Laplacian Eigenvalues Research Articles

Computing the Kirchhoff index of a family of phenylene chain networks

Abstract The Kirchhoff index is a fundamental topological metric that provides insights into the structural and electrical characteristics of networks. It is defined as the sum of resistance distances between all pairs of nodes, serving as a key factor in understanding the dynamics within networks. To investigate the impact of structural variations on the Kirchhoff index, we select a family of phenylene chain networks as our model and establish a methodology to explore the Kirchhoff index using the Laplacian spectrum. By analyzing the network structure, we introduce a parameter to control the number of iterations, providing a recursive relationship between the Laplacian matrix and its eigenvalues at intervals of generations. This approach enables the derivation of an analytical expression for both the sum of the reciprocals of all nonzero Laplacian eigenvalues and the Kirchhoff index.

Numerical optimisation of Dirac eigenvalues

Abstract Motivated by relativistic materials, we develop a numerical scheme to support existing or state new conjectures in the spectral optimisation of eigenvalues of the Dirac operator, subject to infinite-mass boundary conditions. We numerically study the optimality of the regular polygon (respectively, disk) among all polygons of a given number of sides (respectively, arbitrary sets), subject to area or perimeter constraints. We consider the three lowest positive eigenvalues and their ratios. Roughly, we find results analogous to known or expected for the Dirichlet Laplacian, except for the third eigenvalue which does not need to be minimised by the regular polygon (respectively, the disk) for all masses. In addition to the numerical results, a new, mass-dependent upper bound to the lowest eigenvalue in rectangles is proved and its extension to arbitrary quadrilaterals is conjectured.

A reverse Faber-Krahn inequality for the magnetic Laplacian

We consider the first eigenvalue of the magnetic Laplacian in a bounded and simply connected planar domain, with uniform magnetic field and Neumann boundary conditions. We investigate the reverse Faber-Krahn inequality conjectured by S. Fournais and B. Helffer, stating that this eigenvalue is maximized by the disk for a given area. Using the method of level lines, we prove the conjecture for small enough values of the magnetic field (those for which the corresponding eigenfunction in the disk is radial).

On comparison between the distance energies of a connected graph

Let G be a simple connected graph of order n having Wiener index W(G). The distance, distance Laplacian and the distance signless Laplacian energies of G are respectively defined asDE(G)=∑i=1n|υiD|,DLE(G)=∑i=1n|υiL−Tr‾|andDSLE(G)=∑i=1n|υiQ−Tr‾|, where υiD,υiL and υiQ,1≤i≤n are respectively the distance, distance Laplacian and the distance signless Laplacian eigenvalues of G and Tr‾=2W(G)n is the average transmission degree. In this paper, we will study the relation between DE(G), DLE(G) and DSLE(G). We obtain some necessary conditions for the inequalities DLE(G)≥DSLE(G),DLE(G)≤DSLE(G),DLE(G)≥DE(G) and DSLE(G)≥DE(G) to hold. We will show for graphs with one positive distance eigenvalue the inequality DSLE(G)≥DE(G) always holds. Further, we will show for the complete bipartite graphs the inequality DLE(G)≥DSLE(G)≥DE(G) holds. We end this paper by computational results on graphs of order at most 6.

Riemannian invariants for warped product submanifolds in Q ε m × R {{\mathbb{Q}}}_{\varepsilon }^{m}\times {\mathbb{R}} and their applications

Abstract This article investigates the geometric and topologic of warped product submanifolds in Riemannian warped product Q ε m × R {{\mathbb{Q}}}_{\varepsilon }^{m}\times {\mathbb{R}} . In this respect, we obtain the first Chen inequality that involves extrinsic invariants like the length of the warping functions and the mean curvature. This inequality involves two intrinsic invariants (sectional curvature and δ \delta -invariant). In addition, an integral bound is provided for the Bochner operator formula of compact warped product submanifolds in terms of the Ricci curvature gradient. We aim to apply this theory to many structures and obtain Dirichlet eigenvalues for problem applications. Some new results regarding the vanishing mean curvature are presented as a partial solution, and this can be considered for the well-known problem given by Chern.

Graphs whose Laplacian eigenvalues are almost all 1 or 2

Abstract We explicitly determine all connected graphs whose Laplacian matrices have at most four eigenvalues different from 1 and 2.

Resonant Solitary States in Complex Networks

Abstract Partially synchronized solitary states occur frequently when a synchronized system of networked oscillators with inertia is perturbed locally. Several asymptotic states of different frequencies can coexist at the same node.
Here, we reveal the mechanism behind this multistability: additional solitary frequencies arise from the coupling between network modes and the solitary oscillator's frequency, leading to significant energy transfer. This can cause the solitary node's frequency to resonate with a Laplacian eigenvalue. We analyze which network structures enable this resonance and explain longstanding numerical observations.
Another solitary state that is known in the literature is characterized by the effective decoupling of the synchronized network and the solitary node at the natural frequency. Our framework unifies the description of solitary states near and far from resonance, allowing to predict the behavior of complex networks from their topology.

Analysis of Graph Inverses and Algebraic Connectivity in a Systematic Way

an exhaustive and methodical study of algebraic connectedness and graph inverses. In many contexts, graph theory plays a crucial role, particularly when understanding intricate systems. This paper delves into the fundamental concepts of graph inverses and explains their significance for network analysis and connectivity evaluation. The Laplacian lattice's second-smallest eigenvalue, or algebraic connectivity µN−1, plays a crucial role in some features like network heartiness, synchronisation security, and diffusion processes. In this study, we focus on the algebraic connectedness in the network-of-networks (NoN), which is the general context of linked networks. A key component of science is the graph hypothesis. Algebraic graph hypothesis refers to the use of algebraic techniques to graph problems. This article concludes a focus on algebraic graph hypothesis and presents and examines a different algebraic and mathematical variety idea to regard as the greatest matching of an undirected graph.

A Distributed Algorithm for Reaching Average Consensus in Unbalanced Tree Networks

In this paper, a distributed algorithm for reaching average consensus is proposed for multi-agent systems with tree communication graph, when the edge weight distribution is unbalanced. First, the problem is introduced as a key topic of core algorithms for several modern scenarios. Then, the relative solution is proposed as a finite-time algorithm, which can be included in any application as a preliminary setup routine, and it is well-suited to be integrated with other adaptive setup routines, thus making the proposed solution useful in several practical applications. A special focus is devoted to the integration of the proposed method with a recent Laplacian eigenvalue allocation algorithm, and the implementation of the overall approach in a wireless sensor network framework. Finally, a worked example is provided, showing the significance of this approach for reaching a more precise average consensus in uncertain scenarios.

Spectra of the Dirichlet Laplacian in 3-dimensional polyhedral layers

The structure of the spectrum of the three-dimensional Dirichlet Laplacian in a 3D polyhedral layer of fixed width is studied. It turns out that the essential spectrum is determined by the smallest dihedral angle that forms the boundary of the layer while the discrete spectrum is always finite. An example of a layer with empty discrete spectrum is constructed. The spectrum is proved to be nonempty in regular polyhedral layer.

The first Dirichlet eigenvalue and the width

For a geodesic ball with non-negative Ricci curvature and mean convex boundary, it is known that the first Dirichlet eigenvalue of this geodesic ball has a sharp lower bound in term of its radius. We show a quantitative explicit inequality, which bounds the width of geodesic ball in terms of the spectral gap between the first Dirichlet eigenvalue and the corresponding sharp lower bound.

An improved spectral lower bound of treewidth

We show that for every n-vertex graph with at least one edge, its treewidth is greater than or equal to nλ2/(Δ+λ2)−1, where Δ and λ2 are the maximum degree and the second smallest Laplacian eigenvalue of the graph, respectively. This lower bound improves the one by Chandran and Subramanian [Inf. Process. Lett., 2003] and the subsequent one by the authors of the present paper [IEICE Trans. Inf. Syst., 2024]. The new lower bound is almost tight in the sense that there is an infinite family of graphs such that the lower bound is only 1 less than the treewidth for each graph in the family. Additionally, using similar techniques, we also present a lower bound of treewidth in terms of the largest and the second smallest Laplacian eigenvalues.

On the eigenvalues of the distance signless Laplacian matrix of graphs

Let G be a connected graph and let DQ(G) be the distance signless Laplacian matrix of G with eigenvalues ρ1≥ ρ2≥…≥ ρn. The spread of the matrix DQ}(G) is defined as s(DQ(G)) := maxi,j| ρi-ρj| = ρ1- ρn. We derive new bounds for the distance signless Laplacian spectral radius ρ1 of G. We establish a relationship between the distance signless Laplacian energy and the spread of DQ(G). For a real number α ≠ 0, the graph invariant mα (G) is the sum of the α -th power of the distance signless Laplacian eigenvalues of G. Finally, we obtain various bounds for the graph invariant mα(G).

Sharp lower bounds for the Laplacian Estrada index of graphs

Let G be a simple graph on n vertices, and let λ 1 , λ 2 , … , λ n be the Laplacian eigenvalues of G. The Laplacian Estrada index of G is defined as LEE ( G ) = ∑ i = 1 n e λ i . Consider a graph G with n ≥ 3 vertices, m edges, c connected components, and the largest Laplacian eigenvalue λ n . Let K n , S n , and K p , q (p + q = n) denote the complete graph, the star graph, and the complete bipartite graph on n vertices, respectively. In this paper, we establish that LEE ( G ) ≥ n e 2 m n + c + e λ n − ( c + 1 ) e λ n c + 1 . Furthermore, we show that the equality holds if and only if G ≅ K ¯ n (the complement of K n ), G ≅ ∪ i = 1 c − 1 K 1 ∪ S c + 1 if n = 2c, or G ≅ K n 2 , n 2 if G is a connected graph on an even number of vertices. As a consequence of this lower bound, we derive sharp lower bounds for the Laplacian Estrada index of a graph, considering its well-known graph parameters. This leads to improvements to some previously known lower bounds for the Laplacian Estrada index of a graph. Notably, we establish a sharp lower bound for the Laplacian Estrada index of a graph in terms of its maximum vertex degree. As an application, we demonstrate that the lower bound for the Laplacian Estrada index presented by Khosravanirad in [A Lower Bound for Laplacian Estrada Index of a Graph, MATCH Commun Math Comput Chem. 2013;70:175–180.] is not complete. Consequently, we provide a complete version of this lower bound.

Laplacian eigenvalue distribution for unicyclic graphs

Let G be a graph and let mG[0,1) denote the number of Laplacian eigenvalues of G in the interval [0,1). For a tree T with diameter d, Guo, Xue, and Liu proved that mT[0,1)≥(d+1)/3. In this paper, we provide a lower bound for mG[0,1) when G is a unicyclic graph, in terms of the diameter and girth of G. Moreover, for the lollipop graph, under certain conditions on its diameter and girth, we give a formula for the exact value of mG[0,1).

Net Laplacian eigenvalues of certain corona-like products of signed graphs

Net Laplacian eigenvalues of certain corona-like products of signed graphs

Dynamics of a susceptible-infected-recovered model on complex networks with interregional migration.

We present a susceptible-infected-recovered model based on a dynamic flow network that describes the epidemic process on complex metapopulation networks. This model views population regions as interconnected nodes and describes the evolution of each region using a system of differential equations. The next-generation matrix method is used to derive the global basic reproduction number for three cases: a general network with homogeneous infection rates in all regions, a fully connected network, and a star network with heterogeneous infection and recovery rates. For the homogeneous case, we show that this global basic reproduction number is independent of the migration rates between regions. However, the rate of convergence of each region to an equilibrium state exhibits a much larger variance in random (Erdős-Rényi) networks compared to small-scale (Barabási-Albert) networks. For the general heterogeneous case, we report interesting results, namely that the global basic reproduction number decays exponentially with respect to the smallest nonzero Laplacian eigenvalue (algebraic connectivity). Furthermore, we demonstrate both analytically and numerically that as the network's algebraic connectivity increases, either by increasing the average node degree of each region or the global migration rate, the global basic reproduction number decreases and converges to the ratio of the average local infection rate to the average local recovery rate, meaning that the lower bound of the global basic reproduction rate does not equal the mean of local basic reproduction rates.

Laplacian eigenvalue distribution, diameter and domination number of trees

For a graph G with domination number γ, Hedetniemi, Jacobs and Trevisan (2016) proved that m G [ 0 , 1 ) ≤ γ , where m G [ 0 , 1 ) means the number of Laplacian eigenvalues of G in the interval [ 0 , 1 ) . Let T be a tree with diameter d. In this paper, we show that m T [ 0 , 1 ) ≥ ( d + 1 ) / 3 . All trees achieving the lower bound are completely characterized. Moreover, we prove that the domination number of a tree is ( d + 1 ) / 3 if and only if it has exactly ( d + 1 ) / 3 Laplacian eigenvalues less than one. As an application, it also provides a new type of tree, which shows the sharpness of the inequality due to Hedetniemi, Jacobs and Trevisan.

The Role of Topology and Capacity in Some Bounds for Principal Frequencies

We prove a lower bound on the sharp Poincaré–Sobolev embedding constants for general open sets, in terms of their inradius. We consider the following two situations: planar sets with given topology; open sets in any dimension, under the restriction that points are not removable sets. In the first case, we get an estimate which optimally depends on the topology of the sets, thus generalizing a result by Croke, Osserman and Taylor, originally devised for the first eigenvalue of the Dirichlet–Laplacian. We also consider some limit situations, like the sharp Moser–Trudinger constant and the Cheeger constant. As a byproduct of our discussion, we also obtain a Buser-type inequality for open subsets of the plane, with given topology. An interesting problem on the sharp constant for this inequality is presented.

Inequalities for eigenvalues of the bi-drifting Laplacian on bounded domains in complete noncompact Riemannian manifolds and related results

Inequalities for eigenvalues of the bi-drifting Laplacian on bounded domains in complete noncompact Riemannian manifolds and related results