Title: A Coxeter--Gram Classification of Positive Simply Laced Edge-Bipartite Graphs
Abstract: Following the spectral graph theory, a graph coloring technique, and algebraic methods in graph theory, we study the category ${\cal U}{\cal B} igr_n$ of loop-free edge-bipartite (signed) graphs $\Delta$, with $n\geq 2$ vertices, up to $\mathbb{Z}$-congruences $\sim_\mathbb{Z}$ and $\approx_\mathbb{Z}$ (defined in the paper), by means of the Gram matrix $ \check G_\Delta \in \mathbb{M}_n(\mathbb{Z})$, the Coxeter--Gram matrix ${\rm Cox}_\Delta= - \check G_\Delta\cdot \check G_\Delta^{-tr} \in \mathbb{M}_n(\mathbb{Z})$, the spectrum $ {\bf specc}_\Delta$ of $\mbox{\rm Cox}_\Delta$, the Coxeter polynomial $\mbox{\rm cox}_\Delta(t)\in \mathbb{Z}[t]$, the Coxeter number ${\bf c}_\Delta \geq 2$, and the $\mathbb{Z}$-bilinear Gram form $b_\Delta: \mathbb{Z}^n\times \mathbb{Z}^n \to \mathbb{Z}$ of $\Delta$. Our main inspiration for the study comes from the representation theory of posets, groups and algebras, Lie theory, and Diophantine geometry problems. One of our aims is to compute the set ${\cal C}{\cal G} pol^+_n$ of all polynomials $\mbox{\rm cox}_\Delta(t)$, with positive connected graphs $\Delta$ in ${\cal U}{\cal B} igr_n$, for all $n\geq 2$, and to present a framework for a classification of graphs $\Delta$ in ${\cal U}{\cal B} igr_n$, up to the congruence $\approx_\mathbb{Z}$, by means of their Coxeter polynomials $\mbox{\rm cox}_\Delta(t)\in \mathbb{Z}[t]$ and Coxeter spectra. In particular, the Coxeter spectral analysis question, whether the congruence $\Delta\approx_\mathbb{Z}\Delta{'}$ holds, for any pair of connected positive graphs $\Delta, \Delta{'}\in {\cal U}{\cal B} igr_n$ such that ${\bf specc}_\Delta={\bf specc}_{\Delta '}$, is studied in the paper. One of our main results contains an affirmative answer to the question and a description of the finite set ${\cal C} {\cal G} pol^+_n$ for $n \leq 8$. One of the tools we apply is an inflation algorithm $ \Delta \mapsto D\Delta $ that associates with any connected positive graph $\Delta\in {\cal U}{\cal B} igr_n$ a simply laced Dynkin diagram $D\Delta$ such that $\Delta\sim_\mathbb{Z} D\Delta$, and $\mbox{\rm cox}_\Delta(t)$ and ${\bf c}_\Delta$ coincide with the Coxeter polynomial $\mbox{\rm cox}_A(t)$ and the Coxeter number ${\bf c}_A$ of a matrix morsification $A\in \mbox{\bf Mor}_{D\Delta }\subseteq\mathbb{M}_n(\mathbb{Z})$ for the diagram $D\Delta$ and, in case $n\leq 8$, all pairs ($\mbox{\rm cox}_\Delta(t),{\bf c}_\Delta$) are listed.
Publication Year: 2013
Publication Date: 2013-01-01
Language: en
Type: article
Indexed In: ['crossref']
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Cited By Count: 66
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