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Upper bounds on the (signless) Laplacian eigenvalues of graphs.
- Source :
-
Linear Algebra & its Applications . Oct2014, Vol. 459, p334-341. 8p. - Publication Year :
- 2014
-
Abstract
- Let G be a simple graph of order n with vertex set V = { v 1 , v 2 , … , v n } . Also let μ 1 ( G ) ≥ μ 2 ( G ) ≥ ⋯ ≥ μ n − 1 ( G ) ≥ μ n ( G ) = 0 and q 1 ( G ) ≥ q 2 ( G ) ≥ ⋯ ≥ q n ( G ) ≥ 0 be the Laplacian eigenvalues and signless Laplacian eigenvalues of G , respectively. In this paper we obtain μ i ( G ) ≤ i − 1 + min U i max { | N H ( v k ) ∪ N H ( v j ) | : v k v j ∈ E ( H ) } , where N H ( v k ) is the set of neighbors of vertex v k in V ( H ) = V ( G ) \ U i , U i is any ( i − 1 ) -subset of V ( G ) (here, we agree that i ∈ { 1 , … , n − 1 } and μ i ( G ) ≤ i − 1 if E ( H ) = ∅ ). For any graph G , this bound does not exceed the order of G . Moreover, we prove that max { μ i ( G ) , q i ( G ) } ≤ max i ≤ k ≤ n { d G ( v k ) + ∑ v j ∈ N G ( v k ) ∩ N d G ( v j ) d G ( v k ) } ≤ 2 d G ( v i ) , where d G ( v i ) is the i -th largest degree of G and N = { v i , v i + 1 , … , v n } . [ABSTRACT FROM AUTHOR]
- Subjects :
- *MATHEMATICAL bounds
*LAPLACIAN matrices
*EIGENVALUES
*GRAPH theory
*SET theory
Subjects
Details
- Language :
- English
- ISSN :
- 00243795
- Volume :
- 459
- Database :
- Academic Search Index
- Journal :
- Linear Algebra & its Applications
- Publication Type :
- Academic Journal
- Accession number :
- 97522255
- Full Text :
- https://doi.org/10.1016/j.laa.2014.07.018