圖與重邊圖結合各式的設計之研究

Abstract

Bose 首先提出強正則重邊圖的概念，接著Neumair 和Metsch 利用強正則重邊圖的概念進一步地解決準剩餘2-設計的問題。近來，不完全幾何設計的概念被van Dam和Spence 使用在具有2 個奇異值的組合設計。我們將Neumair 和Metsch 兩篇論文中的定義與結果做整理，並以統一形式呈現在此論文裡，進而舉出一些2-設計及其對應的強正則重邊圖。藉由這些圖，研究具有3 或4 個相異特徵值的連通正則圖之 特性。

The notion of strongly regular multigraphs was first introduced by R. C. Bose, followed by Neumaier for characterizing quasi-residual 2-designs, and further by Metsch for embeddings of residual 2-designs. Recently, the notion of partial geometric designs was also used by van Dam and Spence over combinatorial designs with two singular values. The basic de‾nitions and most results regarding strongly regular multigraphs and partial geometric designs covered in the works of Neumaier and Metsch are given in a uni‾ed way in this thesis. The associated multigraphs or graphs of 2-designs are then studied, followed by a few examples of 2-designs and their corresponding strongly regular multigraphs. Motivated by these graphs, connected regular graphs with 3 or 4 distinct eigenvalues are also studied.