圖的譜差值

Abstract

中文摘要 對一n × n 方陣M 而言，其「值距」φ(M) 通常定義為M 的最大與最小特 徵值的差距，也就是 φ(M) := max i;j |ρi − ρj |。 上式的最大值是對任兩個M 的特徵值差考慮，不過有時特徵值0 被排除。我們考 慮圖論上被廣泛使用的三種方陣：鄰接方陣、拉普拉斯方陣、正拉普拉斯方陣。我 們探討圖與此三種值距的關係，特別研究點數固定時，能得到最大或最小值距的 圖

Abstract Given an n×n matrixM, the spread, φ(M), is essentially the diameter of its spectrum: φ(M) := max i;j |ρi − ρj |, where the maximal is taken over all pairs of eigenvalues (or nonzero eigenvalues in some cases) of M. We consider adjacent matrices, Laplacian and signless Laplacian matrices which are commonly used in graph theory. After discussing relatedness on the graphs and their corresponding spreads, we discover the boundary which affects the spread, and use this result to find the graphs that may have the maximal or minimal spread.