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Cartan matrices, quasi-Cartan matrices and associated upper triangular Gram matrices control important combinatorial aspects of Lie theory and representation theory of associative algebras. We provide a graph theoretic proof of the fact that the absolute values of the coefficients of a non-negative quasi-Cartan matrix A as well as of its (minimal) symmetrizer D are bounded by 4, and that the analogous bound in case of the associated Gram matrix G ˇ A is 8. Moreover, we show that D (and G ˇ A) has at least one diagonal coefficient equal to 1. We describe some other restrictions and interrelations between the coefficients of A , D and G ˇ A , and the corank and other properties of A relevant in Lie theory. We apply our results to construct an algorithm by which we classify all non-negative quasi-Cartan matrices of small sizes. [ABSTRACT FROM AUTHOR]

Cartan matrices and quasi-Cartan matrices play an important role in such areas as Lie theory, representation theory, and algebraic graph theory. It is known that each (connected) positive definite quasi-Cartan matrix A ∈ Mn(Z) is Z-equivalent with the Cartan matrix of a Dynkin diagram, called the Dynkin type of A. We present a symbolic, graph-theoretic algorithm to compute the Dynkin type of A, of the pessimistic arithmetic (word) complexity O(n2), significantly improving the existing algorithms. As an application we note that our algorithm can be used as a positive definiteness test for an arbitrary quasi-Cartan matrix, more efficient than standard tests. Moreover, we apply the algorithm to study a class of (symmetric and non-symmetric) quasi-Cartan matrices related to Nakayama algebras. [ABSTRACT FROM AUTHOR]

Abstract The paper is devoted to a Gram classification of an important class of signed graphs with loops playing an important role in many branches of mathematics, physics and computer science including game theory, crystallography, Diophantine geometry, Lie theory, spectral graph theory, a graph coloring technique, algebraic methods in graph theory. They have many deep applications in spectral machine learning methods, electrical networks, to link sign prediction in signed unipartite and bipartite networks, and community partition in social networks. We continue the study of finite connected edge-bipartite graphs Δ (bigraphs, for short), with m ≥ 2 vertices (a class of signed graphs), started in Simson (2013) and developed in Simson (2016) by means of the non-symmetric Gram matrix G ˇ Δ ∈ M m (Z) defining Δ , its symmetric Gram matrix G Δ : = 1 2 [ G ˇ Δ + G ˇ Δ t r ] ∈ M m (1 2 Z) , and the Gram quadratic form q Δ : Z m → Z. In the present paper, given n ≥ 1 , we mainly study connected principal Cox-regular edge-bipartite graphs Δ with loops and n + 1 ≥ 2 vertices, in the sense that the symmetric Gram matrix G Δ ∈ M n + 1 (Z) of Δ is positive semi-definite of rank n ≥ 1. Our aim is to classify such edge-bipartite graphs by means of an inflation algorithm, up to the weak Gram Z -congruence Δ ∼ Z Δ ′ , where Δ ∼ Z Δ ′ means that G Δ ′ = B t r ⋅ G Δ ⋅ B , for some B ∈ M n + 1 (Z) with det B = ± 1. Our main result of the paper asserts that, given a principal Cox-regular connected edge-bipartite graph Δ with n + 1 ≥ 2 vertices, there exists an edge-bipartite Euclidean graph D n + 1 of Tables A and B, and a suitably chosen sequence t • − of the inflation operators, each of one of the forms Δ ′ ↦ t a − Δ ′ or Δ ′ ↦ t a b − Δ ′ , such that the composite operator Δ ↦ t • − Δ reduces Δ to the bigraph D n + 1 such that: (i) Δ ∼ Z D n + 1 and (ii) the bigraphs Δ , and D n + 1 have the same number of loops. The algorithm does not change loops and computes a matrix B ∈ M n + 1 (Z) , with det B = ± 1 , defining the weak Gram Z -congruence Δ ∼ Z D n + 1 , that is, satisfying the equation G D n + 1 = B t r ⋅ G Δ ⋅ B. [ABSTRACT FROM AUTHOR]