A Gram classification of non-negative corank-two 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].…”

“…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.…”

Effective Nondeterministic Positive Definiteness Test for Unidiagonal Integral Matrices

“…We refer to [19] for more complete bibliography and more detailed introduction to the topic. On the other hand, inflations of bigraphs, matrices and integral quadratic forms are an important ingredient of the new lively area -the Coxeter spectral analysis of bigraphs and their morsifications [8], [11], [12], [13], [18], [19], [20], [27], [28], [29], [30], cf. [1], as well as the Coxeter type study of finite dimensional algebras and their bounded derived categories, see [20], [22], [23], [27].…”