A Coxeter--Gram Classification of Positive Simply Laced Edge-Bipartite Graphs

Coxeter energy of graphs

“…(a) The matrix C G := C G ∈ M n (Z) is called the (upper-triangular) Gram matrix [57] of G, where C G is the Gram matrix of the digraph G as in Definition 2.1 (with respect to the labeling…”

“…To keep the paper self-contained we sketch the main arguments of the proof (cf. [2,VII] and [57,49], also [14]). …”

“…Then, since D = D is a forest, by [2, Lemma VII.5.2] there exists a sequence of sink-source quiver reflections in the sense of Bernstein-Gelfand-Ponomariev [4] which transform D to D (see [2,VII.5] for more details). One shows that this digraph transformation yields a matrix B ∈ Gl n (Z) such that B tr · C D · B = C D (see [57,Proposition 2.2] and [2,VII.4]). Now, by a similar calculation as in (a) one can easily check that the matrix B defines the similarity of the Coxeter matrices Cox D and Cox D .…”

“…Recall that the Coxeter spectral properties of graphs (including edge-bipartite graphs [57], a class of signed multigraphs) are studied by many authors [3,14,15,17,18,33,36,45,47,49,[57][58][59]62] and the study is motivated by the Coxeter formalism appearing in group theory, Lie theory, representation theory of algebras, their derived categories and K-theory, geometry, mathematical physics and other contexts, see [2,4,8,10,13,[30][31][32]34,37,38,[44][45][46][47][49][50][51]55,62,64] and Remark 2.4. Observe that the entries under the absolute values in (1) (resp.…”

“…It can be considered interesting that the key role in the description of lowest Coxeter energies of trees is played by Dynkin graphs, Euclidean (i.e., extended Dynkin) graphs and hyperbolic Dynkin (i.e., over-extended Dynkin) graphs, combinatorial objects appearing in classification results in many branches of mathematics and physics (see [2,7,8,28,34,57,62] and Section 2).…”