The n-Queens graph and its generalizations - Costa, Inês Filipa Serôdio (2024)
This thesis presents some spectral results related to the n-Queens graph – a graph associated with the n × n chessboard, with n ∈ N – and its generalizations. A lower (and upper) bound is established for the least eigenvalue of a graph, using the concept of edge clique partition. Applying this bound, throughout the thesis, lower bounds are established for the least eigenvalue of the n-Queens graph and of some of its generalizations – for m × n rectangular chessboards ( Q (m, n )) and n × n × n chessboards ( Q (n, n, n )), m, n ∈ N. Other integer eigenvalues for these graphs, as well as eigenvectors associated with them, are studied. Finally, an integral graph – a graph whose spectrum is entirely composed of integer eigenvalues – is introduced which is, like Q (n, n ) and Q (m, n ), an induced subgraph of Q (n, n, n ).