Algebraic graph theory, combinatorial matrix analysis and their applications to quantum information and computation. More specific areas include continuous-time quantum walks, graph spectra, graphs with well-structured eigenbases, inverse eigenvalue problems for graphs, and matrix decomposition problems.

• Continuous quantum walks

A continuous quantum walk on a graph models 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 (usually subatomic particles) and their interactions in a network. Perfect state transfer refers to the accurate transmission of quantum states, while fractional revival represents quantum entanglement amongst some qubits in the network. These two phenomenon allow quantum algorithms to outperform classical ones. Using methods from combinatorics and linear algebra, I am working to unravel the role of the underlying structure and algebraic properties of graphs in the existence of perfect state transfer and fractional revival in spin networks.

• Graph spectra

The eigenvalues and eigenvectors of the adjacency and Laplacian matrices of a graph reveal many of its structural properties (such as expansion, walk counts, forbidden subgraphs etc) and provide useful bounds for numerous graph parameters (such as the independence number, chromatic number etc). Presently, I am interested on extremal problems on graph eigenvalues, the goal of which is to determine graphs that attain the maximum/minimum value of linear combinations of certain graph eigenvalues. I also work on graphs with well-structured eigenbases, such as Hadamard diagonalizable graphs and their generalizations. I am keen to explore Kemeny's constant, a graph parameter 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) refers to determining 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 (that is, 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 trees with prescribed spectral property and its extensions to general graphs. I am also interested in IEPG problems involving achievable multiplicity partitions, strong properties and rectangular matrices.

• H. Monterde, On proper and balanced fractional revival, submitted.

• S. Kirkland, S. Mohapatra, H. Monterde, and H. Pal, Pair and plus state transfer on bounded graphs, submitted.

• H. Monterde, New results on vertex sedentariness, submitted.

• H. Monterde, H. Pal and S. Kirkland, Laplacian quantum walks on blow-up graphs, accepted pending minor revisions in Linear Algebra and its Applications.

• S. Kirkland and H. Monterde, Quantum walks on join graphs, accepted pending minor revisions in Discrete Mathematics.

• D. McLaren, H. Monterde, and S. Plosker,  Weakly Hadamard diagonalizable graphs and quantum state transfer, to appear in Linear and Multilinear Algebra. [arXiv]

• C. Godsil, S. Kirkland and H. Monterde, Perfect state transfer between real pure states, SIAM Journal on Matrix Analysis and Applications, 46 (3), 2093-2115 (2025). [arXiv]

• S. Kim, H. Monterde, B. Ahmadi, A. Chan, S. Kirkland, and S. Plosker, A generalization of quantum pair state transfer, Quantum Information Processing 23, 369 (2024). [arXiv]

• B. Bhattacharjya, H. Monterde, and H. Pal, Quantum walks on blow-up graphs, Journal of Physics A: Mathematical and Theoretical, 57 (33), 335303  (2024). [arXiv]

• S. Kirkland, Monterde, H. and S. Plosker, Quantum state transfer between twins in weighted graphs, Journal of Algebraic Combinatorics 58, 623–649 (2023).  [arXiv]

• H. Monterde, Fractional revival between twin vertices, Linear Algebra and its Applications 676, 25-43 (2023).  [arXiv]

• H. Monterde, Sedentariness in quantum walks, Quantum Information Processing 22, 273 (2023). [arXiv]

• H. Monterde, Strong cospectrality and twin vertices in weighted graphs, The Electronic Journal of Linear Algebra 38, 494-518 (2022).  [arXiv]

• H. Monterde, and A. Paras, On the sum of strictly k-zero matrices, Science Diliman 29: 1, 37-52 (2017).

• Quantum pure state transfer, PhD Dissertation, UManitoba, 2025

• Quantum state transfer between twins in graphs, MSc Thesis, UManitoba, 2021

• On the sum of strictly k-zero matrices, MSc Thesis, UP Diliman, 2015

• Quantum walks on graphs, PIMS First-Year Interest Groups

Bahman Ahmadi - Bikash Bhattacharjya - Ada Chan - Chris Godsil - Hiranmoy Pal - Sooyeong Kim - Stephen Kirkland - Darian McLaren - Sarojini Mohapatra - Agnes Paras - Sarah Plosker