Sharp upper bounds for the Laplacian graph eigenvalues

Let G = (V ,E) be a simple connected graph and λ1(G) be the largest Laplacian eigenvalue of G. In this paper, we prove that: 1. λ1(G) = max{du +mu : u ∈ V } if and only if G is a regular bipartite or a semiregular bipartite graph, where du and mu denote the degree of u and the average of the degrees of the vertices adjacent to u, respectively. 2. λ1(G) = 2 + √ (r − 2)(s − 2) if and only if G is a regular bipartite graph or a semiregular bipartite graph, or a path with four vertices, where r = max{du + dv : uv ∈ E} and suppose xy ∈ E satisfies dx + dy = r , s = max{du + dv : uv ∈ E − {xy}}. 3. λ1(G) = max { du(du +mu)+ dv(dv +mv) du + dv : uv ∈ E } if and only if G is a regular bipartite graph or a semiregular bipartite graph. 4. λ1(G) 2 + √ (t − 2)(b − 2) with equality if and only if G is a regular bipartite graph or a semiregular bipartite graph, or a path with four vertices, where t = max { du(du +mu)+ dv(dv +mv) du + dv : uv ∈ E } and suppose xy ∈ E satisfies dx(dx +mx)+ dy(dy +my) dx + dy = t, Supported by NSF of the People’s Republic of China and Anhui Province. E-mail address: [email protected] (Y.-L. Pan). 0024-3795/02/$ see front matter 2002 Elsevier Science Inc. All rights reserved. PII: S0024 -3795(02)00353-1 288 Y.-L. Pan / Linear Algebra and its Applications 355 (2002) 287–295 b = max { du(du +mu)+ dv(dv +mv) du + dv : uv ∈ E − {xy} }