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, d ₂ are the highest and the second highest degree, respectively. 2. λ₁ (G) = max ⌈√2(d_u²+d _um′_u): u ∈ V⌉ if and only if G is a bipartite regular graph, where m′_u = ∑_v{d _v-|N_u∩N_v|:uvE}, d_u denotes the degree of u and |N_u ∩ N_v| is the number of common neighbors of u and v. 3. λ₁(G) ≤ max ⌈(d _u+d_v)+√(d_u-d_v) ²+4m_um_v2: uv ∈ E⌉ with equality if and only if G is a bipartite regular graph or a bipartite semiregular 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.