強正則重邊圖及其應用之研究

Abstract

為了研究準剩餘2-設計的問題，Bose首先於1976年提出強正則重邊圖的概念。接著Neumaier和Metsch分別在1982年及1995年也利用強正則重邊圖的概念進一步的解決準剩餘2-設計的問題。Neumaier在1976年的論文提到，若強正則重邊圖的參數滿足某些條件，這個強正則重邊圖就會是唯一的3/2-設計的共線性圖。 我們在第三節證明強正則重邊圖和強正則圖相同，均為具有三個相異的特徵值所刻畫。針對具有三個相異特徵值的圖，我們找出其分別對應於強正則圖和強正則重邊圖的條件。我們在第四節，詳細回顧Bose，Neumaier及Metsch等的三篇論文，比較他們的結論，及其證明所用的方法。我們在第五節給出一類源於交錯圖的強正則重邊圖的例子。同時，我們根據Neumaier的定理證明他們是唯一的3/2-設計的共線性圖，這項結論有助於交錯圖的幾何刻畫。

The conception of strongly regular multigraph was ‾rst proposed by Bose in 1976, followed by Neumaier and Metsch in 1982 and 1995 respectively for the problem of embedding of quasi-residual 2-design. In particular, Neumaier asserted that the collinearity graph of a unique 3/2-design if it meets some constraints over its parameters. The spectral properties of strongly regular multigraphs are studied in Section 3, we show that they can be haracterized as multigraphs with exactly three distinct eigenvalues, we show further when they are strongly regular graphs in terms of their eigenvalues. For reference purpose, the results together with the arguments for the proofs of the papers of Bose, Neumaier and etsch are summarized is Section 4. A class of strongly regular multigraphs ssociated with the alternating forms graphs is studied in Section 5. Under some numerical constraints, they are the collinearity graphs of uniquely determined 3/2-designs, which provide some information for the geometric characterization of the alternating forms graphs.