scholarly journals Bounds of the Spectral Radius and the Nordhaus-Gaddum Type of the Graphs

2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
Tianfei Wang ◽  
Liping Jia ◽  
Feng Sun
Keyword(s):  
Spectral Radius ◽  
Chemical Properties ◽  
Laplacian Matrix ◽  
Upper Bounds ◽  
Maximum Energy ◽  
Laplacian Spectral Radius ◽  
Laplacian Spectra ◽  
Molecular Stability ◽  
Vertex Degrees ◽  
Better Than

The Laplacian spectra are the eigenvalues of Laplacian matrixL(G)=D(G)-A(G), whereD(G)andA(G)are the diagonal matrix of vertex degrees and the adjacency matrix of a graphG, respectively, and the spectral radius of a graphGis the largest eigenvalue ofA(G). The spectra of the graph and corresponding eigenvalues are closely linked to the molecular stability and related chemical properties. In quantum chemistry, spectral radius of a graph is the maximum energy level of molecules. Therefore, good upper bounds for the spectral radius are conducive to evaluate the energy of molecules. In this paper, we first give several sharp upper bounds on the adjacency spectral radius in terms of some invariants of graphs, such as the vertex degree, the average 2-degree, and the number of the triangles. Then, we give some numerical examples which indicate that the results are better than the mentioned upper bounds in some sense. Finally, an upper bound of the Nordhaus-Gaddum type is obtained for the sum of Laplacian spectral radius of a connected graph and its complement. Moreover, some examples are applied to illustrate that our result is valuable.

scholarly journals Sharp Upper Bounds for the Laplacian Spectral Radius of Graphs

2013 ◽  
Vol 2013 ◽  
pp. 1-4 ◽  
Author(s):  
Houqing Zhou ◽  
Youzhuan Xu
Keyword(s):  
Spectral Radius ◽  
Transient Stability ◽  
Laplacian Matrix ◽  
Schwarz Inequality ◽  
Upper Bounds ◽  
Power Network ◽  
Laplacian Eigenvalue ◽  
Laplacian Spectral Radius ◽  
Wide Range ◽  
Simple Connected Graph

The spectrum of the Laplacian matrix of a network plays a key role in a wide range of dynamical problems associated with the network, from transient stability analysis of power network to distributed control of formations. LetG=(V,E)be a simple connected graph onnvertices and letμ(G)be the largest Laplacian eigenvalue (i.e., the spectral radius) ofG. In this paper, by using the Cauchy-Schwarz inequality, we show that the upper bounds for the Laplacian spectral radius ofG.

TWO SHARP UPPER BOUNDS FOR THE SIGNLESS LAPLACIAN SPECTRAL RADIUS OF GRAPHS

2011 ◽  
Vol 03 (02) ◽  
pp. 185-191 ◽  
Author(s):  
YA-HONG CHEN ◽  
RONG-YING PAN ◽  
XIAO-DONG ZHANG
Keyword(s):  
Spectral Radius ◽  
Upper Bound ◽  
Adjacency Matrix ◽  
Laplacian Matrix ◽  
Upper Bounds ◽  
Extremal Graphs ◽  
Laplacian Spectral Radius ◽  
Signless Laplacian Matrix ◽  
Signless Laplacian Spectral Radius ◽  
Signless Laplacian

The signless Laplacian matrix of a graph is the sum of its degree diagonal and adjacency matrices. In this paper, we present a sharp upper bound for the spectral radius of the adjacency matrix of a graph. Then this result and other known results are used to obtain two new sharp upper bounds for the signless Laplacian spectral radius. Moreover, the extremal graphs which attain an upper bound are characterized.

Bounds for the skew Laplacian spectral radius of oriented graphs

2019 ◽  
Vol 35 (1) ◽  
pp. 31-40 ◽  
Author(s):  
BILAL A. CHAT ◽  
◽  
HILAL A. GANIE ◽  
S. PIRZADA ◽  
◽  
...  
Keyword(s):  
Spectral Radius ◽  
Upper Bound ◽  
Laplacian Matrix ◽  
Upper Bounds ◽  
Extremal Graphs ◽  
Arbitrary Direction ◽  
Laplacian Spectral Radius ◽  
Underlying Graph ◽  
Signless Laplacian Spectral Radius ◽  
Signless Laplacian

We consider the skew Laplacian matrix of a digraph −→G obtained by giving an arbitrary direction to the edges of a graph G having n vertices and m edges. We obtain an upper bound for the skew Laplacian spectral radius in terms of the adjacency and the signless Laplacian spectral radius of the underlying graph G. We also obtain upper bounds for the skew Laplacian spectral radius and skew spectral radius, in terms of various parameters associated with the structure of the digraph −→G and characterize the extremal graphs.

scholarly journals On Distance Signless Laplacian Spectral Radius and Distance Signless Laplacian Energy

Mathematics ◽  
2020 ◽  
Vol 8 (5) ◽  
pp. 792
Author(s):  
Luis Medina ◽  
Hans Nina ◽  
Macarena Trigo
Keyword(s):  
Spectral Radius ◽  
Lower Bounds ◽  
Connected Graph ◽  
Laplacian Matrix ◽  
Upper Bounds ◽  
Laplacian Spectral Radius ◽  
Signless Laplacian Matrix ◽  
Signless Laplacian Spectral Radius ◽  
Laplacian Energy ◽  
Signless Laplacian

In this article, we find sharp lower bounds for the spectral radius of the distance signless Laplacian matrix of a simple undirected connected graph and we apply these results to obtain sharp upper bounds for the distance signless Laplacian energy graph. The graphs for which those bounds are attained are characterized.

On the distance and distance signless Laplacian eigenvalues of graphs and the smallest Gersgorin disc

2018 ◽  
Vol 34 ◽  
pp. 191-204 ◽  
Author(s):  
Fouzul Atik ◽  
Pratima Panigrahi
Keyword(s):  
Spectral Radius ◽  
Lower Bounds ◽  
Nonnegative Matrix ◽  
Laplacian Matrix ◽  
Upper And Lower Bounds ◽  
Laplacian Spectral Radius ◽  
Laplacian Eigenvalues ◽  
Signless Laplacian Matrix ◽  
Signless Laplacian Spectral Radius ◽  
Signless Laplacian

The \emph{distance matrix} of a simple connected graph $G$ is $D(G)=(d_{ij})$, where $d_{ij}$ is the distance between the $i$th and $j$th vertices of $G$. The \emph{distance signless Laplacian matrix} of the graph $G$ is $D_Q(G)=D(G)+Tr(G)$, where $Tr(G)$ is a diagonal matrix whose $i$th diagonal entry is the transmission of the vertex $i$ in $G$. In this paper, first, upper and lower bounds for the spectral radius of a nonnegative matrix are constructed. Applying this result, upper and lower bounds for the distance and distance signless Laplacian spectral radius of graphs are given, and the extremal graphs for these bounds are obtained. Also, upper bounds for the modulus of all distance (respectively, distance signless Laplacian) eigenvalues other than the distance (respectively, distance signless Laplacian) spectral radius of graphs are given. These bounds are probably first of their kind as the authors do not find in the literature any bound for these eigenvalues. Finally, for some classes of graphs, it is shown that all distance (respectively, distance signless Laplacian) eigenvalues other than the distance (respectively, distance signless Laplacian) spectral radius lie in the smallest Ger\^sgorin disc of the distance (respectively, distance signless Laplacian) matrix.

Sharp upper bounds on the spectral radius of the signless Laplacian matrix of a graph

2013 ◽  
Vol 219 (10) ◽  
pp. 5025-5032 ◽  
Author(s):  
A. Dilek (Güngör) Maden ◽  
Kinkar Ch. Das ◽  
A. Sinan Çevik
Keyword(s):  
Spectral Radius ◽  
Laplacian Matrix ◽  
Upper Bounds ◽  
Signless Laplacian Matrix ◽  
Signless Laplacian

On spectra and spectral radius of Signless Laplacian of power graphs of some finite groups

2020 ◽  
pp. 2150090
Author(s):  
Subarsha Banerjee ◽  
Avishek Adhikari
Keyword(s):  
Spectral Radius ◽  
Finite Groups ◽  
Upper Bounds ◽  
Lower And Upper Bounds ◽  
Laplacian Spectral Radius ◽  
Power Graph ◽  
Signless Laplacian Spectral Radius ◽  
Signless Laplacian ◽  
Extremal Values ◽  
A Finite Group

Let [Formula: see text] denote the power graph of a finite group [Formula: see text]. Let [Formula: see text] denote the Signless Laplacian spectral radius of [Formula: see text]. In this paper, we give lower and upper bounds on [Formula: see text] for any [Formula: see text] and find those graphs for which the extremal values are attained. We give a comparison between the bounds obtained and exact value of [Formula: see text] for any [Formula: see text]. We then find the eigenvalues of [Formula: see text] and give lower and upper bounds on the spectral radius of [Formula: see text]. When [Formula: see text] and [Formula: see text] where [Formula: see text] are primes and [Formula: see text] is a positive integer, we obtain sharper bounds on [Formula: see text]. Finally, we make a conjecture on [Formula: see text] for any [Formula: see text].

LAPLACIAN EIGENVALUES OF GRAPHS WITH GIVEN DOMINATION NUMBER

2009 ◽  
Vol 02 (01) ◽  
pp. 71-76 ◽  
Author(s):  
Lihua Feng ◽  
Guihai Yu ◽  
Xiqin Lin
Keyword(s):  
Spectral Radius ◽  
Domination Number ◽  
Upper Bounds ◽  
Algebraic Connectivity ◽  
Laplacian Spectral Radius ◽  
Laplacian Eigenvalues ◽  
Eigenvalues Of Graphs

In this paper, we study the Laplacian eigenvalues of graphs on n vertices with domination number γ and present upper bounds for the Laplacian spectral radius and algebraic connectivity as well, which improve old results apparently.

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].

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