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A Horizontal Mesh Algorithm for a Class of Edge-bipartite Graphs and their Matrix Morsifications

Kaniecki M., Kosakowska J., Malicki P., Marczak G.

Fundamenta Informaticae

2015

Vol. 136, nr 4

345--379

We construct a horizontal mesh algorithm for a study of a special type of mesh root systems of connected positive loop-free edge-bipartite graphs Δ, with n ≥ 2 vertices, in the sense of [SIAM J. Discrete Math. 27 (2013), 827–854] and [Fund. Inform. 124 (2013), 309-338]. Given such a loop-free edge-bipartite graph Δ, with the non-symmetric Gram matrix ˇGΔ ∈ Mn(Z) and the Coxeter transformation ΦA : Zn → Zn defined by a quasi-triangular matrix morsification A ∈ Mn(Z) of Δ satisfying a non-cycle condition, our combinatorial algorithm constructs a ΦA-mesh root system structure Γ(RΔ,ΦA) on the finite set of all ΦA-orbits of the irreducible root system RΔ := {v ∈ Zn; v · ˇGΔ · vtr = 1}. We apply the algorithm to a graphical construction of a ΦI - mesh root system structure Γ(RI ,ΦI ) on the finite set of ΦI -orbits of roots of any poset I with positive definite Tits quadratic form bqI : ZI → Z.

Inflation Algorithms for Positive and Principal Edge-bipartite Graphs and Unit Quadratic Forms

Kosakowska J.

Fundamenta Informaticae

2012

Vol. 119, nr 2

149-162

We describe combinatorial algorithms that compute the Dynkin type (resp. Euclidean type) of any positive (resp. principal) unit quadratic form q : Z^n →Z and of any positive (resp. principal) edge-bipartite connected graph Δ. The study of the problem is inspired by applications of the algorithms in the representation theory, in solving a class of Diophantine equations, in the study of mesh geometries of roots, in the spectral analysis of graphs, and in the Coxeter-Gram classification of edge-bipartite graphs.