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. if and only if G is a bipartite regular graph, where , du denotes the degree of u and Nu∩Nv is the number of common neighbors of u and v.3. 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.