On graphs with distance Laplacian eigenvalues of multiplicity n−4 (original) (raw)

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...

Graphs with few distinct distance eigenvalues irrespective of the diameters

The Electronic Journal of Linear Algebra, 2015

The distance matrix of a simple connected graph GGG is D(G)=(dij)D(G)=(d_{ij})D(G)=(dij), where dijd_{ij}dij is the distance between iiith and jjjth vertices of GGG. The multiset of all eigenvalues of D(G)D(G)D(G) is known as the distance spectrum of GGG. Lin et al.(On the distance spectrum of graphs. \newblock {\em Linear Algebra Appl.}, 439:1662-1669, 2013) asked for existence of graphs other than strongly regular graphs and some complete kkk-partite graphs having exactly three distinct distance eigenvalues. In this paper some classes of graphs with arbitrary diameter and satisfying this property is constructed. For each kin4,5,ldots,11k\in \{4,5,\ldots,11\}kin4,5,ldots,11 families of graphs that contain graphs of each diameter grater than k−1k-1k−1 is constructed with the property that the distance matrix of each graph in the families has exactly kkk distinct eigenvalues. While making these constructions we have found the full distance spectrum of square of even cycles, square of hypercubes, corona of a transmission regular graph with $K_...

The Laplacian spectral radius of some graphs

Linear Algebra and its Applications, 2009

The Laplacian spectral radius of a graph is the largest eigenvalue of the associated Laplacian matrix. In this paper, we determine those graphs which maximize the Laplacian spectral radius among all bipartite graphs with (edge-)connectivity at most k. We also characterize graphs of order n with k cut-edges, having Laplacian spectral radius equal to n.

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

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.

Eigenvalues, diameter, and mean distance in graphs

Graphs and Combinatorics, 1991

It is well-known that the second smallest eigenvalue 22 of the difference Laplacian matrix of a graph G is related to the expansion properties of G. A more detailed analysis of this relation is given. Upper and lower bounds on the diameter and the mean distance in G in terms of 22 are derived.

Laplacian and Normalized Laplacian Spectral Distances of Graphs

Suppose 1 and 2 are −vertex graphs with Laplacian and normalized Laplacian eigenvalues () and (), = 1, 2 and 1 ≤ ≤ , respectively. We also assume that 1() ≤ 2() ≤ ⋅ ⋅ ⋅ ≤ (), 1() ≤ 2() ≤ ⋅ ⋅ ⋅ ≤ (). The Laplacian and normalized Laplacian spectral distances between 1 and 2 are defined as follows: (1, 2) = =1 ∣(1) − (2)∣ and ℓ(1, 2) = =1 ∣(1) − (2)∣. In this paper, we compute the Laplacian and normalized Laplacian spectral distances of some particular classes of graphs.

The Laplacian spectral radius of graphs

Czechoslovak Mathematical Journal, 2010

On Generalized Distance Spectral Radius of a Bipartite Graph

2020

For a simple connected graph G, let D(G), Tr(G), D(G) and D(G) respectively be the distance matrix, the diagonal matrix of the vertex transmissions, the distance Laplacian matrix and the distance signless Laplacian matrix of a graph G. The convex linear combination Dα(G) of Tr(G) and D(G) is defined as Dα(G) = αTr(G) + (1 − α)D(G), 0 ≤ α ≤ 1. As D0(G) = D(G), 2D 1 2 (G) = D(G), D1(G) = Tr(G), this matrix reduces to merging the distance spectral, signless distance Laplacian spectral theories. In this paper, we study the spectral radius of the generalized distance matrix Dα(G) of a graph G. We obtain bounds for the generalized distance spectral radius of a bipartite graph in terms of various parameters associated with the structure of the graph and characterize the extremal graphs. For α = 0, our results improve some previously known bounds.

On the second largest Laplacian eigenvalues of graphs

Linear Algebra and its Applications, 2013

The second largest Laplacian eigenvalue of a graph is the second largest eigenvalue of the associated Laplacian matrix. In this paper, we study extremal graphs for the extremal values of the second largest Laplacian eigenvalue and the Laplacian separator of a connected graph, respectively. All simple connected graphs with second largest Laplacian eigenvalue at most 3 are characterized. It is also shown that graphs with second largest Laplacian eigenvalue at most 3 are determined by their Laplacian spectrum. Moreover, the graphs with maximum and the second maximum Laplacian separators among all connected graphs are determined.

On graphs with distance Laplacian eigenvalues of multiplicity n−4 (original) (raw)

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