
Natural sciences
 Linear and multilinear algebra, matrix theory ^{ }
 Applied mathematics in specific fields not elsewhere classified ^{ }
 Other mathematical sciences not elsewhere classified ^{ }
Spectral graph theory studies the relation between structural properties of a graph and the eigenvalues of associated matrices (spectrum). While there is a huge amount of results concerning the spectral properties of the adjacency and the Laplacian matrix, for the distance matrix many problems remain open, and we seem to have a lack of good tools to prove which graph properties follow from the distance spectra. In the research component of this proposal, the PI proposes a threeyear plan focusing on developing new algebraic tools to further the spectral understanding of the distance matrix. This proposal introduces a novel framework (the socalled qdistance matrix, which generalizes the classical distance matrix) to tackle the study of spectral properties of the distance matrix of a graph. Mathematically, the project include the following directions: (i) Development of a new tool for the construction of graphs having the same qdistance spectrum in order to understand which graph properties cannot be deduced from the eigenvalues of the qdistance matrix. (ii) Investigate whether simple graph structural properties, such as being bipartite, can be seen from the spectrum of the qdistance matrix. (iii) Study and characterize the qdistance spectrum of an important class of graphs: distance regular graphs with classical parameters.