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 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.

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 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.

On the Largest Distance (Signless Laplacian) Eigenvalue of Non-transmission-regular Graphs

2018 ◽  
Vol 34 ◽  
pp. 459-471 ◽  
Author(s):  
Shuting Liu ◽  
Jinlong Shu ◽  
Jie Xue
Keyword(s):  
Spectral Radius ◽  
Regular Graph ◽  
Connected Graph ◽  
Distance Matrix ◽  
Laplacian Matrix ◽  
Wiener Index ◽  
Regular Graphs ◽  
Laplacian Eigenvalue ◽  
Signless Laplacian ◽  
Sum Of Distances

Let $G=(V(G),E(G))$ be a $k$-connected graph with $n$ vertices and $m$ edges. Let $D(G)$ be the distance matrix of $G$. Suppose $\lambda_1(D)\geq \cdots \geq \lambda_n(D)$ are the $D$-eigenvalues of $G$. The transmission of $v_i \in V(G)$, denoted by $Tr_G(v_i)$ is defined to be the sum of distances from $v_i$ to all other vertices of $G$, i.e., the row sum $D_{i}(G)$ of $D(G)$ indexed by vertex $v_i$ and suppose that $D_1(G)\geq \cdots \geq D_n(G)$. The $Wiener~ index$ of $G$ denoted by $W(G)$ is given by $W(G)=\frac{1}{2}\sum_{i=1}^{n}D_i(G)$. Let $Tr(G)$ be the $n\times n$ diagonal matrix with its $(i,i)$-entry equal to $TrG(v_i)$. The distance signless Laplacian matrix of $G$ is defined as $D^Q(G)=Tr(G)+D(G)$ and its spectral radius is denoted by $\rho_1(D^Q(G))$ or $\rho_1$. A connected graph $G$ is said to be $t$-transmission-regular if $Tr_G(v_i) =t$ for every vertex $v_i\in V(G)$, otherwise, non-transmission-regular. In this paper, we respectively estimate $D_1(G)-\lambda_1(G)$ and $2D_1(G)-\rho_1(G)$ for a $k$-connected non-transmission-regular graph in different ways and compare these obtained results. And we conjecture that $D_1(G)-\lambda_1(G)>\frac{1}{n+1}$. Moreover, we show that the conjecture is valid for trees.

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.

Ordering cacti with signless Laplacian spread

2018 ◽  
Vol 34 ◽  
pp. 609-619 ◽  
Author(s):  
Zhen Lin ◽  
Shu-Guang Guo
Keyword(s):  
Connected Graph ◽  
Laplacian Matrix ◽  
Largest Eigenvalue ◽  
Smallest Eigenvalue ◽  
Signless Laplacian Matrix ◽  
Signless Laplacian ◽  
The Difference ◽  
The Largest Eigenvalue

A cactus is a connected graph in which any two cycles have at most one vertex in common. The signless Laplacian spread of a graph is defined as the difference between the largest eigenvalue and the smallest eigenvalue of the associated signless Laplacian matrix. In this paper, all cacti of order n with signless Laplacian spread greater than or equal to n − 1/2 are determined.

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.

scholarly journals The Laplacian Spread of Tricyclic Graphs

10.37236/169 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Yanqing Chen ◽  
Ligong Wang
Keyword(s):  
Laplacian Matrix ◽  
Largest Eigenvalue ◽  
Smallest Eigenvalue ◽  
Fixed Order ◽  
Tricyclic Graphs ◽  
The Difference ◽  
The Largest Eigenvalue

The Laplacian spread of a graph is defined to be the difference between the largest eigenvalue and the second smallest eigenvalue of the Laplacian matrix of the graph. In this paper, we investigate Laplacian spread of graphs, and prove that there exist exactly five types of tricyclic graphs with maximum Laplacian spread among all tricyclic graphs of fixed order.

scholarly journals On the Ger\v{s}gorin disks of distance matrices of graphs

2021 ◽  
Vol 37 ◽  
pp. 709-717
Author(s):  
Mustapha Aouchiche ◽  
Bilal A. Rather ◽  
Issmail El Hallaoui
Keyword(s):  
Spectral Radius ◽  
Connected Graph ◽  
Infinite Family ◽  
Distance Matrix ◽  
Laplacian Matrix ◽  
Negative Answer ◽  
Diagonal Matrix ◽  
Signless Laplacian Matrix ◽  
Signless Laplacian ◽  
Simple Connected Graph

For a simple connected graph $G$, let $D(G)$, $Tr(G)$, $D^{L}(G)=Tr(G)-D(G)$, and $D^{Q}(G)=Tr(G)+D(G)$ be the distance matrix, the diagonal matrix of the vertex transmissions, the distance Laplacian matrix, and the distance signless Laplacian matrix of $G$, respectively. Atik and Panigrahi [2] suggested the study of the problem: Whether all eigenvalues, except the spectral radius, of $ D(G) $ and $ D^{Q}(G) $ lie in the smallest Ger\v{s}gorin disk? In this paper, we provide a negative answer by constructing an infinite family of counterexamples.

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