The study of the characteristic polynomial of graphs

• Main Authors: Ming-Feng Li, 李明峯
• Other Authors: 高金美
• Format: Others
• Language: zh-TW
• Published: 2011
• Online Access: http://ndltd.ncl.edu.tw/handle/66437375837774865184

碩士 === 淡江大學 === 中等學校教師在職進修數學教學碩士學位班 === 99 === 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.