Bounds for the Largest Laplacian Eigenvalue of Weighted Graphs

Bounds for the Largest Laplacian Eigenvalue of Weighted Graphs

Let G = (V, E) be simple graphs, as graphs which have no loops or parallel edges such that V is a finite set of vertices and E is a set of edges. A weighted graph is a graph each edge of which has been assigned to a square matrix called the weight of the edge. All the weightmatrices are assumed to be of same order and to be positive matrix. In this paper, by “weighted graph” we mean “a weighted...

Graphs with small second largest Laplacian eigenvalue

Let L(G) be the Laplacian matrix of G. In this paper, we characterize all of the connected graphs with second largest Laplacian eigenvalue no more than l; where l . = 3.2470 is the largest root of the equation μ3 − 5μ2 + 6μ − 1 = 0. Moreover, this result is used to characterize all connected graphs with second largest Laplacian eigenvalue no more than three. © 2013 Elsevier Ltd. All rights rese...

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...

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

On the bounds for the largest Laplacian eigenvalues of weighted graphs