Parsons graphs of matrices on Lpn

✍ Scribed by Zhaoji Zhang

• Publisher: Elsevier Science
• Year: 1996
• Tongue: English
• Weight: 112 KB
• Volume: 148
• Category: Article
• ISSN: 0012-365X
• DOI: 10.1016/0012-365x(94)00252-e

✦ Synopsis

Let R be a finite commutative ring with q elements, d an even integer, and SLd(R) the special linear group on R of dimension d. For any b in R, let Tb(d,q) denote the following graph:

(1) V = V(Tb(d,q)) = SLd(R), that is the collection of all the d x d matrices A over R for which det(A) = 1.

(2) E = E(Tb(d,q)) is the collection of all the pairs (A,B) of elements of V for which det(A -B) = b.

When R = GF(q) is a finite field with q elements, Zaks [2] called Tb(d, q) a Parsons graph, and proposed the following conjecture:

Conjecture. Every Parsons graph, except for T1(2,2), is connected.

In this paper, we will generalize the concept of a Parsons graph on GF(q) to that on Lrp~ (p a prime), called here the Parsons graph also, and discuss the connectivity of Tb(d, pn). Our main results are the following two theorems.