Applications of matrix morsifications to Coxeter spectral study of loop-free edge-bipartite graphs

Coxeter energy of graphs

“…[19,53]) in the sense of Simson [57]. Coxeter matrices and their spectra defined for bigraphs and posets are studied in [3,15,17,18,33,45,55,[57][58][59] in context of applications in Gram's classifications of integral quadratic and bilinear forms, problems of Diophantine geometry, computational number theory, representation theory of groups, algebras and posets and spectral graph theory. For an interesting discussion on symmetrizable Cartan matrices in the context of Coxeter-Gram study of bigraphs and valued bigraphs we refer to [59,Section 3].…”

“…, λ n ∈ C denote all eigenvalues of the Coxeter matrix Cox C := −C −1 · C tr of C (cf. Definition 2.1(b), and [15,45,57], where matrix morsifications of bigraphs and their Coxeter matrices are studied). Then by the inequality of arithmetic and geometric means we have:…”

“…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.…”