Remoteness and distance eigenvalues of a graph

Lin, Huiqiu; Das, Kinkar Ch.; Wu, Baoyindureng

doi:10.1016/j.dam.2016.07.018

Cited by 19 publications

(6 citation statements)

References 11 publications

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“…The spectrum of distance matrix, arose from a data communication problem studied in [7] by Graham and Pollack in 1971, has been studied extensively (see [1,9,10,11,12,13,22]). The eigenvalues, eigenvectors and spectrum of D(G) are the D-eigenvalues, D-eigenvectors and D-spectrum of G, respectively.…”

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confidence: 99%

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

Liu

Shu

Xue

2018

ELA

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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.

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confidence: 99%

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

Liu

Shu

Xue

2018

ELA

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“…Furthermore, they also proposed two conjectures. Lin et al [ 12 ] confirmed these two conjectures. They also gave lower bounds on and when and the extremal graphs were characterized.…”

Section: Introductionmentioning

confidence: 75%

Remoteness and distance, distance (signless) Laplacian eigenvalues of a graph

Jia

Song

2018

J Inequal Appl

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“…Recently, the distance matrix of a graph has received increasing attention. Aouchiche and Hansen [2] and Lin, Das and Wu [12] proved some results on the relations between the distance eigenvalues and some graphic parameters. Lin [11] proved an upper bound on the least distance eigenvalue of a graph in terms of it order and diameter.…”

Section: Introductionmentioning

confidence: 94%