Let G be a simple graph of order n. The matrix L(G) = D (G) - A (G) is called the Laplacian matrix of G, where D (G) and A (G) denote the diagonal matrix of vertex degrees and the adjacency matrix of G, respectively. Let l(1) (G), l(n-1) (G) be the largest eigenvalue, the second smallest eigenvalue of L (G) respectively, and lambda(1) (G) be the largest eigenvalue of A (G). In this paper, we will present sharp upper and lower bounds for l(1) (G) and l(n-1) (G). Moreover, we investigate the relation between I-1 (G) and lambda(1) (G).
展开▼
