In this paper, we investigate how the smallest signless Laplacian eigenvalue of a graph behaves when the graph is perturbed by deleting a vertex, subdividing edges or moving edges.

Let B(n, g) be the class of bicyclic graphs on n vertices with girth g. In this paper, the graphs in B(n, g) with the largest signless Laplacian spectral radius are characterized.

Let G be a k-degenerate graph of order n. It is well-known that G has no more edges than Sn,k, the join of a complete graph of order k and an independent set of order n−k. In this note, it is shown that Sn,k is extremal for some spectral parameters of G as well. More precisely, letting μ (H) and q (H) denote the largest eigenvalues of the adjacency matrix and the signless Laplacian of a graph H...

In this paper, we develop a regularization framework for image deblurring based on a new definition of the normalized graph Laplacian. We apply a fast scaling algorithm to the kernel similarity matrix to derive the symmetric, doubly stochastic filtering matrix from which the normalized Laplacian matrix is built. We use this new definition of the Laplacian to construct a cost function consisting...

Abstract. Let G be a k-degenerate graph of order n. It is well-known that G has no more edges than Sn,k, the join of a complete graph of order k and an independent set of order n−k. In this note, it is shown that Sn,k is extremal for some spectral parameters of G as well. More precisely, letting μ (H) and q (H) denote the largest eigenvalues of the adjacency matrix and the signless Laplacian of...

Spectral graph theory gives an algebraical approach to analyze the dynamics of a network by using the matrix that represents the network structure. However, it is not easy for social networks to apply the spectral graph theory because the matrix elements cannot be given exactly to represent the structure of a social network. The matrix element should be set on the basis of the relationship betw...

The spectrum of the normalized Laplacian matrix cannot determine the number of edges in a graph, however finding constructions of cospectral graphs with differing number of edges has been elusive. In this paper we use basic properties of twins and scaling to show how to construct such graphs. We also give examples of families of graphs which are cospectral with a subgraph for the normalized Lap...

The eccentricity of a vertex is the maximum distance from it to another vertex and the average eccentricity ecc (G) of a graph G is the mean value of eccentricities of all vertices of G. The harmonic index H (G) of a graph G is defined as the sum of 2 di+dj over all edges vivj of G, where di denotes the degree of a vertex vi in G. In this paper, we determine the unique tree with minimum average...

, 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...