scholarly journals Bounds of Laplacian Energy of a Hypercube Graph

2018 ◽  
Vol 7 (4.10) ◽  
pp. 582
Author(s):  
K. Ameenal Bibi ◽  
B. Vijayalakshmi ◽  
R. Jothilakshmi
Keyword(s):  
Lower Bounds ◽  
Adjacency Matrix ◽  
Laplacian Matrix ◽  
Positive Semidefinite ◽  
Laplacian Energy ◽  
Eigen Values ◽  
Degree Matrix

Let  Qn denote  the n – dimensional  hypercube  with  order   2n and  size n2n-1. The  Laplacian  L  is defined  by  L = D  where D is  the  degree  matrix and  A is  the  adjacency  matrix  with  zero  diagonal  entries.  The  Laplacian  is a  symmetric  positive  semidefinite.  Let  µ1 ≥ µ2 ≥ ....µn-1 ≥ µn = 0 be the eigen values of  the Laplacian matrix.  The  Laplacian  energy is defined as  LE(G) = . In  this  paper, we  defined  Laplacian  energy  of  a  Hypercube  graph  and  also attained  the  lower  bounds.   

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.

scholarly journals Signless Laplacian determinations of some graphs with independent edges

2018 ◽  
Vol 10 (1) ◽  
pp. 185-196 ◽  
Author(s):  
R. Sharafdini ◽  
A.Z. Abdian
Keyword(s):  
Natural Number ◽  
Complete Graph ◽  
Adjacency Matrix ◽  
Undirected Graph ◽  
Laplacian Matrix ◽  
Laplacian Spectrum ◽  
Signless Laplacian Spectrum ◽  
Laplacian Spectra ◽  
Signless Laplacian ◽  
Degree Matrix

Let $G$ be a simple undirected graph. Then the signless Laplacian matrix of $G$ is defined as $D_G + A_G$ in which $D_G$ and $A_G$ denote the degree matrix and the adjacency matrix of $G$, respectively. The graph $G$ is said to be determined by its signless Laplacian spectrum (DQS, for short), if any graph having the same signless Laplacian spectrum as $G$ is isomorphic to $G$. We show that $G\sqcup rK_2$ is determined by its signless Laplacian spectra under certain conditions, where $r$ and $K_2$ denote a natural number and the complete graph on two vertices, respectively. Applying these results, some DQS graphs with independent edges are obtained.

ON LOWER BOUNDS FOR THE COMMUNICATION VOLUME IN DISTRIBUTED SYSTEMS

1991 ◽  
Vol 01 (02) ◽  
pp. 125-133
Author(s):  
U. A. RANAWAKE ◽  
P. M. LENDERS ◽  
S. M. GOODNICK
Keyword(s):  
Distributed Systems ◽  
Lower Bound ◽  
Lower Bounds ◽  
Adjacency Matrix ◽  
Task Graph ◽  
Task Graphs ◽  
Total Communication ◽  
Eigen Values

In this paper we derive a lower bound for the total communication volume when mapping arbitrary task graphs onto a distributed processor system. For a K processor system this lower bound can be computed with only the K (possibly) largest eigen values of the adjacency matrix of the task graph and the eigen values of the adjacency matrix of the processor graph. We also derive the eigen values of the adjacency matrix of the processor graph for a hypercube multiprocessor and illustrate the concept with a simple example for the two processor case.

scholarly journals On the laplacian estrada index of a graph

2009 ◽  
Vol 3 (1) ◽  
pp. 147-156 ◽  
Author(s):  
Jianxi Li ◽  
Chee Shiu ◽  
An Chang
Keyword(s):  
Adjacency Matrix ◽  
Laplacian Matrix ◽  
Graph Laplacian ◽  
Estrada Index ◽  
Laplacian Energy ◽  
Eigenvalues Of The Laplacian

Let G be a graph of order n. Let ?1 , ?2 , . . . , ?n be the eigenvalues of the adjacency matrix of G, and let ?1 , ?2 , . . . , ?n be the eigenvalues of the Laplacian matrix of G. Much studied Estrada index of the graph G is defined n as EE = EE(G)= ?n/i=1 e?i . We define and investigate the Laplacian Estrada index of the graph G, LEE=LEE(G)= ?n/i=1 e(?i - 2m/n). Bounds for LEE are obtained, as well as some relations between LEE and graph Laplacian energy.

scholarly journals Sharp Bounds on (Generalized) Distance Energy of Graphs

Mathematics ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 426 ◽  
Author(s):  
Abdollah Alhevaz ◽  
Maryam Baghipur ◽  
Kinkar Ch. Das ◽  
Yilun Shang
Keyword(s):  
Lower Bounds ◽  
Distance Matrix ◽  
Laplacian Matrix ◽  
Upper And Lower Bounds ◽  
Extremal Graphs ◽  
Graph Invariants ◽  
Laplacian Energy ◽  
Signless Laplacian ◽  
Simple Connected Graph ◽  
Generalized Distance

Given a simple connected graph G, let D ( G ) be the distance matrix, D L ( G ) be the distance Laplacian matrix, D Q ( G ) be the distance signless Laplacian matrix, and T r ( G ) be the vertex transmission diagonal matrix of G. We introduce the generalized distance matrix D α ( G ) = α T r ( G ) + ( 1 − α ) D ( G ) , where α ∈ [ 0 , 1 ] . Noting that D 0 ( G ) = D ( G ) , 2 D 1 2 ( G ) = D Q ( G ) , D 1 ( G ) = T r ( G ) and D α ( G ) − D β ( G ) = ( α − β ) D L ( G ) , we reveal that a generalized distance matrix ideally bridges the spectral theories of the three constituent matrices. In this paper, we obtain some sharp upper and lower bounds for the generalized distance energy of a graph G involving different graph invariants. As an application of our results, we will be able to improve some of the recently given bounds in the literature for distance energy and distance signless Laplacian energy of graphs. The extremal graphs of the corresponding bounds are also characterized.

On the distance signless Laplacian Estrada index of graphs

2021 ◽  
pp. 2250078
Author(s):  
Abdollah Alhevaz ◽  
Maryam Baghipur ◽  
Harishchandra Ramane ◽  
Xueliang Li
Keyword(s):  
Lower Bounds ◽  
Connected Graph ◽  
Distance Matrix ◽  
Laplacian Matrix ◽  
Upper And Lower Bounds ◽  
Graph Invariants ◽  
Laplacian Eigenvalues ◽  
Estrada Index ◽  
Laplacian Energy ◽  
Signless Laplacian

The distance signless Laplacian eigenvalues [Formula: see text] of a connected graph [Formula: see text] are the eigenvalues of the distance signless Laplacian matrix of [Formula: see text], 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]. In this paper, we define and investigate the distance signless Laplacian Estrada index of a graph [Formula: see text] as [Formula: see text], and obtain some upper and lower bounds for [Formula: see text] in terms of other graph invariants. We also obtain some relations between [Formula: see text] and the auxiliary distance signless Laplacian energy of [Formula: see text].

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 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 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.

Bounds for the Eigen Values and Energy of Degree Product Adjacency Matrix of A Graph

2019 ◽  
Vol 10 (3) ◽  
pp. 565-573
Author(s):  
Keerthi G. Mirajkar ◽  
Bhagyashri R. Doddamani
Keyword(s):  
Adjacency Matrix ◽  
Eigen Values
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