CHARACTERIZATION OF SOME CLASSES OF INTEGRAL HYPERGRAPHS

• Main Author: ASTUTI, MULIA
• Format: Dissertations
• Language: Indonesia
• Online Access: https://digilib.itb.ac.id/gdl/view/69634
• Institution: Institut Teknologi Bandung

is a structure H(V; E), where V = fv1; v2; :::; vng is a finite set and E = fE1;E2; :::;Emg is a family of subsets of V such that Ei 6= ; 8i 2 f1; 2; : : : ;mg and [ i Ei = V: The elements v1; v2; :::; vn are called the vertices, and the sets E1;E2; :::;Em are called the hyperedges. Several results of hypergraphs theory have been studied in the literature. We can found that the results on the hypergraph theory is a generalization of the graph theory. Biggs (1993) has interpreted the relations between some coefficients of the characteristic and the Laplacian polynomial of a graph. In the first part of this dissertation, we obtained the relations for a simple and linear hypergraph. Then we will be concerned with integral hypergraphs as a generalization of integral graphs. A hypergraph is called integral if all the eigenvalues of its adjacency matrix are integers. In general, the problem of characterizing integral hypergraphs seems to be very difficult. Since there is no general characterization (besides the definition) of these hypergraphs, the problem of finding (or characterizing) integral hypergraphs has to be treated in some special interesting classes of hypergraphs. In this dissertation, we characterize six classes of hypergraphs to be integral. These classes of hypergraphs are divided into three parts. The first part, we find two classes of hypergraph that are integral, namely projective plane of order n and complete r-uniform hypergraph. In the second part, we find a sufficient and necessary condition for three classes of hypergraph that are integral. The classes of hypergraph are hyperstars, sunflower hypergraphs, and complete tripartite hypergraph. In the third part, we show a class of hypergraph which is not integral, namely hyper-regular. Hypergraph products have been a good way to construct a larger hypergraph from small ones. The most commonly studied hypergraph products are Cartesian product, direct product and strong product. In this dissertation, we characterize these hypergraph products to be integral. We prove that these three operations preserve the integrality.