Research Projects

In this project, we focus on various graphs associated with groups, including the power graph, enhanced power graph, commuting graph, and generating graph. One of the central problems in graph theory is determining the existence of Hamiltonian paths or cycles. A cycle C in a graph is called a Hamiltonian cycle if it passes through every vertex exactly once. A graph is said to be Hamiltonian if it contains such a cycle. The problem of determining whether a Hamiltonian path or cycle exists in a given graph is known to be NP-complete. Notably, graph theory does not offer any simple necessary and sufficient conditions for a graph to be Hamiltonian. The structure of the power graph of a finite abelian group or a nilpotent group is not fully understood. As such, the problem of determining Hamiltonicity in the power graph of such groups is both significant and challenging. Additionally, problems like the domination number and matching number are of intrinsic interest in graph theory. This project also aims to explore explicit formulas for the matching number and domination number of the commuting graph and the generating graph in terms of group-theoretic parameters. Moreover, we aim to investigate whether certain classes of groups (such as nilpotent or solvable groups) can be characterized uniquely using these graph-theoretic invariants. Let Γ = (V,E) be a graph. A graph automorphism is a bijection on V (Γ) that preserves adjacency. The set of all such automorphisms forms a group under composition, called the automorphism group of Γ, and is denoted by Aut(Γ). This group is a permutation group on V (Γ) and is sometimes referred to as the full automorphism group of Γ. Several important classes of graphs such as asymmetric graphs, vertex-transitive graphs, edge-transitive graphs, symmetric graphs, and distance-transitive graphs are defined or characterized in terms of their automorphism groups. Accordingly, in this project, we are interested in determining the full automorphism groups of the enhanced power graph, the commuting graph, and the generating graph of a group. Apart from the power graph, enhanced power graph and the commuting graph, graph-theoretic features of the generating graphs of finite groups are one of the primary interest to us. Numerous theorems in graph theory and combinatorial theory, such as the matrix-tree theorem is proved using spectral techniques.