Sharp upper bounds for the Laplacian graph eigenvalues

Abstract

Let G=( V, E) be a simple connected graph and λ 1( G) be the largest Laplacian eigenvalue of G. In this paper, we prove that: 1. λ 1( G)=max{ d u + m u : u∈ V} if and only if G is a regular bipartite or a semiregular bipartite graph, where d u and m u denote the degree of u and the average of the degrees of the vertices adjacent to u, respectively. 2. λ 1(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{ d u + d v : uv∈ E} and suppose xy∈ E satisfies d x + d y = r, s=max{ d u + d v : uv∈ E−{ xy}}. 3. λ 1(G)= max d u(d u+m u)+d v(d v+m v) d u+d v :uv∈E if and only if G is a regular bipartite graph or a semiregular bipartite graph. 4. λ 1(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 d u(d u+m u)+d v(d v+m v) d u+d v :uv∈E and suppose xy∈ E satisfies d x(d x+m x)+d y(d y+m y) d x+d y =t, b= max d u(d u+m u)+d v(d v+m v) d u+d v :uv∈E−{xy} .