Bounds of Eigenvalues of <inline-formula> <graphic file="1029-242X-2009-852406-i1.gif"/></inline-formula>-Minor Free Graphs

• Main Author: Fang Kun-Fu
• Format: Article
• Language: English
• Published: SpringerOpen 2009-01-01
• Series: Journal of Inequalities and Applications
• Online Access: http://www.journalofinequalitiesandapplications.com/content/2009/852406

<p/> <p>The spectral radius <inline-formula> <graphic file="1029-242X-2009-852406-i2.gif"/></inline-formula> of a graph <inline-formula> <graphic file="1029-242X-2009-852406-i3.gif"/></inline-formula> is the largest eigenvalue of its adjacency matrix. Let <inline-formula> <graphic file="1029-242X-2009-852406-i4.gif"/></inline-formula> be the smallest eigenvalue of <inline-formula> <graphic file="1029-242X-2009-852406-i5.gif"/></inline-formula>. In this paper, we have described the <inline-formula> <graphic file="1029-242X-2009-852406-i6.gif"/></inline-formula>-minor free graphs and showed that (A) let <inline-formula> <graphic file="1029-242X-2009-852406-i7.gif"/></inline-formula> be a simple graph with order <inline-formula> <graphic file="1029-242X-2009-852406-i8.gif"/></inline-formula>. If <inline-formula> <graphic file="1029-242X-2009-852406-i9.gif"/></inline-formula> has no <inline-formula> <graphic file="1029-242X-2009-852406-i10.gif"/></inline-formula>-minor, then <inline-formula> <graphic file="1029-242X-2009-852406-i11.gif"/></inline-formula>. (B) Let <inline-formula> <graphic file="1029-242X-2009-852406-i12.gif"/></inline-formula> be a simple connected graph with order <inline-formula> <graphic file="1029-242X-2009-852406-i13.gif"/></inline-formula>. If <inline-formula> <graphic file="1029-242X-2009-852406-i14.gif"/></inline-formula> has no <inline-formula> <graphic file="1029-242X-2009-852406-i15.gif"/></inline-formula>-minor, then <inline-formula> <graphic file="1029-242X-2009-852406-i16.gif"/></inline-formula>, where equality holds if and only if <inline-formula> <graphic file="1029-242X-2009-852406-i17.gif"/></inline-formula> is isomorphic to <inline-formula> <graphic file="1029-242X-2009-852406-i18.gif"/></inline-formula>.</p>