Eigenvalues of Transmission Graph Laplacians (original) (raw)

Relationship between scattering matrix and spectrum of quantum graphs

Transactions of the American Mathematical Society, 2010

We investigate the equivalence between spectral characteristics of the Laplace operator on a metric graph, and the associated unitary scattering operator. We prove that the statistics of level spacings, and moments of observations in the eigenbases coincide in the limit that all bond lengths approach a positive constant value.

Geometric properties of quantum graphs and vertex scattering matrices

Opuscula Mathematica, 2010

Differential operators on metric graphs are investigated. It is proven that vertex matching (boundary) conditions can be successfully parameterized by the vertex scattering matrix. Two new families of matching conditions are investigated: hyperplanar Neumann and hyperplanar Dirichlet conditions. Using trace formula it is shown that the spectrum of the Laplace operator determines certain geometric properties of the underlying graph.

Laplacian spread of graphs: Lower bounds and relations with invariant parameters

Linear Algebra and its Applications, 2015

The spread of an n × n complex matrix B with eigenvalues β 1 , β 2 ,. .. , β n is defined by s (B) = max i,j |β i − β j | , where the maximum is taken over all pairs of eigenvalues of B. Let G be a graph on n vertices. The concept of Laplacian spread of G is defined by the difference between the largest and the second smallest Laplacian eigenvalue of G. In this work, by combining old techniques of interlacing eigenvalues and rank 1 perturbation matrices new lower bounds on the Laplacian spread of graphs are deduced, some of them involving invariant parameters of graphs, as it is the case of the bandwidth, independence number and vertex connectivity.

On Distribution of Laplacian Eigenvalues of Graphs

2021

The work in this thesis concerns the investigation of eigenvalues of the Laplacian matrix, normalized Laplacian matrix, signless Laplacian matrix and distance signless Laplacian matrix of graphs. In Chapter 1, we present a brief introduction of spectral graph theory with some definitions. Chapter 222 deals with the sum of $ k $ largest Laplacian eigenvalues $ S_{k}(G) $ of graph $ G $ and Brouwer's conjecture. We obtain the upper bounds for $ S_{k}(G) $ for some classes of graphs and use them to verify Brouwer's conjecture for these classes of graphs. Also, we prove Brouwer's conjecture for more general classes of graphs. In Chapter 333, we investigate the Laplacian eigenvalues of graphs and the Laplacian energy conjecture for trees. We prove the Laplacian energy conjecture completely for trees of diameter $ 4 .Further,weprovethisconjectureforalltreeshavingatmost. Further, we prove this conjecture for all trees having at most .Further,weprovethisconjectureforalltreeshavingatmost \frac{9n}{25}-2 $ non-pendent vertices. Also, we obtain the sufficient conditions for the truth...

On spectral spread of generalized distance matrix of a graph

Linear and Multilinear Algebra, 2020

For a simple connected graph G, let D(G), T r(G), D L (G) and D Q (G), respectively be the distance matrix, the diagonal matrix of the vertex transmissions, distance Laplacian matrix and the distance signless Laplacian matrix of a graph G. The convex linear combinations D α (G) of T r(G) and D(G) is defined as D α (G) = αT r(G) + (1 − α)D(G), 0 ≤ α ≤ 1. As D 0 (G) = D(G), 2D 1 2 (G) = D Q (G), D 1 (G) = T r(G) and D α (G) − D β (G) = (α − β)D L (G), this matrix reduces to merging the distance spectral, distance Laplacian spectral and distance signless Laplacian spectral theories. Let ∂ 1 (G) ≥ ∂ 2 (G) ≥ • • • ≥ ∂ n (G) be the eigenvalues of D α (G) and let D α S(G) = ∂ 1 (G) −∂ n (G) be the generalized distance spectral spread of the graph G. In this paper, we obtain some bounds for the generalized distance spectral spread D α (G). We also obtain relation between the generalized distance spectral spread D α (G) and the distance spectral spread S D (G). Further, we obtain the lower bounds for D α S(G) of bipartite graphs involving different graph parameters and we characterize the extremal graphs for some cases. We also obtain lower bounds for D α S(G) in terms of clique number and independence number of the graph G and characterize the extremal graphs for some cases.

The Laplacian spectrum of graphs

The paper is essentially a survey of known results about the spectrum of the Laplacian matrix of graphs with special emphasis on the second smallest Laplacian eigenvalue λ 2 and its relation to numerous graph invariants, including connectivity, expanding properties, isoperimetric number, maximum cut, independence number, genus, diameter, mean distance, and bandwidth-type parameters of a graph. Some new results and generalizations are added.

A new spectral invariant for quantum graphs

Scientific Reports, 2021

The Euler characteristic i.e., the difference between the number of vertices |V| and edges |E| is the most important topological characteristic of a graph. However, to describe spectral properties of differential equations with mixed Dirichlet and Neumann vertex conditions it is necessary to introduce a new spectral invariant, the generalized Euler characteristic \chi _G:= |V|-|V_D|-|E|χG:=∣V∣−∣VD∣−∣E∣,withχ G : = | V | - | V D | - | E | , withχG:=∣V∣−∣VD∣−∣E∣,with|V_D|∣VD∣denotingthenumberofDirichletvertices.WedemonstratetheoreticallyandexperimentallythatthegeneralizedEulercharacteristic| V D | denoting the number of Dirichlet vertices. We demonstrate theoretically and experimentally that the generalized Euler characteristic∣VD∣denotingthenumberofDirichletvertices.WedemonstratetheoreticallyandexperimentallythatthegeneralizedEulercharacteristic\chi _GχGofquantumgraphsandmicrowavenetworkscanbedeterminedfromsmallsetsoflowesteigenfrequencies.Ifthetopologyofthegraphisknown,thegeneralizedEulercharacteristicχ G of quantum graphs and microwave networks can be determined from small sets of lowest eigenfrequencies. If the topology of the graph is known, the generalized Euler characteristicχGofquantumgraphsandmicrowavenetworkscanbedeterminedfromsmallsetsoflowesteigenfrequencies.Ifthetopologyofthegraphisknown,thegeneralizedEulercharacteristic\chi _GχGcanbeusedtodeterminethenumberofDirichletvertices.ThatmakesthegeneralizedEulercharacteristicχ G can be used to determine the number of Dirichlet vertices. That makes the generalized Euler characteristicχGcanbeusedtodeterminethenumberofDirichletvertices.ThatmakesthegeneralizedEulercharacteristic\chi _G$$ χ G a new powerful tool for studying of physical systems modeled by differential eq...

Bounds for the Generalized Distance Eigenvalues of a Graph

Symmetry, 2019

Let G be a simple undirected graph containing n vertices. Assume G is connected. Let D ( G ) be the distance matrix, D L ( G ) be the distance Laplacian, D Q ( G ) be the distance signless Laplacian, and T r ( G ) be the diagonal matrix of the vertex transmissions, respectively. Furthermore, we denote by D α ( G ) the generalized distance matrix, i.e., D α ( G ) = α T r ( G ) + ( 1 − α ) D ( G ) , where α ∈ [ 0 , 1 ] . In this paper, we establish some new sharp bounds for the generalized distance spectral radius of G, making use of some graph parameters like the order n, the diameter, the minimum degree, the second minimum degree, the transmission degree, the second transmission degree and the parameter α , improving some bounds recently given in the literature. We also characterize the extremal graphs attaining these bounds. As an special cases of our results, we will be able to cover some of the bounds recently given in the literature for the case of distance matrix and distance s...

On the Laplacian eigenvalue 222 of graphs

2019

Let GGG be a graph. The Laplacian matrix of GGG is L(G)=D(G)−A(G)L(G)=D(G)-A(G)L(G)=D(G)−A(G), where D(G)=diag(d(v1),ldots,d(vn))D(G)=diag(d(v_{1}),\ldots , d(v_{n}))D(G)=diag(d(v1),ldots,d(vn)) is a diagonal matrix and d(v)d(v)d(v) denotes the degree of the vertex vvv in GGG and A(G)A(G)A(G) is the adjacency matrix of GGG. Let G1G_1G1 and G2G_2G2 be two (unicyclic) graphs. We study the multiplicity of the Laplacian eigenvalue 222 of G=G1odotG2G=G_1\odot G_2G=G1odotG2 where the graphs G1G_1G1 or G2G_2G2 may have perfect matching and Laplacian eigenvalue 222 or not. We initiate the Laplacian characteristic polynomial of G1G_1G1, G2G_2G2 and G=G1odotG2G=G_1\odot G_2G=G1odotG2. It is also investigated that Laplacian eigenvalue 222 of G=G1odotG2G=G_1\odot G_2G=G_1odotG2 for some graphs G1G_1G1 and G2G_2G_2 under the conditions.

On distance Laplacian and distance signless Laplacian eigenvalues of graphs

Les Cahiers du GERAD, 2017

Let D(G), D L (G) = Diag(Tr) − D(G) and D Q (G) = Diag(Tr) + D(G) be, respectively, the distance matrix, the distance Laplacian matrix and the distance signless Laplacian matrix of graph G, where Diag(Tr) denotes the diagonal matrix of the vertex transmissions in G. The eigenvalues of D L (G) and D Q