The spectrum and toughness of regular graphs

In 1995, Brouwer proved that the toughness of a connected kk-regular graph GG is at least k/λ−2k/λ−2, where λλ is the maximum absolute value of the non-trivial eigenvalues of GG. Brouwer conjectured that one can improve this lower bound to k/λ−1k/λ−1 and that many graphs (especially graphs attaining equality in the Hoffman ratio bound for the independence number) have toughness equal to k/λk/λ. In this paper, we improve Brouwer’s spectral bound when the toughness is small and we determine the exact value of the toughness for many strongly regular graphs attaining equality in the Hoffman ratio bound such as Lattice graphs, Triangular graphs, complements of Triangular graphs and complements of point-graphs of generalized quadrangles. For all these graphs with the exception of the Petersen graph, we confirm Brouwer’s intuition by showing that the toughness equals k/(−λmin)k/(−λmin), where λminλmin is the smallest eigenvalue of the adjacency matrix of the graph.