No abstract

The scheme associated with a graph is an association scheme iff the graph is strongly regular. Consider the problem of extending such an association scheme to a superscheme. The obstacles can be expressed in terms of t-vertex conditions. If a graph does not satisfy the t-vertex condition, a presuperscheme associated with it cannot be erected beyond the (t − 3)rd level. We give an example of an association scheme which is not extendible to a superscheme: it cannot be extended beyond the bottom level of a presuperscheme.

The scheme associated with a graph is an association scheme i the graph is strongly regular. Consider the problem of extending such an association scheme to a superscheme in the case of a colored, directed graph. If a presuperscheme associated with a graph is of height t, then the graph satis®es the t 3-vertex condition. On the other hand, the current paper provides an example of a regular and 3-regular graph, satisfying the 4-vertex condition, whose association scheme cannot be extended to a presuperscheme of height 1.

A surface of revolution is a surface that can be generated by rotating a planar curve (the directrix) around a straight line (the axis) in the same plane. Using the mathematics of quaternions, we provide a parametric equation of a surface of revolution generated by rotating a directrix about an axis by quaternion multiplication of the parametric representations of the directrix curve and the line of axis. Then, we describe an algorithm to determine whether a parametric surface is a surface of revolution, and identify the axis and the directrix. Examples are provided to illustrate our algorithm.

In this paper, we prove that the family of binomials $x_1^{a_1} \cdots x_m^{a_m}-y_1^{b_1}\cdots y_n^{b_n}$ with $\gcd(a_1, \ldots, a_m, b_1, \ldots, b_n)=1$ is irreducible by identifying the connection between the irreducibility of a binomial in ${\mathbb C}[x_1, \ldots, x_m, y_1, \ldots, y_n]$ and ${\mathbb C}(x_2, \ldots, x_m, y_1, \ldots, y_n)[x_1]$. Then we show that the necessary and sufficient conditions for the irreducibility of this family of binomials is equivalent to the existence of a unimodular matrix $U_i$ with integer entries such that $(a_1, \ldots, a_m, b_1, \ldots, b_n)^T=U_i \be_i$ for $i\in \{1, \ldots, m+n\}$, where $\be_i$ is the standard basis vector.