arXiv subjects

S. Morteza Mirafzal

Publications and source records attributed to S. Morteza Mirafzal.

22 records · Page 2Linked to original sources

arXiv2017

A note on the automorphism groups of Johnson graphs

The Johnson graph $J(n, i)$ is defined as the graph whose vertex set is the set of all $i$-element subsets of $\{1, . . ., n \}$, and two vertices are adjacent whenever the cardinality of their intersection is equal to $i$-1. In Ramras and Donovan [SIAM J. Discrete Math, 25(1): 267-270, 2011], it is proved that if $ n \neq 2i$, then the automorphism group of $J(n, i)$ is isomorphic with the group $Sym(n)$ and it is conjectured that if $n = 2i$, then the automorphism group of $J(n, i)$ is isomorphic with the group $ Sym(n) \times \mathbb{Z}_2$. In this paper, we will find these results by different methods. We will prove the conjecture in the affirmative.

math.CO↗

arXiv2017

More odd graph theory from another point of view

The Kneser graph $K(n, k)$ has as vertices all $k$-element subsets of $[n]=\{1,2,...,n \}$ and an edge between any two vertices that are disjoint. If $n=2k+1$, then $K(n, k)$ is called an odd graph. Let $ n >4$ and $1< k < \frac{n}{2} $. In the present paper, we show that if the Kneser graph $K(n,k)$ is of even order where $n$ is an odd integer or both of the integers $n,k $ are even, then $K(n,k)$ is a vertex-transitive non Cayley graph. Although, these are special cases of Godsil [8], unlike his proof that uses some very deep group-theoretical facts, ours uses no heavy group-theoretic facts. We obtain our results by using some rather elementary facts of number theory and group theory. We show that "almost all" odd graphs are of even order, and consequently are vertex-transitive non Cayley graphs. Finally, we show that if $ k > 4 $ is an even integer such that $k$ is not of the form $ k = 2^t $ for some $ t> 2 $, then the line graph of the odd graph $O_{k+1}$ is a vertex-transitive non Cayley graph.

math.GR↗

arXiv2011

Some other algebraic properties of folded hypercubes

We construct explicity the automorphism group of the folded hypercube $FQ_n$ of dimension $n>3$, as a semidirect product of $N$ by $M$, where $N$ is isomorphic to the Abelian group $Z_2^n$, and $M$ is isomorphic to $Sym(n+1)$, the symmetric group of degree $n+1$, then we will show that the folded hypercube $FQ_n$ is a symmetric graph.

math.GR↗

arXiv2011

On the Automorphism Groups of Regular Hyper-Stars and Folded Hyper-Stars

The hyper-star graph $HS(n,k)$ is defined as follows : its vertex-set is the set of $ {0,1} $-sequences of length $n$ with weight $k$, where the weight of a sequence $v$ is the number of $1^,s$ in $v$, and two vertices are adjacent if and only if one can be obtained from the other by exchanging the first symbol with a different symbol (1 with 0, or 0 with 1) in another position. In this paper, we will find the automorphism groups of regular hyper-star and folded hyper-star graphs. Then, we will show that, only the graphs HS(4,2) and FHS(4,2) are Cayley graphs.

math.GR↗