scholarly journals A note on distance spectral radius of trees

2017 ◽  
Vol 5 (1) ◽  
pp. 296-300
Author(s):  
Yanna Wang ◽  
Rundan Xing ◽  
Bo Zhou ◽  
Fengming Dong
Keyword(s):  
Spectral Radius ◽  
Connected Graph ◽  
Distance Matrix ◽  
Largest Eigenvalue ◽  
Maximal Distance ◽  
The Largest Eigenvalue

Abstract The distance spectral radius of a connected graph is the largest eigenvalue of its distance matrix. We determine the unique non-starlike non-caterpillar tree with maximal distance spectral radius.

Distance spectral radius of unicyclic graphs with fixed maximum degree

2019 ◽  
Vol 19 (04) ◽  
pp. 2050068
Author(s):  
Hezan Huang ◽  
Bo Zhou
Keyword(s):  
Spectral Radius ◽  
Connected Graph ◽  
Maximum Degree ◽  
Distance Matrix ◽  
The Other ◽  
Connected Graphs ◽  
Largest Eigenvalue ◽  
Unicyclic Graphs ◽  
The Largest Eigenvalue

The distance spectral radius of a connected graph is the largest eigenvalue of its distance matrix. For integers [Formula: see text] and [Formula: see text] with [Formula: see text], we prove that among the connected graphs on [Formula: see text] vertices of given maximum degree [Formula: see text] with at least one cycle, the graph [Formula: see text] uniquely maximizes the distance spectral radius, where [Formula: see text] is the graph obtained from the disjoint star on [Formula: see text] vertices and path on [Formula: see text] vertices by adding two edges, one connecting the star center with a path end, and the other being a chord of the star.

scholarly journals On graft transformations decreasing distance spectral radius of graphs

2021 ◽  
Author(s):  
Yanna Wang ◽  
Bo Zhou
Keyword(s):  
Spectral Radius ◽  
Connected Graph ◽  
Minimum Distance ◽  
Distance Matrix ◽  
Largest Eigenvalue ◽  
Unique Graph ◽  
The Largest Eigenvalue

The distance spectral radius of a connected graph  is the largest eigenvalue of its distance matrix. In this paper, we give several less restricted graft transformations that decrease the distance spectral radius, and determine the unique   graph   with   minimum   distance  spectral radius among homeomorphically irreducible unicylic graphs on $n\geq 6$ vertices, and the unique tree with minimum distance spectral radius among trees on $n$  vertices with given number of  vertices of degree two, respectively.

scholarly journals The Distance Laplacian Spectral Radius of Clique Trees

2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
Xiaoling Zhang ◽  
Jiajia Zhou
Keyword(s):  
Spectral Radius ◽  
Connected Graph ◽  
Minimum Distance ◽  
Distance Matrix ◽  
Laplacian Matrix ◽  
Diagonal Matrix ◽  
Laplacian Spectral Radius ◽  
Largest Eigenvalue ◽  
Clique Trees ◽  
The Largest Eigenvalue

The distance Laplacian matrix of a connected graph G is defined as ℒ G = Tr G − D G , where D G is the distance matrix of G and Tr G is the diagonal matrix of vertex transmissions of G . The largest eigenvalue of ℒ G is called the distance Laplacian spectral radius of G . In this paper, we determine the graphs with maximum and minimum distance Laplacian spectral radius among all clique trees with n vertices and k cliques. Moreover, we obtain n vertices and k cliques.

A note on a conjecture for the distance Laplacian matrix

2016 ◽  
Vol 31 ◽  
pp. 60-68 ◽  
Author(s):  
Celso Marques da Silva ◽  
Maria Aguieiras Alvarez de Freitas ◽  
Renata Raposo Del-Vecchio
Keyword(s):  
Connected Graph ◽  
Distance Matrix ◽  
Laplacian Matrix ◽  
Mathematical Journal ◽  
Laplacian Eigenvalue ◽  
Induced Subgraph ◽  
Largest Eigenvalue ◽  
The Largest Eigenvalue

In this note, the graphs of order n having the largest distance Laplacian eigenvalue of multiplicity n −2 are characterized. In particular, it is shown that if the largest eigenvalue of the distance Laplacian matrix of a connected graph G of order n has multiplicity n − 2, then G = S_n or G = K_(p,p), where n = 2p. This resolves a conjecture proposed by M. Aouchiche and P. Hansen in [M. Aouchiche and P. Hansen. A Laplacian for the distance matrix of a graph. Czechoslovak Mathematical Journal, 64(3):751–761, 2014.]. Moreover, it is proved that if G has P_5 as an induced subgraph then the multiplicity of the largest eigenvalue of the distance Laplacian matrix of G is less than n − 3.

scholarly journals Extremal properties of the distance spectral radius of hypergraphs

2020 ◽  
Vol 36 (36) ◽  
pp. 411-429
Author(s):  
Yanna Wang ◽  
Bo Zhou
Keyword(s):  
Spectral Radius ◽  
Minimum Distance ◽  
Distance Matrix ◽  
Maximum Distance ◽  
Matching Number ◽  
Largest Eigenvalue ◽  
Extremal Properties ◽  
The Largest Eigenvalue

The distance spectral radius of a connected hypergraph is the largest eigenvalue of its distance matrix. The unique hypertrees with minimum distance spectral radii are determined in the class of hypertrees of given diameter, in the class of hypertrees of given matching number, and in the class of non-hyperstar-like hypertrees, respectively. The unique hypergraphs with minimum and second minimum distance spectral radii are determined in the class of unicylic hypergraphs. The unique hypertree with maximum distance spectral radius is determined in the class of $k$-th power hypertrees of given matching number.

scholarly journals Distance spectral radius of series-reduced trees with parameters

2020 ◽  
Author(s):  
Yuyuan Deng ◽  
Dangui Li ◽  
Hongying Lin ◽  
Bo Zhou
Keyword(s):  
Spectral Radius ◽  
Connected Graph ◽  
Maximum Degree ◽  
Symmetric Matrix ◽  
Domination Number ◽  
Distance Matrix ◽  
Internal Vertex ◽  
Largest Eigenvalue ◽  
Real Symmetric Matrix ◽  
Fixed Order

For a connected graph $G$, the distance matrix is a real-symmetric matrix where the $(u,v)$-entry is the distance between vertex $u$ and vertex $v$ in $G$. The distance spectral radius of $G$ is the largest eigenvalue of the distance matrix. A series-reduced tree is a tree with at least one internal vertex and all internal vertices having degree at least three. Those series-reduced trees that maximize the distance spectral radius are determined over all series-reduced trees with fixed order and maximum degree and over all series-reduced trees with fixed order and domination number, respectively.

scholarly journals The normalized distance Laplacian

2021 ◽  
Vol 9 (1) ◽  
pp. 1-18
Author(s):  
Carolyn Reinhart
Keyword(s):  
Spectral Radius ◽  
Characteristic Polynomial ◽  
Adjacency Matrix ◽  
Connected Graph ◽  
Distance Matrix ◽  
Laplacian Matrix ◽  
Diagonal Matrix ◽  
The Matrix

Abstract The distance matrix 𝒟(G) of a connected graph G is the matrix containing the pairwise distances between vertices. The transmission of a vertex vi in G is the sum of the distances from vi to all other vertices and T(G) is the diagonal matrix of transmissions of the vertices of the graph. The normalized distance Laplacian, 𝒟𝒧(G) = I−T(G)−1/2 𝒟(G)T(G)−1/2, is introduced. This is analogous to the normalized Laplacian matrix, 𝒧(G) = I − D(G)−1/2 A(G)D(G)−1/2, where D(G) is the diagonal matrix of degrees of the vertices of the graph and A(G) is the adjacency matrix. Bounds on the spectral radius of 𝒟 𝒧 and connections with the normalized Laplacian matrix are presented. Twin vertices are used to determine eigenvalues of the normalized distance Laplacian. The distance generalized characteristic polynomial is defined and its properties established. Finally, 𝒟𝒧-cospectrality and lack thereof are determined for all graphs on 10 and fewer vertices, providing evidence that the normalized distance Laplacian has fewer cospectral pairs than other matrices.

Trees with given maximum degree minimizing the spectral radius

2016 ◽  
Vol 31 ◽  
pp. 335-361
Author(s):  
Xue Du ◽  
Lingsheng Shi
Keyword(s):  
Spectral Radius ◽  
Adjacency Matrix ◽  
Maximum Degree ◽  
Equal Length ◽  
Disjoint Paths ◽  
Common Vertex ◽  
Largest Eigenvalue ◽  
Large N ◽  
The Largest Eigenvalue

The spectral radius of a graph is the largest eigenvalue of the adjacency matrix of the graph. Let $T^*(n,\Delta ,l)$ be the tree which minimizes the spectral radius of all trees of order $n$ with exactly $l$ vertices of maximum degree $\Delta $. In this paper, $T^*(n,\Delta ,l)$ is determined for $\Delta =3$, and for $l\le 3$ and $n$ large enough. It is proven that for sufficiently large $n$, $T^*(n,3,l)$ is a caterpillar with (almost) uniformly distributed legs, $T^*(n,\Delta ,2)$ is a dumbbell, and $T^*(n,\Delta ,3)$ is a tree consisting of three distinct stars of order $\Delta $ connected by three disjoint paths of (almost) equal length from their centers to a common vertex. The unique tree with the largest spectral radius among all such trees is also determined. These extend earlier results of Lov\' asz and Pelik\'an, Simi\' c and To\u si\' c, Wu, Yuan and Xiao, and Xu, Lin and Shu.

scholarly journals The Perturbation Bound for the Spectral Radius of a Nonnegative Tensor

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Wen Li ◽  
Michael K. Ng
Keyword(s):  
Spectral Radius ◽  
Perturbation Analysis ◽  
Absolute Difference ◽  
Nonnegative Tensor ◽  
Backward Error ◽  
Perturbation Bound ◽  
Error Matrix ◽  
Largest Eigenvalue ◽  
The Stability ◽  
The Largest Eigenvalue

We study the perturbation bound for the spectral radius of an mth-order n-dimensional nonnegative tensor A. The main contribution of this paper is to show that when A is perturbed to a nonnegative tensor A~ by ΔA, the absolute difference between the spectral radii of A and A~ is bounded by the largest magnitude of the ratio of the ith component of ΔAxm-1 and the ith component xm-1, where x is an eigenvector associated with the largest eigenvalue of A in magnitude and its entries are positive. We further derive the bound in terms of the entries of A only when x is not known in advance. Based on the perturbation analysis, we make use of the NQZ algorithm to estimate the spectral radius of a nonnegative tensor in general. On the other hand, we study the backward error matrix ΔA and obtain its smallest error bound for its perturbed largest eigenvalue and associated eigenvector of an irreducible nonnegative tensor. Based on the backward error analysis, we can estimate the stability of computation of the largest eigenvalue of an irreducible nonnegative tensor by the NQZ algorithm. Numerical examples are presented to illustrate the theoretical results of our perturbation analysis.

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