A NEW UPPER BOUND ON THE LARGEST NORMALIZED LAPLACIAN EIGENVALUE

作者：Rojo Oscar^*; Soto Ricardo L

来源：Operators and Matrices, 2013, 7(2): 323-332.

DOI：10.7153/oam-07-19

• normalized Laplacian matrix
• Randic matrix
• upper bound
• largest eigenvalue
• subdominant eigenvalue

Let G be a simple undirected connected graph on n vertices. Suppose that the vertices of G are labelled 1,2, ... ,n. Let d(i) be the degree of the vertex i. The Randic matrix of G, denoted by R, is the n x n matrix whose (i,j)-entry is 1/root d(i)d(j) if the vertices i and j are adjacent and 0 otherwise. The normalized Laplacian matrix of G is L = I - R, where I is the n x n identity matrix. In this paper, by using an upper bound on the maximum modulus of the subdominant Randic eigenvalues of G, we obtain an upper bound on the largest eigenvalue of L. We also obtain an upper bound on the largest modulus of the negative Randic eigenvalues and, from this bound, we improve the previous upper bound on the largest eigenvalue of L.

• 出版日期2013-6