DOI: https://doi.org/10.11568/kjm.2012.20.4.

### 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.

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ISSN: 1976-8605 (Print), 2288-1433 (Online)

Copyright(c) 2013 By The Kangwon-Kyungki Mathematical Society, Department of Mathematics, Kangwon National University Chuncheon 21341, Korea Fax: +82-33-259-5662 E-mail: kkms@kangwon.ac.kr