21 papers · 1 filter
Threshold Graphs Allow Few Distinct Eigenvalues: A New Approach
Jane Breen, Shaun Fallat, Johnna Parenteau
For any graph , we associate a family of real symmetric matrices, , where for any , the location of the nonzero off-diagonal entries of are governed by the…
Strictly Interlaced Spectral Data for the Weighted Matching Polynomial of a Graph
Shaun Fallat, Johnna Parenteau
Interlacing of the real roots of a weighted matching polynomial for a graph and that of a vertex-deleted subgraph is classical and well-known. In the context of strict interlac…
The Strong Spectral Property and the Jacobian Method for Weighted Laplacian Matrices
Minerva Catral, Shaun Fallat, Himanshu Gupta +1
Strong matrix properties, roughly speaking, refer to generic conditions on a matrix such that its spectral perturbation and pattern perturbation interact nicely to cover a neighbor…
The Weak Version of the Graph Complement Conjecture and Partial Results for the Delta Conjecture
Francesco Barioli, Shaun M. Fallat, Himanshu Gupta +1
Since the transformative workshop by the American Institute of Mathematics on the minimum rank of a graph, two longstanding open problems have captivated the community interested i…
Inverse eigenvalue problem for Laplacian matrices of a graph
Shaun Fallat, Himanshu Gupta, Jephian C. -H. Lin
For a given graph , we aim to determine the possible realizable spectra for a generalized (or sometimes referred to as a weighted) Laplacian matrix associated with . This new…
Minimum number of distinct eigenvalues of distance-regular and signed Johnson graphs
Shaun Fallat, Himanshu Gupta, Allen Herman +1
We study the minimum number of distinct eigenvalues over a collection of matrices associated with a graph. Lower bounds are derived based on the existence or non-existence of certa…