Abstract
The purpose of this paper is the study of simple graphs that are generalized Cayley graphs over LA-polygroups (GCLAP - graphs). In this regard, we construct two new extensions for building LA-polygroups. Then, we define Cayley graph over LA-group and GCLAP-graph. Further, we investigate a few properties of them to show that each simple graph of order three, four, and five (except cycle graph of order five which may or may not be a GCLAP-graph) is a GCLAP-graph and then we prove this result.