We introduce the concept of the asymmetry number for finite digraphs, as a natural generalization of that for undirected graphs by Erdős and Rényi in 1963. We prove an upper bound for the asymmetry number of finite digraphs and give a condition for equality. We show that our bound is asymptotically best for digraphs with sufficiently large order. We also consider the random oriented graph $RO$, and make some remarks on Aut$(RO)$.

We study the strong instability of ground-state standing waves $e^{i\omega t}{\varphi }_{\omega }(x)$ for $N$-dimensional nonlinear Schrödinger equations with focusing double power nonlinearity. One is $L^2$-subcritical, and the other is $L^2$-supercritical. The strong instability of standing waves with positive energy was proven by Ohta and Yamaguchi (2015). In this paper, we improve the previous result, that is, we prove that if ${\partial }_{\lambda }^2{S}_{\omega }({\varphi }_{\omega }^{\lambda }{)|}_{\lambda =1}\le 0$, the standing wave is strongly unstable, where $S_{\omega }$ is the action, and ${\varphi }_{\omega }^{\lambda }(x):={\lambda }^{N/2}{\varphi }_{\omega }(\lambda x)$ is the $L^2$-invariant scaling.

We consider the initial value problem for a two-component system of cubic nonlinear Schrödinger equations in one space dimension. We provide a detailed lower bound estimate for the lifespan of the solution to the system, which can be computed explicitly from the initial data, the masses and the nonlinear term.

An edge cutset $E\subset E(G)$ of a graph $G$ is called a restricted edge cutset if every component of $G-E$ has order at least $2$. We let ${\lambda }^{\prime }(G)$ denote the minimum cardinality of a restricted edge cutset of $G$, and let ${\delta }^{\prime }(G)$ denote the minimum of ${\mathrm{deg}}_G(x)+{\mathrm{deg}}_G(y)-2$ as $x$ and $y$ range over all adjacent vertices of $G$. We let $\lambda (G)$ and $\delta (G)$ denote the edge connectivity and the minimum degree of $G$, respectively. Among other results, we show that if $G_1$ and $G_2$ are graphs such that $\lambda (G_i)=\delta (G_i)\ge 2$ and ${\lambda }^{\prime }(G_i)={\delta }^{\prime }(G_i)\ge 2$ for each $i=1,2$, then ${\lambda }^{\prime }(G_1\otimes G_2)={\delta }^{\prime }(G_1\otimes G_2)=\min \{{\delta }^{\prime }(G_1)+2\delta (G_2),{\delta }^{\prime }(G_2)+2\delta (G_1)\}$, where $G_1\otimes G_2$ denotes the cartesian product of $G_1$ and $G_2$.

We pose a new problem of collinear central configuration in Newtonian $n$-body problem. For a given three-body of collinear central configuration, we ask whether we can add another body in a way such that (a) the total four-body is also in a state of collinear central configuration and (b) the initial three-body keeps its motion without any change during the process. We find four solutions to the above problem having zero mass. We also discuss a similar but ‘positive’ problem by modifying the conditions such that $(\text{a}’)$ the four-body is in a state of collinear central configuration and the initial three bodies keep their position without any change, and $(\text{b}’)$ the masses of the two out of the three of initial bodies are invariant with the rest having a slight change and the mass of the added body is positive.

We also find explicit solutions to the second problem.