Eigenvalues and forbidden subgraphs I

Abstract

Suppose a graph G have n vertices, m edges, and t triangles. Letting λ n ( G) be the largest eigenvalue of the Laplacian of G and μ n ( G) be the smallest eigenvalue of its adjacency matrix, we prove that λ n ( G ) ⩾ 2 m 2 - 3 nt m ( n 2 - 2 m ) n , μ n ( G ) ⩽ 3 n 3 t - 4 m 3 nm ( n 2 - 2 m ) , with equality if and only if G is a regular complete multipartite graph. Moreover, if G is K r+1 -free, then λ n ( G ) ⩾ 2 mn ( r - 1 ) ( n 2 - 2 m ) with equality if and only if G is a regular complete r-partite graph.