A new upper bound on the largest normalized Laplacian eigenvalue

A New Upper Bound on the Largest Normalized Laplacian Eigenvalue

Abstract. Let G be a simple undirected connected graph on n vertices. Suppose that the vertices of G are labelled 1,2, . . . ,n. Let di be the degree of the vertex i. The Randić matrix of G , denoted by R, is the n× n matrix whose (i, j)−entry is 1 √ did 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× n identity matrix. ...

A sharp upper bound on the largest Laplacian eigenvalue of weighted graphs

We consider weighted graphs, where the edge weights are positive definite matrices. The Laplacian of the graph is defined in the usual way. We obtain an upper bound on the largest eigenvalue of the Laplacian and characterize graphs for which the bound is attained. The classical bound of Anderson and Morley, for the largest eigenvalue of the Laplacian of an unweighted graph follows as a special ...

The (normalized) Laplacian Eigenvalue of Signed Graphs

Abstract. A signed graph Γ = (G, σ) consists of an unsigned graph G = (V, E) and a mapping σ : E → {+,−}. Let Γ be a connected signed graph and L(Γ),L(Γ) be its Laplacian matrix and normalized Laplacian matrix, respectively. Suppose μ1 ≥ · · · ≥ μn−1 ≥ μn ≥ 0 and λ1 ≥ · · · ≥ λn−1 ≥ λn ≥ 0 are the Laplacian eigenvalues and the normalized Laplacian eigenvalues of Γ, respectively. In this paper, ...

A lower bound for the second largest Laplacian eigenvalue of weighted graphs

Ela a Lower Bound for the Second Largest Laplacian Eigenvalue of Weighted Graphs

, where deg(vi) is the sum of weights of all edges connected to vi. The signless Laplacian matrix Q(G) is defined by D(G) + A(G). We denote by 0 = λ1(G) ≤ λ2(G) ≤ · · · ≤ λn(G) the eigenvalues of L(G), and by μ1(G) ≤ μ2(G) ≤ · · · ≤ μn(G) the eigenvalues of Q(G). We order the degrees of the vertices of G as d1(G) ≤ d2(G) ≤ · · · ≤ dn(G). Various bounds for the Laplacian eigenvalues of unweighte...