Locating the Eigenvalues of Trees
โ Scribed by David P. Jacobs; Vilmar Trevisan
- Publisher
- Elsevier Science
- Year
- 2011
- Tongue
- English
- Weight
- 497 KB
- Volume
- 434
- Category
- Article
- ISSN
- 0024-3795
No coin nor oath required. For personal study only.
๐ SIMILAR VOLUMES
Let ฮป 1 (T ) and ฮป 2 (T ) be the largest and the second largest eigenvalues of a tree T , respectively. We obtain the following sharp lower bound for ฮป 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 i
Very little is known about upper bounds for the largest eigenvalues of a tree that depend only on the vertex number. Starting from a classical upper bound for the largest eigenvalue, some refinements can be obtained by successively removing trees from consideration. The results can be used to charac
Let T c n be the set of the complements of trees of order n. In this paper, we characterize the unique graph whose least eigenvalue attains the minimum among all graphs in T c n .
## Abstract Let ฮป~__k__~(__G__) be the __k__th Laplacian eigenvalue of a graph __G__. It is shown that a tree __T__ with __n__ vertices has $\lambda\_{k}(T)\le \lceil { {n}\over{k}}\rceil$ and that equality holds if and only if __k__ < __n__, __k__|__n__ and __T__ is spanned by __k__ vertex disjoin