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 matrixǦ A is 8. Moreover, we show that D (andǦ A ) has at

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... least one diagonal coefficient equal to 1. We describe some other restrictions and interrelations between the coefficients of A, D andǦ 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. Gram matrices are independently studied by means of associated signed graphs (bigraphs) and integral quadratic forms (in the sense of Roiter [6, 21] ) by Simson in [27]-[30], and in [2, 9, 10, 34] . This study is inspired by problems of representation theory of associative algebras, their derived categories and Grothendieck lattices, see [1, 11, 18, 20, 27] . In this paper we follow the interrelation between quasi-Cartan matrices and so-called Cox-regular Gram matrices observed recently in [12] and [31] . The set of quasi-Cartan matrices is equipped with the Z-equivalence relation ∼ Z (see (2.1)) which by the result of Pérez-Rivera [24] corresponds to the isomorphism of associated semisimple Lie algebras in the positive case. On the other hand, positive quasi-Cartan matrices are classified with respect to ∼ Z by the Cartan matrices associated with Dynkin diagrams, see Theorem 3.5(a) and [5, 22, 23, 29] . In contrast to the positive case, a complete classification of non-negative quasi-Cartan matrices with respect to the Z-equivalence remains unknown. We recall some partial results in this direction in Theorem 3.5 and Remark 3.6. The study of the present paper can be treated as a step towards a classification (up to ∼ Z ) of all non-negative quasi-Cartan matrices. Namely, our main goal is to face the following natural question: given a non-negative quasi-Cartan matrix , what are the possible values of the coefficients of A, its minimal symmetrizer D ∈ M n (Z), and the associated Gram matrixǦ A ∈ M n (Z)? Besides the motivations which follow from the remarks we made before, we point out also the following stimuli (of a computational character) for the study of this question: • existence of bounds for the coefficients of non-negative quasi-Cartan matrices (resp. Gram matrices) proves the finiteness of the set of these matrices of fixed size; moreover, it provides a basis for an efficient algorithmic method to determine all such matrices (see Section 6), • knowing a precise characterization of the coefficients as above can increase the efficiency of algorithmic definiteness tests as well as Dynkin type recognition algorithms, cf. [2, 12, 13, 19, 22] , Bounds for the coefficients of non-negative quasi-Cartan matrices appear implicitly in the literature and they are relatively easy to prove (see Lemma 4.1). However, the problem of determining analogous bounds in case of symmetrizers and Gram matrices seems to be more difficult. Main results of the paper provide finite lists of the possible values (hence also bounds) for the coefficients of a non-negative quasi-Cartan matrix A, its minimal symmetrizer D, and of the associated Gram matrixǦ A , see Theorem 5.3 and Corollary 5.4. The proof applies combinatorial analysis of bigraphs associated with Gram matrices. Moreover, we show that D (andǦ A ) has at least one diagonal coefficient equal to 1 and that certain configurations of the diagonal coefficients cannot appear. We also point out some interrelations between the coefficients of A, D andǦ A , and the properties of A which are important from the point of view of Lie theory and representation theory. The paper is organized as follows. In Section 2 (resp. Section 3) we collect the basic properties of quasi-Cartan matrices and their Z-equivalences (resp. Gram matrices and associated bigraphs) supplemented with few new general observations and some examples showing, among others, that the bounds we found are strict. In Section 4 we survey results from various papers concerning the coefficients of considered matrices. In some cases we provide alternative (simpler) proofs and/or some slight improvements. Section 5 contains the main results mentioned above and their proofs. The final section contains an application of the main results in the form of a computational classification of all non-negative quasi-Cartan matrices of small sizes with respect to their coranks and Dynkin/Euclidean types in the coranks 0 and 1. This classification largely generalizes some numerical results on positive quasi-Cartan matrices from the recent papers [10] and [31]. Generalities on quasi-Cartan matrices By N ⊆ Z ⊆ Q ⊆ R we denote the set of natural numbers (without zero), the ring of integers and the fields of rational and real numbers, respectively. Given an integer n ≥ 1, we denote by n the finite set {1, 2, . . . , n} ⊆ N. Given a permutation σ : n → n, we denote by P(σ ) = [p ij ] ∈ M n (Z) the permutation matrix of σ with p σ (i)i = 1 for i ∈ n, and the remaining p ij 's are zero. By rk(A) (resp. crk(A)) we denote the rank (resp. the corank n − rk(A)) of a matrix A ∈ M n (R). Following [22,23], two quasi-Cartan matrices A, A ′ ∈ M n (Z) are called Z-equivalent if there exist symmetrizers D, D ′ of A and A ′ , respectively, and a matrix M ∈ Gl n (Z) := {B ∈ M n (Z) : det(B) = ±1} such that M tr DAM = D ′ A ′ , (2.1) and D ′