scholarly journals Properties and Applications of the Eigenvector Corresponding to the Laplacian Spectral Radius of a Graph

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Haizhou Song ◽  
Qiufen Wang
Keyword(s):  
Spectral Radius ◽  
Upper Bound ◽  
Laplacian Spectrum ◽  
Laplacian Spectral Radius

We study the properties of the eigenvector corresponding to the Laplacian spectral radius of a graph and show some applications. We obtain some results on the Laplacian spectral radius of a graph by grafting and adding edges. We also determine the structure of the maximal Laplacian spectrum tree among trees withnvertices andkpendant vertices (n,kfixed), and the upper bound of the Laplacian spectral radius of some trees.

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 The Signless Laplacian Spectral Radius of Unicyclic and Bicyclic Graphs with a Given Girth

10.37236/670 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Ke Li ◽  
Ligong Wang ◽  
Guopeng Zhao
Keyword(s):  
Spectral Radius ◽  
Upper Bound ◽  
Extremal Graph ◽  
Laplacian Spectral Radius ◽  
Unicyclic Graphs ◽  
Signless Laplacian Spectral Radius ◽  
Disjoint Cycles ◽  
Signless Laplacian ◽  
Unique Graph ◽  
Bicyclic Graphs

Let $\mathcal{U}(n,g)$ and $\mathcal{B}(n,g)$ be the set of unicyclic graphs and bicyclic graphs on $n$ vertices with girth $g$, respectively. Let $\mathcal{B}_{1}(n,g)$ be the subclass of $\mathcal{B}(n,g)$ consisting of all bicyclic graphs with two edge-disjoint cycles and $\mathcal{B}_{2}(n,g)=\mathcal{B}(n,g)\backslash\mathcal{B}_{1}(n,g)$. This paper determines the unique graph with the maximal signless Laplacian spectral radius among all graphs in $\mathcal{U}(n,g)$ and $\mathcal{B}(n,g)$, respectively. Furthermore, an upper bound of the signless Laplacian spectral radius and the extremal graph for $\mathcal{B}(n,g)$ are also given.

On distance and distance Laplacian spectra of corona of two graphs

2016 ◽  
Vol 08 (01) ◽  
pp. 1650007
Author(s):  
Somnath Paul
Keyword(s):  
Graph Theory ◽  
Spectral Radius ◽  
Graph Transformation ◽  
Laplacian Spectrum ◽  
Laplacian Spectral Radius ◽  
Cospectral Graphs ◽  
Laplacian Spectra

Corona of two graphs has been defined in [F. Harary, Graph Theory (Addison-Wesley, 1969)]. In this paper, we study the distance and the distance Laplacian spectra of corona of two graphs and describe the complete distance (distance Laplacian) spectrum for some particular cases. As an application, we show that the corona operation can be used to create distance singular graphs. We also show that these results enable us to construct infinitely many pairs of distance (respectively, distance Laplacian) cospectral graphs. Last, we give a graph transformation and discuss its effect on the distance Laplacian spectral radius.

scholarly journals An improved upper bound for the Laplacian spectral radius of graphs

2009 ◽  
Vol 309 (21) ◽  
pp. 6318-6321 ◽  
Author(s):  
Mei Lu ◽  
Huiqing Liu ◽  
Feng Tian
Keyword(s):  
Spectral Radius ◽  
Upper Bound ◽  
Laplacian Spectral Radius
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