“Beyond the distance matrix”: A new framework to study and extend spectral properties of the distance matrix of a graph

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 three-year plan focusing on developing new algebraic tools to further the spectral understanding of the distance matrix. This proposal introduces a novel framework (the socalled q-distance 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 q-distance spectrum in order to understand which graph properties cannot be deduced from the eigenvalues of the q-distance matrix. (ii) Investigate whether simple graph structural properties, such as being bipartite, can be seen from the spectrum of the q-distance matrix. (iii) Study and characterize the q-distance spectrum of an important class of graphs: distance regular graphs with classical parameters.