Research
My research interests include spectral graph theory, operators associated with hypergraphs, and dynamical networks.
A hypergraph is actually a generalization of graphs in which instead of being two-element subsets of the vertex set, an edge (hyperedge) can contain more than two elements.
A hypergraph H with V(H)={1,2,...,11} and E(H)={e,f,g,h}.
Spectral graph theory evolves around spectra of some matrices and operators associated with graphs. Similarly, there are multiple concepts of Matrices and operators associated with hypergraphs. In my recent work (joint work with my supervisor), named 'On Some General Operators of Hypergraphs', we introduce connectivity operators such as diffusion operators, general Laplacian operators, and general adjacency operators associated with hypergraphs. These operators are generalizations of several conventional ideas of seemingly disparate connectivity matrices related to hypergraphs. In fact, we present here a unified framework for simultaneously studying various versions of the connectivity operators associated with hypergraphs. Eigenvalues and eigenspaces of general connectivity operators associated with some types of hypergraphs are computed. From the standpoint of our newly introduced operators, applications such as random walks on hypergraphs, dynamical networks, and disease transmission on hypergraphs are investigated. We also calculate spectral constraints for hypergraphs' weak connectivity number, degree of vertices, maximum cut, bipartition width, and isoperimetric constant.
In another project (joint work with my supervisor) we are trying to incorporate multi-body interaction in dynamical network using hypergraphs. Preprint of this project can be found here.
I am also interested in the interaction of graphs and groups. A commuting graph of a group is a graph with the group as its set of vertices and two elements in the group are adjacent if they commute in the group. In recent work (joint work with Dr Gargi Ghosh), we have provided complete spectra of commuting graphs of a group in terms of the group's property. See a preprint of this work here.
Currently, I am working on some binary and unary operations of hypergraphs with my supervisor and Rajiv Mishra.
My other ongoing project is on hyperflower hypergraphs.
Highlights
On Some General operators of Hypergraphs.
Key highlights:
inner product spaces on the space of all the functions of the vertex set and the same on the hyperedge set.
Diffusion operator associated with hypergraphs.
Spectra of diffusion operator
General Laplacian operator
General Adjacency operator
Complete spectra of the operators associated with hyperflower
Applications
Multi-body Interaction in dynamical network
Key highlights:
Incorporate multi- body interaction in dynamical network by using hypergraphs
Multibody-interaction in uniform groups of agents.
Multibody-interaction in non-uniform groups of agents.
Multi-body interaction in groups with heterogeneous impact.
Synchronization and the spectra of underlying hypergraphs.
Applications
On The Laplacian spectra of Commuting Graphs
Key highlights:
Commuting Graphs
Laplacian spectra of commuting graphs
Necessary conditions for a commuting graph
Isoperimetric number of a commuting graph
Independent number of commutating graphs