On symmetric Moore geometries
A generalized Moore geometry is an incidence structure with \(a + 1\) points on each line and \(b + 1\) lines through each point such that: Two distrinct points are incident with at most one line and if they are not incident then there is required a sequence of at most \(d\) consecutive lines that give a bridge between them. \(d\) is called the diameter of the geometry. To any two points that require a sequence of \(d\) lines there exist \(c\) distinct sequences of \(d\) lines connecting them. The sequence is unique if it is shorter than \(d\). [\textit{C. Roos} and \textit{A. J. van Zanten}, J. Comb. Theory B 33, 197-212 (1982; Zbl 0488.05055 )]. The author proved earlier that \(d \leq 13\). Now, here it is shown that for the case \(a = b\) the boundary can be reduced to 3 if the geometry is not trivial. The proof is done by investigating the characteristic equation of the intersection matrix and its eigenvalues. For this computer algebra programs are used.
scientific article; zbMATH DE number 617600
scientific article