scholarly journals On the Spectrum of Laplacian Matrix

2021 ◽  
Vol 2021 ◽  
pp. 1-4
Author(s):  
Akbar Jahanbani ◽  
Seyed Mahmoud Sheikholeslami ◽  
Rana Khoeilar
Keyword(s):  
Lower Bounds ◽  
Adjacency Matrix ◽  
Laplacian Matrix ◽  
Simple Graph ◽  
Upper And Lower Bounds ◽  
Largest Eigenvalue ◽  
Smallest Eigenvalue ◽  
The Matrix ◽  
Vertex Degrees ◽  
The Largest Eigenvalue

Let G be a simple graph of order n . The matrix ℒ G = D G − A G is called the Laplacian matrix of G , where D G and A G denote the diagonal matrix of vertex degrees and the adjacency matrix of G , respectively. Let l 1 G , l n − 1 G be the largest eigenvalue, the second smallest eigenvalue of ℒ G respectively, and λ 1 G be the largest eigenvalue of A G . In this paper, we will present sharp upper and lower bounds for l 1 G and l n − 1 G . Moreover, we investigate the relation between l 1 G and λ 1 G .

scholarly journals On the Main Eigenvalues of Universal Adjacency Matrices and U-Controllable Graphs

2015 ◽  
Vol 30 ◽  
pp. 812-826
Author(s):  
Alexander Farrugia ◽  
Irene Sciriha
Keyword(s):  
Adjacency Matrix ◽  
Sufficient Conditions ◽  
Laplacian Matrix ◽  
Simple Graph ◽  
Identity Matrix ◽  
The Matrix ◽  
Vertex Degrees ◽  
Definition Of ◽  
Main Eigenvalues ◽  
Necessary And Sufficient

A universal adjacency matrix U of a graph G is a linear combination of the 0–1 adjacency matrix A, the diagonal matrix of vertex degrees D, the identity matrix I and the matrix J each of whose entries is 1. A main eigenvalue of U is an eigenvalue having an eigenvector that is not orthogonal to the all–ones vector. It is shown that the number of distinct main eigenvalues of U associated with a simple graph G is at most the number of orbits of any automorphism of G. The definition of a U–controllable graph is given using control–theoretic techniques and several necessary and sufficient conditions for a graph to be U–controllable are determined. It is then demonstrated that U–controllable graphs are asymmetric and that the converse is false, showing that there exist both regular and non–regular asymmetric graphs that are not U–controllable for any universal adjacency matrix U. To aid in the discovery of these counterexamples, a gamma–Laplacian matrix L(gamma) is used, which is a simplified form of U. It is proved that any U-controllable graph is a L(gamma)–controllable graph for some parameter gamma.

scholarly journals The Laplacian Spread of Tricyclic Graphs

10.37236/169 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Yanqing Chen ◽  
Ligong Wang
Keyword(s):  
Laplacian Matrix ◽  
Largest Eigenvalue ◽  
Smallest Eigenvalue ◽  
Fixed Order ◽  
Tricyclic Graphs ◽  
The Difference ◽  
The Largest Eigenvalue

The Laplacian spread of a graph is defined to be the difference between the largest eigenvalue and the second smallest eigenvalue of the Laplacian matrix of the graph. In this paper, we investigate Laplacian spread of graphs, and prove that there exist exactly five types of tricyclic graphs with maximum Laplacian spread among all tricyclic graphs of fixed order.

scholarly journals Laplacian integral graphs with a given degree sequence constraint

2021 ◽  
Vol 40 (6) ◽  
pp. 1431-1448
Author(s):  
Ansderson Fernandes Novanta ◽  
Carla Silva Oliveira ◽  
Leonardo de Lima
Keyword(s):  
Adjacency Matrix ◽  
Degree Sequence ◽  
Laplacian Matrix ◽  
Bipartite Graphs ◽  
Diagonal Matrix ◽  
Connected Graphs ◽  
Sequence Constraint ◽  
The Matrix ◽  
Vertex Degrees ◽  
Integral Graphs

Let G be a graph on n vertices. The Laplacian matrix of G, denoted by L(G), is defined as L(G) = D(G) −A(G), where A(G) is the adjacency matrix of G and D(G) is the diagonal matrix of the vertex degrees of G. A graph G is said to be L-integral if all eigenvalues of the matrix L(G) are integers. In this paper, we characterize all Lintegral non-bipartite graphs among all connected graphs with at most two vertices of degree larger than or equal to three.

A note on the relationship between graph energy and determinant of adjacency matrix

2019 ◽  
Vol 11 (01) ◽  
pp. 1950001
Author(s):  
Igor Ž. Milovanović ◽  
Emina I. Milovanović ◽  
Marjan M. Matejić ◽  
Akbar Ali
Keyword(s):  
Lower Bounds ◽  
Adjacency Matrix ◽  
Bipartite Graphs ◽  
Simple Graph ◽  
Graph Invariants ◽  
Graph Energy ◽  
The Matrix ◽  
Matrix Formula ◽  
The Relationship

Let [Formula: see text] be a simple graph of order [Formula: see text], without isolated vertices. Denote by [Formula: see text] the adjacency matrix of [Formula: see text]. Eigenvalues of the matrix [Formula: see text], [Formula: see text], form the spectrum of the graph [Formula: see text]. An important spectrum-based invariant is the graph energy, defined as [Formula: see text]. The determinant of the matrix [Formula: see text] can be calculated as [Formula: see text]. Recently, Altindag and Bozkurt [Lower bounds for the energy of (bipartite) graphs, MATCH Commun. Math. Comput. Chem. 77 (2017) 9–14] improved some well-known bounds on the graph energy. In this paper, several inequalities involving the graph invariants [Formula: see text] and [Formula: see text] are derived. Consequently, all the bounds established in the aforementioned paper are improved.

Laplacian integral graphs with a given degreee sequence constraint

2021 ◽  
Author(s):  
Anderson Fernandes Novanta ◽  
Carla Silva Oliveira ◽  
Leonardo Silva de Lima
Keyword(s):  
Adjacency Matrix ◽  
Laplacian Matrix ◽  
Bipartite Graphs ◽  
Diagonal Matrix ◽  
Connected Graphs ◽  
Sequence Constraint ◽  
The Matrix ◽  
Vertex Degrees ◽  
Integral Graphs

Let G be a graph on n vertices. The Laplacian matrix of G, denoted by L(G), is defined as L(G) = D(G) − A(G), where A(G) is the adjacency matrix of G and D(G) is the diagonal matrix of the vertex degrees of G. A graph G is said to be L-integral is all eigenvalues of the matrix L(G) are integers. In this paper, we characterize all L-integral non-bipartite graphs among all connected graphs with at most two vertices of degree larger than or equal to three.

scholarly journals Eigenvalue Bounds for the Signless $p$-Laplacian

10.37236/6683 ◽  
2018 ◽  
Vol 25 (2) ◽  
Author(s):  
Elizandro Max Borba ◽  
Uwe Schwerdtfeger
Keyword(s):  
Quadratic Form ◽  
Lower Bounds ◽  
Unit Sphere ◽  
Laplacian Matrix ◽  
Upper And Lower Bounds ◽  
Eigenvalue Bounds ◽  
Smallest Eigenvalue ◽  
Signless Laplacian Matrix ◽  
Signless Laplacian ◽  
Largest Eigenvalues

We consider the signless $p$-Laplacian $Q_p$ of a graph, a generalisation of the quadratic form of the signless Laplacian matrix (the case $p=2$). In analogy to Rayleigh's principle the minimum and maximum of $Q_p$ on the $p$-norm unit sphere are called its smallest and largest eigenvalues, respectively. We show a Perron-Frobenius property and basic inequalites for the largest eigenvalue and provide upper and lower bounds for the smallest eigenvalue in terms of a graph parameter related to the bipartiteness. The latter result generalises bounds by Desai and Rao and, interestingly, at $p=1$ upper and lower bounds coincide.

scholarly journals Bounds on the $A_{\alpha}$-spread of a graph

2020 ◽  
Vol 36 (36) ◽  
pp. 214-227 ◽  
Author(s):  
Zhen Lin ◽  
Lianying Miao ◽  
Shu-Guang Guo
Keyword(s):  
Real Number ◽  
Adjacency Matrix ◽  
Undirected Graph ◽  
Upper Bounds ◽  
Diagonal Matrix ◽  
Lower And Upper Bounds ◽  
Largest Eigenvalue ◽  
Smallest Eigenvalue ◽  
The Difference ◽  
The Largest Eigenvalue

Let $G$ be a simple undirected graph. For any real number $\alpha \in[0,1]$, Nikiforov defined the $A_{\alpha}$-matrix of $G$ as $A_{\alpha}(G)=\alpha D(G)+(1-\alpha)A(G)$, where $A(G)$ and $D(G)$ are the adjacency matrix and the degree diagonal matrix of $G$, respectively. The $A_{\alpha}$-spread of a graph is defined as the difference between the largest eigenvalue and the smallest eigenvalue of the associated $A_{\alpha}$-matrix. In this paper, some lower and upper bounds on $A_{\alpha}$-spread are obtained, which extend the results of $A$-spread and $Q$-spread. Moreover, the trees with the minimum and the maximum $A_{\alpha}$-spread are determined, respectively.

scholarly journals The Laplacian spread of a tree

2008 ◽  
Vol Vol. 10 no. 1 (Graph and Algorithms) ◽  
Author(s):  
Yi-Zheng Fan ◽  
Jing Xu ◽  
Yi Wang ◽  
Dong Liang
Keyword(s):  
Laplacian Matrix ◽  
Largest Eigenvalue ◽  
Smallest Eigenvalue ◽  
International Audience ◽  
The Difference ◽  
The Largest Eigenvalue

Graphs and Algorithms International audience The Laplacian spread of a graph is defined to be the difference between the largest eigenvalue and the second smallest eigenvalue of the Laplacian matrix of the graph. In this paper, we show that the star is the unique tree with maximal Laplacian spread among all trees of given order, and the path is the unique one with minimal Laplacian spread among all trees of given order.

scholarly journals A Note on Some Bounds of the α-Estrada Index of Graphs

2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Yang Yang ◽  
Lizhu Sun ◽  
Changjiang Bu
Keyword(s):  
Lower Bounds ◽  
Adjacency Matrix ◽  
Simple Graph ◽  
Upper And Lower Bounds ◽  
Estrada Index ◽  
Degree Matrix

Let G be a simple graph with n vertices. Let A~αG=αDG+1−αAG, where 0≤α≤1 and AG and DG denote the adjacency matrix and degree matrix of G, respectively. EEαG=∑i=1neλi is called the α-Estrada index of G, where λ1,⋯,λn denote the eigenvalues of A~αG. In this paper, the upper and lower bounds for EEαG are given. Moreover, some relations between the α-Estrada index and α-energy are established.

scholarly journals Trees with Four and Five Distinct Signless Laplacian Eigenvalues

2019 ◽  
Vol 25 (3) ◽  
pp. 302-313
Author(s):  
Fatemeh Taghvaee ◽  
Gholam Hossein Fath-Tabar
Keyword(s):  
Adjacency Matrix ◽  
Laplacian Matrix ◽  
Simple Graph ◽  
Laplacian Eigenvalues ◽  
Signless Laplacian Matrix ◽  
Signless Laplacian ◽  
The Matrix ◽  
Vertex Set

‎‎Let $G$ be a simple graph with vertex set $V(G)=\{v_1‎, ‎v_2‎, ‎\cdots‎, ‎v_n\}$ ‎and‎‎edge set $E(G)$‎.‎The signless Laplacian matrix of $G$ is the matrix $‎Q‎‎=‎D‎+‎A‎‎$‎, ‎such that $D$ is a diagonal ‎matrix‎%‎‎, ‎indexed by the vertex set of $G$ where‎‎%‎$D_{ii}$ is the degree of the vertex $v_i$ ‎‎‎ and $A$ is the adjacency matrix of $G$‎.‎%‎ where $A_{ij} = 1$ when there‎‎%‎‎is an edge from $i$ to $j$ in $G$ and $A_{ij} = 0$ otherwise‎.‎The eigenvalues of $Q$ is called the signless Laplacian eigenvalues of $G$ and denoted by $q_1$‎, ‎$q_2$‎, ‎$\cdots$‎, ‎$q_n$ in a graph with $n$ vertices‎.‎In this paper we characterize all trees with four and five distinct signless Laplacian ‎eigenvalues.‎‎‎

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