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Sharp bounds on the eigenvalues of trees

✍ Scribed by Shengbiao Hu

Publisher
Elsevier Science
Year
2008
Tongue
English
Weight
289 KB
Volume
56
Category
Article
ISSN
0898-1221

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✦ Synopsis

Let Ξ» 1 (T ) and Ξ» 2 (T ) be the largest and the second largest eigenvalues of a tree T , respectively. We obtain the following sharp lower bound for Ξ» 1 (T ):

where d i is the degree of the vertex v i and m i is the average degree of the adjacent vertices of v i . Equality holds if and only if T is a tree

Let d 1 and d 2 be the highest and the second highest degree of T , respectively. Let r (T ) be the maximum distance between the highest and the second highest degree vertices. We also show that if T is a tree of order (n > 2), then

The equality holds if T is a tree T 1 or a tree T 2 , or T is a tree T 4 and d 1 = d 2 , where T 1 is formed by joining the centers of K 1,d 1 -1 and K 1,d 2 -1 and T 2 is formed by joining the centers of K 1,d 1 -1 and K 1,d 2 -1 to a new vertex, the T 4 is formed by joining a 1-degree vertex of K 1,d 1 and K 1,d 2 to a new vertex.

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