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Congruences of Edge-bipartite Graphs with Applications to Grothendieck Group Recognition. [Part 2], Coxeter Type Study

Mróz A.

Fundamenta Informaticae

2016

Vol. 146, nr 2

145--177

In this two parts article with the same main title we study a problem of Coxeter-Gram spectral analysis of edge-bipartite graphs (bigraphs), a class of signed graphs. We ask for a criterion deciding if a given bigraph Δ is weakly or strongly Gram-congruent with a graph. The problem is inspired by recent works of Simson et al. started in [SIAM J. Discr. Math. 27 (2013), 827-854], and by problems related to integral quadratic forms, bilinear lattices, representation theory of algebras, algebraic methods in graph theory and the isotropy groups of bigraphs. In this Part II we develop general combinatorial techniques, with the use of inflation algorithm discussed in Part I, morsifications and the isotropy group of a bigraph, and we provide a constructive solution of the problem for the class of all positive connected loop-free bigraphs. Moreover, we present an application of our results to Grothendieck group recognition problem: deciding if a given bilinear lattice is the Grothendieck group of some category. Our techniques are tested in a series of experiments for so-called Nakayama bigraphs, illustrating the applications in practice and certain related phenomena. The results show that a computer algebra technique and discrete mathematical computing provide important tools in solving theoretical problems of high complexity.

Congruences of Edge-bipartite Graphs with Applications to Grothendieck Group Recognition. [Part 1], Inflation Algorithm Revisited

Mróz A.

Fundamenta Informaticae

2016

Vol. 146, nr 2

121--144

We study edge-bipartite graphs (bigraphs), a class of signed graphs, by means of the inflation algorithm which relies on performing certain elementary transformations on a given bigraph Δ, or equivalently, on the associated integral quadratic form qΔ: Zn → Z, preserving Gram Z-congruence. The ideas are inspired by classical results of Ovsienko and recent studies of Simson started in [SIAM J. Discr. Math. 27 (2013), 827-854], concerning classifications of integral quadratic and bilinear forms, and their Coxeter spectral analysis. We provide few modifications of the inflation algorithm and new estimations of its complexity for positive and principal loop-free bigraphs. We discuss in a systematic way the behavior and computational aspects of inflation techniques. As one of the consequences we obtain relatively simple proofs of several interesting properties of quadratic forms and their roots, extending known facts. On the other hand, the results are a first step of a solution of a variant of Grothendieck group recognition, a difficult combinatorial problem arising in representation theory of finite dimensional algebras and their derived categories, which we discuss in Part II of this two parts article with the same main title.