On distance Laplacian and distance signless Laplacian eigenvalues of graphs

The distance signless Laplacian matrix of a connected graph [Formula: see text] is defined as [Formula: see text], where [Formula: see text] is the distance matrix of [Formula: see text] and [Formula: see text] is the diagonal matrix of vertex transmissions of [Formula: see text]. If [Formula: see text] are the distance signless Laplacian eigenvalues of a simple graph [Formula: see text] of order [Formula: see text] then we put forward the graph invariants [Formula: see text] and [Formula: see text] for the sum of [Formula: see text]-largest and the sum of [Formula: see text]-smallest distance signless Laplacian eigenvalues of a graph [Formula: see text], respectively. We obtain lower bounds for the invariants [Formula: see text] and [Formula: see text]. Then, we present some inequalities between the vertex transmissions, distance eigenvalues, distance Laplacian eigenvalues, and distance signless Laplacian eigenvalues of graphs. Finally, we give some new results and bounds for the distance signless Laplacian energy of graphs.

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Bounds for peripheral distance signless Laplacian eigenvalues of graphs

Asian-European Journal of Mathematics ◽

10.1142/s1793557120501132 ◽

2019 ◽

Vol 13 (06) ◽

pp. 2050113 ◽

Cited By ~ 1

Author(s):

Abdollah Alhevaz ◽

Maryam Baghipur ◽

Somnath Paul ◽

H. S. Ramane

Keyword(s):

Connected Graph ◽

Laplacian Matrix ◽

Maximum Distance ◽

Laplacian Eigenvalues ◽

Signless Laplacian Matrix ◽

Signless Laplacian ◽

Eigenvalues Of Graphs

The eccentricity of a vertex [Formula: see text] in a graph [Formula: see text] is the maximum distance between [Formula: see text] and any other vertex of [Formula: see text] A vertex with maximum eccentricity is called a peripheral vertex. In this paper, we study the distance signless Laplacian matrix of a connected graph [Formula: see text] with respect to peripheral vertices and define the peripheral distance signless Laplacian matrix of a graph [Formula: see text], denoted by [Formula: see text]. We then give some bounds on various eigenvalues of [Formula: see text] Moreover, we define energy in terms of [Formula: see text] and give some bounds on the energy.

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The sum of the first two largest signless Laplacian eigenvalues of trees and unicyclic graphs

Electronic Journal of Linear Algebra ◽

10.13001/1081-3810.3405 ◽

2019 ◽

Vol 35 ◽

pp. 449-467

Author(s):

Zhibin Du

Keyword(s):

Unicyclic Graph ◽

Unicyclic Graphs ◽

Laplacian Eigenvalues ◽

Signless Laplacian ◽

Pendent Vertices ◽

Eigenvalues Of Graphs

Let $G$ be a graph on $n$ vertices with $e(G)$ edges. The sum of eigenvalues of graphs has been receiving a lot of attention these years. Let $S_2 (G)$ be the sum of the first two largest signless Laplacian eigenvalues of $G$, and define $f(G) = e (G) +3 - S_2 (G)$. Oliveira et al. (2015) conjectured that $f(G) \geqslant f(U_{n})$ with equality if and only if $G \cong U_n$, where $U_n$ is the $n$-vertex unicyclic graph obtained by attaching $n-3$ pendent vertices to a vertex of a triangle. In this paper, it is proved that $S_2(G) < e(G) + 3 -\frac{2}{n}$ when $G$ is a tree, or a unicyclic graph whose unique cycle is not a triangle. As a consequence, it is deduced that the conjecture proposed by Oliveira et al. is true for trees and unicyclic graphs whose unique cycle is not a triangle.

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