Hanni Garminia

Abstracts

2 The Second Smallest Eigenvalue of Complete Tripartite Hypergraph

Authors: M. Salman, Hanni Garminia, Alfi Y. Zakiyyah, A. N. Irawati

Abstract:

In the terminology of the hypergraph, there is a relation with the terminology graph. In the theory of graph, the edges connected two vertices. In otherwise, in hypergraph, the edges can connect more than two vertices. There is representation matrix of a graph such as adjacency matrix, Laplacian matrix, and incidence matrix. The adjacency matrix is symmetry matrix so that all eigenvalues is real. This matrix is a nonnegative matrix. The all diagonal entry from adjacency matrix is zero so that the trace is zero. Another representation matrix of the graph is the Laplacian matrix. Laplacian matrix is symmetry matrix and semidefinite positive so that all eigenvalues are real and non-negative. According to the spectral study in the graph, some that result is generalized to hypergraph. A hypergraph can be represented by a matrix such as adjacency, incidence, and Laplacian matrix. Throughout for this term, we use Laplacian matrix to represent a complete tripartite hypergraph. The aim from this research is to determine second smallest eigenvalues from this matrix and find a relation this eigenvalue with the connectivity of that hypergraph.

Keywords: Connectivity, graph, hypergraph, Laplacian matrix

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1 On Modules over Dedekind Prime Rings

Authors: Pudji Astuti, Elvira Kusniyanti, Hanni Garminia

Abstract:

This research studies an interconnection between finitely generated uniform modules and Dedekind prime rings. The characterization of modules over Dedekind prime rings that will be investigated is an adoption of Noetherian and hereditary concept. Dedekind prime rings are Noetherian and hereditary rings. This property of Dedekind prime rings is a background of the idea of adopting arises. In Noetherian area, it was known that a ring R is Noetherian ring if and only if every finitely generated R-module is a Noetherian module. Similar to that result, a characterization of the hereditary ring is related to its projective modules. That is, a ring R is hereditary ring if and only if every projective R-module is a hereditary module. Due to the above two results, we suppose that characterization of a Dedekind prime ring can be analyzed from finitely generated modules over it. We propose a conjecture: a ring R is a Dedekind prime ring if and only if every finitely generated uniform R-module is a Dedekind module. In this article, we will generalize a concept of the Dedekind module for non-commutative ring case and present a part of the above conjecture.

Keywords: dedekind domains, dedekind prime rings, dedekind modules, uniform modules

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