• Continuous quantum walks

A continuous quantum walk on a graph describes the propagation of quantum states in a spin network. Here, a spin network is modelled by an undirected graph whose vertices and edges represent the qubits and their interactions in a network, respectively. Perfect state transfer (PST) refers to the accurate transmission of quantum states, while fractional revival (FR) represents quantum entanglement amongst some qubits in the network. These two phenomenon allow quantum algorithms to outperform classical counterparts. Using methods from combinatorics and linear algebra, I am working to unravel the role of the structural and algebraic properties of graphs in the occurrence of PST, FR and other types of transfer in spin networks (e.g., pretty good state transfer, uniform mixing and vertex sedentariness).

• Graph spectra and eigenvectors

The eigenvalues and eigenvectors of matrices associated with graphs reveal many of its structural properties and provide useful bounds for numerous graph parameters. Presently, I am interested on the problem of determining graphs that attain the maximum or minimum value of linear combinations of certain eigenvalues of the adjacency matrix. My current projects also include investigating the relationship between the nullity of the adjacency matrix of graph and its structure, as well as developing the theory and applications of graphs whose Laplacian matrices have well-structured eigenbases. I am also keen in exploring Kemeny's constant, a graph parameter (expressible in terms of the eigenvalues of the normalized Laplacian matrix) that provides a measure of how fast a random walker moves around in a graph. 

• Inverse eigenvalue problem for graphs

The inverse eigenvalue problem of a graph (IEPG) determines the possible spectra of real symmetric matrices M whose pattern of nonzero off-diagonal entries is described by the edges of a given graph G. In this case, the (u,v) entry of M is nonzero if and only there is an edge between u and v in G.  At present, I am interested in the hollow version of IEPG for graphs with prescribed spectral property. I am also interested in IEPG problems involving achievable multiplicity partitions, minimum number of distinct eigenvalues, strong properties, as well as inverse eigenvector problems.