Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
4603804 | Linear Algebra and its Applications | 2007 | 5 Pages |
Abstract
We improve some recent results on graph eigenvalues. In particular, we prove that if G is a graph of order n ⩾ 2, maximum degree Δ, and girth at least 5, thenμ(G)⩽mina,n-1},where μ(G) is the largest eigenvalue of the adjacency matrix of G.Also, if G is a graph of order n ⩾ 2 with dominating number γ(G) = γ, thenλ2(G)⩽nifγ=1,n-γifγ⩾2,λn(G)⩾⌈n/γ⌉,where 0 = λ1(G) ⩽ λ2(G) ⩽ ⋯ ⩽ λn(G) are the eigenvalues of the Laplacian of G.We also determine all cases of equality in the above inequalities.
Keywords
Related Topics
Physical Sciences and Engineering
Mathematics
Algebra and Number Theory
Authors
Vladimir Nikiforov,