Abstract:

The symmetry problem for finite graphs has been extensively investigated over the past

few decades. This article is devoted to giving an introduction in one particular family of symmetric

graphs, namely (locally) $s$-geodesic-transitive graphs. Recently, substantial progress has been made on the study of this family of graphs, many

open problems have been solved, and many new research problems have arisen.

The methods used in this area range

from deep group theory, including the finite simple group classification, through to combinatorial

techniques.

Key words: Cayley graph; ,  $s$-geodesic-transitive graph; ,  , Permutation group