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 ≥ . . . ≥ dn, where di is the degree of vi for i = 1, 2, . . . , n and the average of the degrees of the vertices adjacent to vi is denoted by mi. Let mmax be the maximum of mi’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)dn + (dn − 1)mmax, 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 + √∑n i=1di(di−1)− ( 1 2 ∑n i=1 di−1 ) (2dn−2)+(2dn−3)(2d1−2), if dn ≥ 2, 2 + √∑n i=1 di(di − 1) − d1 + 1, if dn = 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 + √ d1 + 4 [ 2e n−1 + n−2 n−1d1 + (d1 − dn) ( 1 − d1 n−1 )] mmax 2 , with equality if and only if G is a regular bipartite graph.