Smallest Eigenvalue Of Laplacian Research Articles

Searching for Key Cycles in a Complex Network.

Searching for key nodes and edges in a network is a long-standing problem. Recently cycle structure in a network has received more attention. Is it possible to propose a ranking algorithm for cycle importance? We address the problem of identifying the key cycles of a network. First, we provide a more concrete definition of importance-in terms of Fiedler value (the second smallest Laplacian eigenvalue). Key cycles are those that contribute most substantially to the dynamical behavior of the network. Second, by comparing the sensitivity of Fiedler value to different cycles, a neat index for ranking cycles is provided. Numerical examples are given to show the effectiveness of this method.

Algebraic connectivity of the second power of a graph

AbstractDenote the Laplacian of a graph by and its second smallest Laplacian eigenvalue by . If is a graph on vertices, then it is shown that the second smallest eigenvalue of is at least 1, where is the complement of the second power of . As a corollary of this result, it is shown that where is the number of vertices of eccentricity at least 3 in .

Superdiffusion criteria on duplex networks.

Diffusion processes widely exist in nature. Some recent papers concerning diffusion processes focus their attention on multiplex networks. Superdiffusion, a phenomenon by which diffusion processes converge to equilibrium faster on multiplex networks than on single networks in isolation, may emerge because diffusion can occur both within and across layers. Some studies have shown that the emergence of superdiffusion depends on the topology of multiplex networks if the interlayer diffusion coefficient is large enough. This paper proposes some superdiffusion criteria relating to the Laplacian matrices of the two layers and provides a construction mechanism for generating a superdiffusible two-layered network. The method we proposed can be used to guide the discovery and construction of superdiffusible multiplex networks without calculating the second smallest Laplacian eigenvalues.

On the lower bound of the sum of the algebraic connectivity of a graph and its complement

For a graph G, let μ2(G) denote its second smallest Laplacian eigenvalue. It was conjectured that μ2(G)+μ2(G‾)≥1, where G‾ is the complement of G. This conjecture has been proved for various families of graphs. Here, we prove this conjecture in the general case. Also, we will show that max⁡{μ2(G),μ2(G‾)}≥1−O(n−13), where n is the number of vertices of G.

An improved result on Laplacian spectral ratio of connected graphs

The Laplacian spectral ratio of a connected graph G, denoted by r L (G), is defined as the quotient between the largest and second smallest Laplacian eigenvalues of G. In 2002, Barahona and Pecora showed that rL(G) play an important role in the network synchronization control. In this paper, we obtain a result on the Laplacian spectral ratio of trees, which improve the known result of You and Liu [Z. You, B. Liu, On the Laplacian spectral ratio of connected graphs, Appl. Math. Lett. 25(2012)1245-1250]. Moreover, some graph operations on Laplacian spectral ratio are given.

A note on the algebraic connectivity of a graph and its complement

For a graph G, let denote its second smallest Laplacian eigenvalue. It was conjectured that , where is the complement of G. In this paper, it is shown that .

Some results on the Laplacian Spread Conjecture

For a graph G of order n, let λ2(G) denote its second smallest Laplacian eigenvalue. It was conjectured that λ2(G)+λ2(G‾)≥1, where G‾ is the complement of G. For any x∈Rn, let ∇x∈R(n2) be the vector whose {i,j}-th entry is |xi−xj|. In this paper, we show the aforementioned conjecture is equivalent to prove that every two orthonormal vectors f,g∈Rn with zero mean satisfy‖∇f−∇g‖2≥2. In this article, it is shown that for the validity of the conjecture it suffices to prove that the conjecture holds for all permutation graphs.

The algebraic connectivity of a graph and its complement

For a graph G, let λ2(G) denote its second smallest Laplacian eigenvalue. It was conjectured that λ2(G)+λ2(G‾)≥1, where G‾ is the complement of G. In this paper, it is shown that max⁡{λ2(G),λ2(G‾)}≥25.

Minimal extremal graphs for addition of algebraic connectivity and independence number of connected graphs

Let G = (V,E) be a simple connected graph. Denote by D(G) the diagonal matrix of its vertex degrees and by A(G) its adjacency matrix. Then the Laplacian matrix of graph G is L(G) = D(G)-A(G). Let a(G) and ?(G), respectively, be the second smallest Laplacian eigenvalue and the independence number of graph G. In this paper, we characterize the extremal graph with second minimum value for addition of algebraic connectivity and independence number among all connected graphs with n ? 6 vertices (Actually, we can determine the p-th minimum value of a(G)+ ?(G) under certain condition when p is small). Moreover, we present a lower bound to the addition of algebraic connectivity and radius of connected graphs.

Spectral integral variation and unicyclic 3-colored digraphs with second smallest eigenvalue 1

In this article, the spectral integral variation of weighted directed graphs occurring in one place by adding a weighted directed edge is studied. Our results extend those of So (1999) [13] for unweighted undirected graphs and Fan (2003) [6] for mixed graphs. We characterize the unicyclic 3-colored digraphs that have 1 as the second smallest Laplacian eigenvalue. Finally, as an application we characterize the unicyclic 3-colored digraphs for which spectral integral variation occurs in one place by adding an edge of color red and where the changed eigenvalue is the second smallest Laplacian eigenvalue.

On the Fiedler value of large planar graphs (Extended abstract)

Abstract The Fiedler value λ 2 , also known as algebraic connectivity, is the second smallest Laplacian eigenvalue of a graph. We study upper bounds on the Fiedler value of planar graphs. Let λ 2 max be the maximum Fiedler value among all planar graphs G with n vertices. We show here the bounds 2 + Θ ( 1 n 2 ) ⩽ λ 2 max ⩽ 2 + O ( 1 n ) . Similar lower and upper bounds on the maximum Fiedler value are obtained for the classes of bipartite planar graphs, bipartite planar graphs with minimum vertex degree 3 and outerplanar graphs. We also derive almost tight bounds on λ 2 max for the classes of graphs of bounded genus and K h -minor-free graphs. The proofs rely on a result of Spielman and Teng stating that λ 2 ⩽ 8 Δ n for any planar graph with n vertices and maximum vertex degree Δ, on a result of Kelner stating that λ 2 = O ( g n ) for genus g graphs of bounded degree, and on the separator theorem.

On trees with Laplacian eigenvalue one

Let T be a tree, and L(T) be its Laplacian matrix. Let p(T), q(T) be the number of pendant and quasipendant (adjacent to a pendant) vertices of T, respectively. We show that given any tree on n>1 vertices and any integer vector y of size n, there is a larger tree T ′ with p(T ′) = q(T ′) and a nonzero integer vector y′ such that L(T ′)y′ =y′ and y′ restricted to T is y. Let mT (1) denote the multiplicity of the Laplacian eigenvalue 1 of the tree T. Let T be a tree obtained by adding an edge between a vertex of a tree T 1 and a vertex of a tree T 2. We discuss the relationship among and mT (1) and give a condition under which which generalizes a known result of Grone, R. Merris, R. Sunder, VS. 1990. The Laplacian spectrum of a graph. SIAM J. Matrix Anal. Appl. 11(2) 218–238. We also give a sufficient condition under which . We give a complete characterization of trees that have 1 as the third smallest Laplacian eigenvalue. As an application we characterize trees in which spectral integral variation occurs in one place by adding an edge where the changed eigenvalue is the third smallest eigenvalue.

On Maximizing the Second Smallest Eigenvalue of a State-Dependent Graph Laplacian

We consider the set G consisting of graphs of fixed order and weighted edges. The vertex set of graphs in G will correspond to point masses and the weight for an edge between two vertices is a functional of the distance between them. We pose the problem of finding the best vertex positional configuration in the presence of an additional proximity constraint, in the sense that, the second smallest eigenvalue of the corresponding graph Laplacian is maximized. In many recent applications of algebraic graph theory in systems and control, the second smallest eigenvalue of Laplacian has emerged as a critical parameter that influences the stability and robustness properties of dynamic systems that operate over an information network. Our motivation in the present work is to assign this Laplacian eigenvalue when relative positions of various elements dictate the interconnection of the underlying weighted graph. In this venue, one would then be able to synthesize information graphs that have desirable system theoretic properties.

Isoperimetric properties of weighted graphs related to the Laplacian spectrum and canonical correlations

The relation between isoperimetric properties and Laplacian spectra of weighted graphs is investigated. The vertices are classified into k clusters with „few" inter-cluster edges of „small" weights (area) and „similar" cluster sizes (volumes). For k=2 the Cheeger constant represents the minimum requirement for the area/volume ratio and it is estimated from above by v?1(2-?1), where ?1 is the smallest positive eigenvalue of the weighted Laplacian. For k?2 we define the k-density of a weighted graph that is a generalization of the Cheeger constant and estimated from below by Si=1k-1?i and from above by c2 Si=1k-1 ?i, where 0<?1=…=Sk-1 are the smallest Laplacian eigenvalues and the constant c?1 depends on the metric classification properties of the corresponding eigenvectors. Laplacian spectra are also related to canonical correlations in a probabilistic setup.

Lower bounds on the third smallest laplacian eigenvalue of a graph

We introduce a new graph-theoretic invariant ω(G) for a simple graph G, and relate it to the third smallest Laplace eigenvalue of G.