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2. On graphs with the largest Laplacian index

Creator:
Liu, Bolian, Chen, Zhibo, and Liu, Muhuo
Format:
bez média and svazek
Type:
model:article and TEXT
Subject:
eigenvalue, Laplacian index, algebraic connectivity, semi-regular graph, regular graph, Hamiltonian graph, and planar graph
Language:
English
Description:
Let $G$ be a connected simple graph on $n$ vertices. The Laplacian index of $G$, namely, the greatest Laplacian eigenvalue of $G$, is well known to be bounded above by $n$. In this paper, we give structural characterizations for graphs $G$ with the largest Laplacian index $n$. Regular graphs, Hamiltonian graphs and planar graphs with the largest Laplacian index are investigated. We present a necessary and sufficient condition on $n$ and $k$ for the existence of a $k$-regular graph $G$ of order $n$ with the largest Laplacian index $n$. We prove that for a graph $G$ of order $n \geq 3$ with the largest Laplacian index $n$, $G$ is Hamiltonian if $G$ is regular or its maximum vertex degree is $\triangle (G)=n/2$. Moreover, we obtain some useful inequalities concerning the Laplacian index and the algebraic connectivity which produce miscellaneous related results.
Rights:
http://creativecommons.org/publicdomain/mark/1.0/ and policy:public

3. The (signless) Laplacian spectral radius of unicyclic and bicyclic graphs with $n$ vertices and $k$ pendant vertices

Creator:
Liu, Muhuo, Tan, Xuezhong, and Liu, Bolian
Format:
bez média and svazek
Type:
model:article and TEXT
Subject:
Laplacian matrix, signless Laplacian matrix, and spectral radius
Language:
English
Description:
In this paper, the effects on the signless Laplacian spectral radius of a graph are studied when some operations, such as edge moving, edge subdividing, are applied to the graph. Moreover, the largest signless Laplacian spectral radius among the all unicyclic graphs with $n$ vertices and $k$ pendant vertices is identified. Furthermore, we determine the graphs with the largest Laplacian spectral radii among the all unicyclic graphs and bicyclic graphs with $n$ vertices and $k$ pendant vertices, respectively.
Rights:
http://creativecommons.org/publicdomain/mark/1.0/ and policy:public

4. The Laplacian spread of graphs

Creator:
You, Zhifu and Liu, Bolian
Format:
bez média and svazek
Type:
model:article and TEXT
Subject:
Laplacian eigenvalues and spread
Language:
English
Description:
The Laplacian spread of a graph is defined as the difference between the largest and second smallest eigenvalues of the Laplacian matrix of the graph. In this paper, bounds are obtained for the Laplacian spread of graphs. By the Laplacian spread, several upper bounds of the Nordhaus-Gaddum type of Laplacian eigenvalues are improved. Some operations on Laplacian spread are presented. Connected $c$-cyclic graphs with $n$ vertices and Laplacian spread $n-1$ are discussed.
Rights:
http://creativecommons.org/publicdomain/mark/1.0/ and policy:public

5. The maximum clique and the signless Laplacian eigenvalues.

Creator:
Liu, Jianping and Liu, Bolian
Format:
bez média and svazek
Type:
model:article and TEXT
Subject:
bound, clique number, independence number, and signless Laplacian eigenvalues
Language:
English
Description:
Lower and upper bounds are obtained for the clique number $\omega (G)$ and the independence number $\alpha (G)$, in terms of the eigenvalues of the signless Laplacian matrix of a graph $G$.
Rights:
http://creativecommons.org/publicdomain/mark/1.0/ and policy:public