THE COMPARISON OF LAPLACIAN OPERATORS SPECTRUM BETWEEN ALGEBRAIC AND METRIC GRAPHS

• Main Author: WAHYU UTAMI (NIM : 20116034), INDAH
• Format: Theses
• Language: Indonesia
• Online Access: https://digilib.itb.ac.id/gdl/view/28027
• Institution: Institut Teknologi Bandung

<p align="justify">Suppose G=(V,E,&#961;) is a weighted and directed graph, with weight &#961;(e)=w. Also suppose that each edge is associated to a closed interval [0,w]. Each vertex is now an identification of some endpoints of the intervals. In this case we said that G is a metric graph. So that a function on a metric graph is function on each edge, i.e. interval, with some compatibility condition on vertices (end points of intervals). Laplacian operators on metric graph is laplacian operator that is defined on each intervals with some additional condition, in our case the Kirchoff condition. Using Kirchoff condition, laplacian operator is self adjoint and positive operator. On the other hand, we can consider discrete laplacian in considering weighted graph. In this case the discrete laplacian is a matrix, therefore finite dimensional operator. We consider spectral gap for laplacian operator and gap between two lowest eigenvalus. Zero is the lowest eigenvalue, then the spectral gap equals with the second eigenvalue. We make comparison between gaps coming from laplacian operators, and from laplacian matrix. For cycle metric graph C_N, we find that both gap almost coincide, limiting to 0.<p align="justify">