Executive Summary : | The study of algebraic structures, using the properties of graphs, becomes an exciting research topic in the last twenty years. Given a group G, the commuting graph of G is the simple undirected graph such that it's vertex set is the set of all non-central elements of G and two distinct vertices x and y are adjacent if xy = yx. Bertram (1983) used several graph-theoretic results to prove theorems on finite groups. Indeed, various combinatorial parameters of some commuting graphs are used to establish long standing conjectures in the theory of division algebras. A variant of commuting graphs on groups has played an important role in classification of finite simple groups. The commuting graphs associated with groups are also used to establish some NSSD (non-singular with a singular deck) molecular graph. The power graph of a group G is the graph whose vertex set is G and two distinct vertices are adjacent if one is a power of the other. In order to measure how much the power graph is close to the commuting graph of a group G, Aalipour et al. (2017) introduced a new graph called enhanced power graph. The enhanced power graph of a group G is the simple undirected graph whose vertex set is G and two distinct vertices x, y are adjacent if x, y lies in the same cyclic subgroup of G. The study of groups by graphs, viz. power graphs and commuting graphs, associated with them gives rise to many interesting results as well as applications. In fact, the enhanced power graph contains the power graph and is a spanning subgraph of the commuting graph. To the best of our knowledge, the enhanced power graph is not much studied so far. Moreover, after various attempts to study power graphs and commuting graphs, there are many gaps in the investigations of such graphs and their connections with groups. This work is proposed to fill this gap. In this proposal, researchers intend to study more interconnections between algebraic properties of groups and their associated graphs, namely: power graphs, enhanced power graphs and commuting graphs. This study will help us to understand some more useful algebraic properties of groups by using their associated graph theoretic invariants. This study will certainly leads to the understanding of other interesting family of graphs associated to groups (or semigroups). The proposed project would also open the area for further studies. |