Eigenvalues and forbidden subgraphs I
Abstract
Suppose a graph G have n vertices, m edges, and t triangles. Letting λn(G) be the largest eigenvalue of the Laplacian of G and μn(G) be the smallest eigenvalue of its adjacency matrix, we prove that(λn (G) ≥ frac(2 m2 - 3 nt, m (n2 - 2 m)) n,; μn (G) ≤ frac(3 n3 t - 4 m3, nm (n2 - 2 m)),)with equality if and only if G is a regular complete multipartite graph. Moreover, if G is Kr+1-free, thenλn (G) ≥ frac(2 mn, (r - 1) (n2 - 2 m))with equality if and only if G is a regular complete r-partite graph. © 2006 Elsevier Inc. All rights reserved.
Publication Title
Linear Algebra and Its Applications
Recommended Citation
Nikiforov, V. (2007). Eigenvalues and forbidden subgraphs I. Linear Algebra and Its Applications, 422 (1), 284-290. https://doi.org/10.1016/j.laa.2006.10.007
