In the present paper we extend the study in [3]. Let $\psi(\xi,\eta,\zeta)$ be a harmonic homogeneous polynomial of degree $s=2\sigma\geq 4$ in three variables $\xi,\eta,\zeta$. Then the bi-symmetric tensor $C$ of bi-degree $(s,s)$ satisfying \[ \psi(\langle J_{1}w,v\rangle,\langle J_{2}w,v\rangle,\langle J_{3}w,v\rangle)=C(v,\ldots,v;w,\ldots,w) \] identically belongs to the linear space $W(3,s)$ of isometric minimal immersions of the three-sphere into spheres. The purpose of the present paper is to study such tensors $C$ and to state some related topics.

In this paper, we get the fact that the identity map of $SU(3)/T(k,l)$ which is a 7-dimensional non-symmetric normal homogeneous space is stable.

A pair $(t,s)$ of nonnegative integers is said to be a connectivity pair for distinct vertices $x$ and $y$ of a graph $G$ if it satisfies the following conditions which were introduced by Beineke and Harary: \begin{enumerate} \item[(1)] For any subset $T\subseteq V(G)-\{x,y\}$ and any subset $S\subseteq E(G)$ with $|T|\leqq t$, $|S|\leqq s$ and $|T|+|S|<t+s$, $G-(T\cup S)$ still contains an $x$-$y$ path. \item[(2)] There exist a subset $T'\subseteq V(G)-\{x,y\}$ and a subset $S'\subseteq E(G)$ with $|T'|=t$ and $|S'|=s$ such that $G-(T'\cup S')$ contains no $x$-$y$ path. \end{enumerate} Let $q$, $r$, $s$ and $t$ be integers with $t\geqq 0$ and $s\geqq 1$ such that $t+s=q(t+1)+r$, $1\leqq r\leqq t+1$, and let $x$ and $y$ be distinct vertices of a graph $G$. It is shown that if $q+r>t$ and if $(t,s)$ is a connectivity pair for $x$ and $y$, then $G$ contains $t+s$ edge-disjoint $x$-$y$ paths $t+1$ of which are openly disjoint.

Let $X^3+AX+B$ be an irreducible abelian cubic polynomial in $Z[X]$. We determine explicitly integers $a_1,\ldots,a_t$, $F$ such that, except for finitely many primes $p$, \[ x^3+Ax+B\equiv 0\pmod{p} \text{ has three solutions} \Leftrightarrow p\equiv a_1,\ldots,a_t\pmod{F}. \]

The totally geodesic 2-sphere and the Clifford torus in $S^3$ are the only compact, minimal surfaces in $S^3$ whose quadric representations yield minimally in hyperquadrics.