Graphs with small diameter determined by their DD-spectra

Let $G$ be a connected graph with vertex set $V(G)=\{v_{1},v_{2},...,v_{n}\}$. The distance matrix $D(G)=(d_{ij})_{n\times n}$ is the matrix indexed by the vertices of $G,$ where $d_{ij}$ denotes the distance between the vertices $v_{i}$ and $v_{j}$. Suppose that $\lambda_{1}(D)\geq\lambda_{2}(D)\geq\cdots\geq\lambda_{n}(D)$ are the distance spectrum of $G$. The graph $G$ is said to be determined by its $D$-spectrum if with respect to the distance matrix $D(G)$, any graph having the same spectrum as $G$ is isomorphic to $G$. In this paper, we give the distance characteristic polynomial of some graphs with small diameter, and also prove that these graphs are determined by their $D$-spectra

Expansion of the Sahara Desert and shrinking of frozen land of the Arctic.

Expansion of the Sahara Desert (SD) and greening of the Arctic tundra-glacier region (ArcTG) have been hot subjects under extensive investigations. However, quantitative and comprehensive assessments of the landform changes in these regions are lacking. Here we use both observations and climate-ecosystem models to quantify/project changes in the extents and boundaries of the SD and ArcTG based on climate and vegetation indices. It is found that, based on observed climate indices, the SD expands 8% and the ArcTG shrinks 16% during 1950-2015, respectively. SD southern boundaries advance 100 km southward, and ArcTG boundaries are displaced about 50 km poleward in 1950-2015. The simulated trends based on climate and vegetation indices show consistent results with some differences probably due to missing anthropogenic forcing and two-way vegetation-climate feedback effect in simulations. The projected climate and vegetation indices show these trends will continue in 2015-2050

B\"acklund-Darboux Transformations and Discretizations of Super KdV Equation

For a generalized super KdV equation, three Darboux transformations and the corresponding B\"acklund transformations are constructed. The compatibility of these Darboux transformations leads to three discrete systems and their Lax representations. The reduction of one of the B\"acklund-Darboux transformations and the corresponding discrete system are considered for Kupershmidt's super KdV equation. When all the odd variables vanish, a nonlinear superposition formula is obtained for Levi's B\"acklund transformation for the KdV equation