Efficient Computation of the Isotropy Group of a Finite Graph: A Combinatorial Approach

Abstract

We continue a Coxeter spectral study of finite posets and edge-bipartite graphs (a class of signed graphs in the sense of Harary and Zaslavsky). Here we are interested in two problems. First: whether the incidence matrices CI and CJ of two connected positive posets I and J are Z-congruent if and only if the Coxeter spectra of I and J coincide. Second: the problem if any square integer matrix A E Mn(Z) is Z-congruent with its transpose Atr. We show that these problems can be effectively solved using the right action * : M_n(Z) × Gl(n, Z)_D → M_n(Z), A → A * B := Btr · A · B, of the isotropy group Gl(n, Z)D of a simply laced Dynkin diagram D E {A_n, D_n, E₆, E₇, E₈}. We present an efficient algorithm for computing the isotropy group Gl(n, Z)D. In particular, we show that symbolic and numerical computer calculations in Python and Cython allow us to present a complete description of the isotropy group Gl(n, Z)D with |D| ≤ 10. Furthermore, we discuss optimisation techniques that are important from the calculation efficiency point of view.