Using multiplicities of eigenvalues of elliptic self-adjoint differential operators on graphs and transversality, we construct some new invariants of graphs which are related to tree-width.
On multiple eigenvalues of trees
β Scribed by P. Rowlinson
- Publisher
- Elsevier Science
- Year
- 2010
- Tongue
- English
- Weight
- 115 KB
- Volume
- 432
- Category
- Article
- ISSN
- 0024-3795
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π SIMILAR VOLUMES
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 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