Half-transitivity of some metacirculants

✍ Scribed by Mateja Šajna

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 897 KB
• Volume: 185
• Category: Article
• ISSN: 0012-365X
• DOI: 10.1016/s0012-365x(97)00168-4

✦ Synopsis

In 1991, Alspach, Maru §ir, and Nowitz proved that there are infinitely many ½-transitive graphs of degree 4. Their graphs were found among metacirculants M(~; m, n), which have vertex set {vj: iEZ,., jCZ,} and edge set {vjvj+~,i: iCZ,,, jEZ,, rE{-1,1}} with the additional condition that ~ E Z* has order m or 2m. Examining only the cases when both m and n are odd, they showed that the graphs M(~;3,n) are ½-transitive when n>~9 and gave a sufficient condition for M(~;m,n) to be ½-transitive when m is composite and n is prime. In this paper, we give a simple generalization of this condition. We also show that the graphs M(~;2,n) are arc-transitive. Then we examine the graphs M(~;4,n). We prove that they are arc-transitive when the order of c~ is 4 with c~ 2 _= -1 (modn) and ½-transitive when either the order of ~ is 8 or the order of ~ is 4 with 0~ 2 ~ --1 (modn) and n is not a multiple of 4. (~