Sharp upper bounds on the spectral radius of the laplacian matrix of graphs

Let G = (V, E) be a simple connected graph with n vertices and e edges. Assume that the vertices are ordered such that d₁ ≥ d₂ ≥ ... ≥ d_n, where d_i is the degree of v_i for i = 1, 2,..., n and the average of the degrees of the vertices adjacent to v_i is denoted by m_i. Let m_max be the maximum of m_i's for i = 1, 2,..., n. Also, let ρ(G) denote the largest eigenvalue of the adjacency matrix and λ(G) denote the largest eigenvalue of the Laplacian matrix of a graph G. In this paper, we present a sharp upper bound on ρ(G): ρ(G) ≤ √ 2e - (n - 1)d_n + (dn - 1)m_max, with equality if and only if G is a star graph or G is a regular graph. In addition, we give two upper bounds for λ(G): 1. λ (G) ≤ { 2 + √∑_i=1ⁿd _i(d_i-1) - (1/2 ∑_i=1ⁿ d _i-1) (2d_n-2) + (2d_n-3) (2d₁-2), if d_n ≥ 2, 2 + √∑_i=1ⁿ d _i(d_i-1) - d₁ + 1, if d_n = 1, where the equality holds if and only if G is a regular bipartite graph or G is a star graph, respectively. 2. λ(G) ≤ d₁ + √d ₁²+4 [2e/n-1 + n-2/n-1 d₁ + (d₁ - d_n) (1-d₁/n-1)] m_max/2, with equality if and only if G is a regular bipartite graph.