2 research outputs found
A note on the automorphism group of the Hamming graph
Let be a -set, where , is an integer. The Hamming graph
, has as its vertex-set, with two vertices are adjacent
if and only if they differ in exactly one coordinate. In this paper, we provide
a proof on the automorphism group of the Hamming graph , by using
elementary facts of group theory and graph theory.Comment: 8 pages, 1 figure
Cayley properties of the line graphs induced by consecutive layers of the hypercube
Let and be integers. In this paper, we
investigate some algebraic properties of the line graph of the graph where is the subgraph of the hypercube
which is induced by the set of vertices of weights and . In the first
step, we determine the automorphism groups of these graphs for all values of
. In the second step, we study Cayley properties of the line graphs of
these graphs. In particular, we show that if and , then
except for the cases and , the line graph of the graph is a vertex-transitive non-Cayley graph. Also, we show that the
line graph of the graph is a Cayley graph if and only if is
a power of a prime . Moreover, we show that for \lq{}almost all\rq{} even
values of , the line graph of the graph is a
vertex-transitive non-Cayley graph