If a cubic surface has a non-trivial automorphism, it specifies the structure of the cubic surface. When the automorphism group is big enough the isomorphic class is uniquely determined. In this paper, we provide equations of specific cubic surfaces and generators of automorphism groups explicitly.

Recently, Chen established a general sharp inequality for warped products in real space forms. As applications, he obtained obstructions to minimal isometric immersions of warped products into real space forms.

In the present paper, we obtain sharp inequalities for warped products isometrically immersed in Sasakian space forms. Some applications are derived.

Frenkel, Khovanov and Kirillov showed that the parabolic Kazhdan-Lusztig basis of Iwahori-Hecke algebra associated to ${\mathfrak{S}}_n$ can be obtained as the canonical basis of a weight subspace of $V^{\otimes n}$, where $V$ is the vector representation of the quantum group $U_v(\mathfrak{s}{\mathfrak{l}}_m)$. In this paper, a similar problem for the case of Ariki-Koike algebra ${\mathscr{H}}_{n,r}$ is discussed. We construct a certain basis of ${\mathscr{H}}_{n,r}$, which is fixed by the involution and is closely related to the canonical basis of $V^{\otimes n}$, by making use of the representation of ${\mathscr{H}}_{n,r}$ on $V^{\otimes n}$. In the case where $r=2$, i.e., in the case of Iwahori-Hecke algebra of type $B_n$, this gives a basis different from the Kazhdan-Lusztig basis of ${\mathscr{H}}_{n,r}$.

A graph $G$ is called edge-magic if it admits a labeling of the vertices and edges by pairwise different integers of $1,2,\mathrm{...},|V(G)|+|E(G)|$ such that the sum of the label of an edge and the labels of its endpoints is constant independent of the choice of edge. A construction of edge-magic labelings of some disconnected graphs is described. Some edge-magic forests are characterized.

We determine the ring homomorphism $H{H}^{*}(\Gamma )\to H^{*}(G,\Gamma )$ explicitly, where $G$ denotes the cyclic group of order $p^v$ and $\Gamma$ denotes the ring of integers of the cyclotomic field $\mathbb{Q}(\zeta )$ for a primitive $p^v$-th root of unity $\zeta$.

Let $S$ denote the graph obtained from $K_4$ by removing two edges which have an endvertex in common. Let $k$ be an integer with $k\ge 2$. Let $G$ be a graph with $|V(G)|\ge 4k$ and ${\sigma }_2(G)\ge |V(G)|/2+2k-1$, and suppose that $G$ contains $k$ vertex-disjoint triangles. In the case where $|V(G)|=4k+2$, suppose further that $G\cancel{\cong }{K}_{4t+3}{{\displaystyle \cup }}^{\text{}}{K}_{4k-4t-1}$ for any $t$ with $0\le t\le k-1$. Under these assumptions, we show that $G$ contains $k$ vertex-disjoint copies of $S$.

Let $k\ge 2$ and $n\ge 1$ be integers, let $G$ be a graph of order $n$ with minimum degree at least $k+1$. Let ${\upsilon }_1,{\upsilon }_2,\dots ,{\upsilon }_k$ be $k$ distinct vertices of $G$, and suppose that there exist $k$ vertex disjoint cycles $C_1,\dots ,C_k$ in $G$ such that ${\upsilon }_i\in V(C_i)$ for each $1\le i\le k$. Suppose further that the minimum value of the sum of the degrees of two nonadjacent distinct vertices is greater than or equal to $n+\frac{k-4}{3}$. Under these assumptions, we show that there is a 2-factor of $G$ with $k$ cycles $D_1,D_2,\dots ,D_k$ such that ${\upsilon }_i\in V(D_i)$ for each $1\le i\le k$.