圖的特徵多項式之探討

The study of the characteristic polynomial of graphs

李明峯(Ming-Feng Li)

指導教授 ： 高金美

淡江大學/理學院/中等學校教師在職進修數學教學碩士學位班/碩士(2011年)

https://doi.org/10.6846/TKU.2011.00229

摘要

假設A是n階方陣，A的特徵多項式為p(A,x)=det(xI-A)，其中I是n階單位方陣。令G為一含有n個點的簡單圖且A(G)為圖G的鄰接矩陣， 我們稱A(G)的特徵多項式為G的特徵多項式，記為p(G,x)=det(xI-A(G))，其中I是n階單位方陣。
在本論文中我們直接計算出完全圖與星圖的特徵多項式，對於路徑與迴圈我們可得其遞迴關係式；由前面的計算我們得到行列式的化簡與圖之間的關係，推導出圖中含有degree 1的點或含有degree 2的點及含有一個橋的特徵多項式，可經由子圖的特徵多項式而獲得。我們利用已經獲得的結果，推得一些特殊圖的特徵多項式或其遞迴關係式。

關鍵字

鄰接矩陣 ； 行列式 ； 遞迴關係 ； 特徵多項式

並列摘要

Let A be a matrix of order n. The characteristic polynomial of A is p(A)=det(xI-A) where I is the identity matrix of order n. Let G be a simple graph and A(G) is the adjacency matrix of G. We call the characteristic polynomial of A(G) is the characteristic polynomial of G, denoted by p(G)=det(xI-A(G)), where I is the identity matrix of order n.
In this thesis, we directly calculate the characteristic polynomial of complete graphs and star graphs. For paths and cycles, we get the recurrence relation of their characteristic polynomial. From the above calculation, we obtain the relation between the simplification of determinants and their subgraphs. We get the characteristic polynomials of the graph with a vertex of degree 1, a vertex of degree 2, or a bridge by the characteristic polynomial of their subgraphs. We use these results to get the characteristic polynomial of some special graphs directly or from the recurrence relation.

並列關鍵字

adjacency matrix ； determinant ； recursive relation ； characteristic polynomial

參考文獻

[2] D.B. West, Graph Theory, 2001.

參考文獻：

[1] D. Cvetković, M. Doob, and H. Sachs, Spectra of Graph, Academic Press, New York, 1980.

[3] Shang-wang Tan, Jiming Guo, The largest eigenvalue on tree,Journal of the University of Petrolum(Edition of Natural Science) , China, 2002, 26(6) : 113-117.

[4] Shang-wang Tan, On formulas calculating the characteristic polynomial of matrices in graph theory, Pure and AppliedMathematics, 2009, 25(2).