A characterization on graphs which achieve the upper bound for the largest Laplacian eigenvalue of graphs

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)=d1+d2, (d1≠d2) if and only if G is a star graph, where d1, d2 are the highest and the second highest degree, respectively. 2. λ1(G)=max2(d2u+dum′u):u∈V if and only if G is a bipartite regular graph, where m′u=∑v{dv−|Nu∩Nv|:uv∈E}du, du denotes the degree of u and |Nu∩Nv| is the number of common neighbors of u and v. 3. λ1(G)⩽max(du+dv)+(du−dv)2+4mumv2:uv∈E with equality if and only if G is a bipartite regular graph or a bipartite semiregular graph, where du and mu denote the degree of u and the average of the degrees of the vertices adjacent to u, respectively.