DIAMETER OF THE DIRECT PRODUCT OF WIELANDT GRAPH

Abstract

A digraph $D$ is primitive if there is a positive integer $k$ such that there is a walk of length $k$ between arbitrary two vertices of $D$. The exponent of a primitive digraph is the least such $k$. Wielandt graph $W_n$ of order $n$ is known as the digraph whose exponent is $n^2 − 2n + 2$, which is the maximum of all the exponents of the primitive digraphs of order $n$. It is known that the diameter of the multiple direct product of a digraph $W_n$ strictly increases according to the multiplicity of the product. And it stops when it attains to the exponent of $W_n$. In this paper, we find the diameter of the direct product of Wielandt graphs.