the following recently established identity [17]: $$\det W_A(G\circ P_m)=\pm (\det A(G))^{\lfloor\frac{m}{2}\rfloor}(\det W_A(G))^m,$$ where $A(G)$ is the adjacency matrix of $G$ and $W_A(G)=[e,A(G)e,\ldots,A^{n-1}(G)e]$. We also propose a conjecture to unify the above two equalities.Comment: 16 pages, 1 figur

Shoshonitic enclaves in the high Sr/Y Nyemo pluton, southern Tibet: Implications for Oligocene magma mixing and the onset of extension of the southern Lhasa terrane

Post-collisional potassic and high Sr/Y magmatism in the Lhasa terrane provides critical constraints on the timing and mechanism of subduction of Indian lithosphere and its role in the uplift of the Tibetan Plateau. Here, we report whole-rock geochemistry, mineral geochemistry, zircon U Pb ages, and in situ zircon Hf isotope ratios for the Nyemo pluton, a representative example of such magmatism. The Nyemo pluton is composed of high Sr/Y host rocks and coeval shoshonitic mafic microgranular enclaves (MMEs). Whole-rock compositions of the host rocks and MMEs form linear trends in Harker diagrams, consistent with modification of both end-members by magma mixing. Although the main high Sr/Y phase of the pluton formed by partial melting of the lower crust of the thickened Lhasa terrane, the MMEs display abnormally enriched light rare earth elements, low whole-rock ε_(Nd)(t) and low zircon ε_(Hf)(t) that suggest derivation from low degree melting of hydrous and enriched mantle. Based on the occurrence of shoshonitic magma and high La/Yb and high Sr/Y with adakitic affinity host rocks around 30 Ma, the Nyemo pluton is best explained as a record of onset of extension that resulted from convective removal of the mantle lithosphere beneath Tibet in the Oligocene