scholarly journals Edge bipartiteness and signless Laplacian spread of graphs

2012 ◽  
Vol 6 (1) ◽  
pp. 31-45 ◽  
Author(s):  
Yi-Zheng Fan ◽  
Shaun Fallat
Keyword(s):  
Connected Graph ◽  
Eigenvalue Bounds ◽  
Signless Laplacian

Let G be a connected graph, and let ?b(G) and SQ(G) be the edge bipartiteness and the signless Laplacian spread of G, respectively. We establish some important relationships between ?b(G) and SQ(G), and prove SQ(G)?2 (1+cos?/n) with equality if and only if G = Pn or G = Cn in case of odd n. In addition, we show that if G?Pn or G?C2k+1; then SQ(G)?4; with equality if and only if G is one of the following graphs: K1,3, K4, two triangles connected by an edge, and Cn for even n. As a consequence, we prove a conjecture of Cvetkovic, Rowlinson and Simic on minimal signless Laplacian spread [Eigenvalue bounds for the signless Laplacian, Publ. Inst. Math. (Beograd), 81(95)(2007), 11-27].

scholarly journals Eigenvalue bounds for the signless laplacian

2007 ◽  
pp. 11-27 ◽  
Author(s):  
Dragos Cvetkovic ◽  
Peter Rowlinson ◽  
Slobodan Simic
Keyword(s):  
Previous Survey ◽  
Eigenvalue Bounds ◽  
Largest Eigenvalue ◽  
Signless Laplacian

We extend our previous survey of properties of spectra of signless Laplacians of graphs. Some new bounds for eigenvalues are given, and the main result concerns the graphs whose largest eigenvalue is maximal among the graphs with fixed numbers of vertices and edges. The results are presented in the context of a number of computer-generated conjectures.

On the distance signless Laplacian spectral radius of graphs and digraphs

2017 ◽  
Vol 32 ◽  
pp. 438-446 ◽  
Author(s):  
Dan Li ◽  
Guoping Wang ◽  
Jixiang Meng
Keyword(s):  
Spectral Radius ◽  
Connected Graph ◽  
Minimum Degree ◽  
Extremal Graph ◽  
Minimal Distance ◽  
Vertex Connectivity ◽  
Laplacian Spectral Radius ◽  
Connected Graphs ◽  
Signless Laplacian Spectral Radius ◽  
Signless Laplacian

Let \eta(G) denote the distance signless Laplacian spectral radius of a connected graph G. In this paper,bounds for the distance signless Laplacian spectral radius of connected graphs are given, and the extremal graph with the minimal distance signless Laplacian spectral radius among the graphs with given vertex connectivity and minimum degree is determined. Furthermore, the digraph that minimizes the distance signless Laplacian spectral radius with given vertex connectivity is characterized.

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.

scholarly journals On three conjectures involving the signless Laplacian spectral radius of graphs

2009 ◽  
Vol 85 (99) ◽  
pp. 35-38 ◽  
Author(s):  
Lihua Feng ◽  
Guihai Yu
Keyword(s):  
Spectral Radius ◽  
Eigenvalue Bounds ◽  
Laplacian Spectral Radius ◽  
Signless Laplacian Spectral Radius ◽  
Signless Laplacian

We study the signless Laplacian spectral radius of graphs and prove three conjectures of Cvetkovic, Rowlinson, and Simic [Eigenvalue bounds for the signless Laplacian, Publ. Inst. Math., Nouv. S?r. 81(95) (2007), 11-27].

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 the Aα-Spectral Radii of Cactus Graphs

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 869
Author(s):  
Chunxiang Wang ◽  
Shaohui Wang ◽  
Jia-Bao Liu ◽  
Bing Wei
Keyword(s):  
Spectral Radius ◽  
Connected Graph ◽  
Laplacian Matrix ◽  
Diagonal Matrix ◽  
Common Vertex ◽  
Extremal Graphs ◽  
Spectral Matrix ◽  
Cactus Graph ◽  
Signless Laplacian ◽  
Cactus Graphs

Let A ( G ) be the adjacent matrix and D ( G ) the diagonal matrix of the degrees of a graph G, respectively. For 0 ≤ α ≤ 1 , the A α -matrix is the general adjacency and signless Laplacian spectral matrix having the form of A α ( G ) = α D ( G ) + ( 1 − α ) A ( G ) . Clearly, A 0 ( G ) is the adjacent matrix and 2 A 1 2 is the signless Laplacian matrix. A cactus is a connected graph such that any two of its cycles have at most one common vertex, that is an extension of the tree. The A α -spectral radius of a cactus graph with n vertices and k cycles is explored. The outcomes obtained in this paper can imply some previous bounds from trees to cacti. In addition, the corresponding extremal graphs are determined. Furthermore, we proposed all eigenvalues of such extremal cacti. Our results extended and enriched previous known results.

On the sum of the distance signless Laplacian eigenvalues of a graph and some inequalities involving them

2019 ◽  
Vol 12 (01) ◽  
pp. 2050006 ◽  
Author(s):  
A. Alhevaz ◽  
M. Baghipur ◽  
E. Hashemi ◽  
S. Paul
Keyword(s):  
Lower Bounds ◽  
Connected Graph ◽  
Distance Matrix ◽  
Laplacian Matrix ◽  
Simple Graph ◽  
Graph Invariants ◽  
Laplacian Eigenvalues ◽  
Laplacian Energy ◽  
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.

Further results on the distance signless Laplacian spectrum of graphs

2018 ◽  
Vol 11 (05) ◽  
pp. 1850066 ◽  
Author(s):  
Abdollah Alhevaz ◽  
Maryam Baghipur ◽  
Ebrahim Hashemi
Keyword(s):  
Spectral Radius ◽  
Connected Graph ◽  
Distance Matrix ◽  
Laplacian Matrix ◽  
Upper And Lower Bounds ◽  
Laplacian Spectral Radius ◽  
Signless Laplacian Spectral Radius ◽  
Signless Laplacian Spectrum ◽  
Signless Laplacian ◽  
Graph Parameters

The distance signless Laplacian matrix [Formula: see text] 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 whose main entries are the vertex transmissions of [Formula: see text], and the spectral radius of a connected graph [Formula: see text] is the largest eigenvalue of [Formula: see text]. In this paper, first we obtain the [Formula: see text]-eigenvalues of the join of certain regular graphs. Next, we give some new bounds on the distance signless Laplacian spectral radius of a graph [Formula: see text] in terms of graph parameters and characterize the extremal graphs. Utilizing these results we present some upper and lower bounds on the distance signless Laplacian energy of a graph [Formula: see text].

scholarly journals On the Harmonic Index and the Signless Laplacian Spectral Radius of Graphs

2021 ◽  
Vol 45 (02) ◽  
pp. 299-307
Author(s):  
HANYUAN DENG ◽  
TOMÁŠ VETRÍK ◽  
SELVARAJ BALACHANDRAN
Keyword(s):  
Spectral Radius ◽  
Adjacency Matrix ◽  
Connected Graph ◽  
Laplacian Matrix ◽  
Laplacian Spectral Radius ◽  
Harmonic Index ◽  
Signless Laplacian Spectral Radius ◽  
Signless Laplacian ◽  
The Matrix ◽  
The Relationship

The harmonic index of a conected graph G is defined as H(G) = ∑ uv∈E(G) 2 d(u)+d-(v), where E(G) is the edge set of G, d(u) and d(v) are the degrees of vertices u and v, respectively. The spectral radius of a square matrix M is the maximum among the absolute values of the eigenvalues of M. Let q(G) be the spectral radius of the signless Laplacian matrix Q(G) = D(G) + A(G), where D(G) is the diagonal matrix having degrees of the vertices on the main diagonal and A(G) is the (0, 1) adjacency matrix of G. The harmonic index of a graph G and the spectral radius of the matrix Q(G) have been extensively studied. We investigate the relationship between the harmonic index of a graph G and the spectral radius of the matrix Q(G). We prove that for a connected graph G with n vertices, we have ( 2 || ----n----- ||{ 2 (n − 1), if n ≥ 6, -q(G-)- ≤ | 16-, if n = 5, H (G ) || 5 |( 3, if n = 4, and the bounds are best possible.

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