Relation between signless Laplacian energy, energy of graph and its line graph

Das, Kinkar Ch.; Mojallal, Seyed Ahmad

doi:10.1016/j.laa.2015.12.006

Cited by 19 publications

(14 citation statements)

References 15 publications

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“…By exhaust computer searching, we can conclude that, in all connected graphs with 1 ≤ n ≤ 10 vertices, there are only five graphs, i.e., G 1 , G 2 , G 3 , G 4 and G 5 (See Figure 1), whose line graphs are borderenergetic, and their adjacency spectra are as follows: S p (G 1 ) = {3.2361, 0 (2) , −1.2361, −2}; 5) , −2 (4) }; 2) , 0 (3) , −2 (5) }; 2) , −1 (2) , −2 (3) }; 4) , −2 (4) };…”

Section: Resultsmentioning

confidence: 99%

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On the borderenergeticity of line graphs

Lv¹,

Deng²,

Li³

2021

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A graph G of order n is borderenergetic if it has the same energy as the complete graph K n . In this paper, we obtain the result that for any connected graph G, except for the five graphs (one of order 5, three of order 6 and one of order 10), the line graph L(G) of G is not borderenergetic. As a consequence, we get that if G is a borderenergetic graph, then the line graph L(G) of G is not borderenergetic. In addition, we observe a relation between the lower bound of the energy of the line graph L(G) of a borderenergetic graph G and the minimum degree δ(G) of G.

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Section: Resultsmentioning

confidence: 99%

“…The energy of the line graph of a graph G and its relations with other graph energies were early studied in [2,16]. In this paper, we shall study the borderenergetic property or borderenergeticity of the line graphs of connected graphs.…”

Section: Introductionmentioning

confidence: 99%

On the borderenergeticity of line graphs

Lv¹,

Deng²,

Li³

2021

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“…The Laplacian matrix is both a positive semidefinite and an -matrix [15]. Since the sum of the degrees of vertices of a graph G is 2m, we note that the trace of L(G) is 2m.…”

Section: Preliminaries Definitionmentioning

confidence: 99%

“…DefinitionThe sign-less Laplacian matrix of a graph, L + (G), is defined as + ( ) = ( ) + ( ), where D(G) is the degree matrix, and A(G) is the adjacency matrix of G[7]. Thus, the entries of the sign-less Laplacian matrix equal the absolute values of those in the Laplacian matrix [15]. Let 1 + , 2 + , … , + be the eigenvalues of + ( ).…”

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confidence: 99%

Correlations Between Network Vulnerability and Laplacian Energies

Balcı¹,

Akgüller²,

Kol³

2019

Mugla Journal of Science and Technology

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In the network analysis, vulnerability plays key role. Similarly, Laplacian matrices are also effective tools in network analysis. In this study, we examine correlations between those two concepts. We first calculate the well-known vulnerability measures called edge connectivity, vertex connectivity, and solitude number. Then, we find correlation between vulnerability measures and energies of Laplacian matrices. As a result, we find strong correlations between Laplacian energies and vertex connectivity of a network.

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“…By linking the edge of a graph to the electron energy of a type of molecule, the energy of a graph is employed in quantum theory and many other applications in the context of energy. Later, Gutman and Zhou [2] defined the Laplacian energy of a graph as the sum of the absolute values of the differences of average vertex degree of G to the Laplacian eigenvalues of G. Details on the properties of graph energy and Laplacian energy can be found in [3][4][5][6][7][8][9][10][11].…”

Section: Introductionmentioning

confidence: 99%

Novel Concept of Energy in Bipolar Single-Valued Neutrosophic Graphs with Applications

Mohamad

Hasni

Smarandache

et al. 2021

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The energy of a graph is defined as the sum of the absolute values of its eigenvalues. Recently, there has been a lot of interest in graph energy research. Previous literature has suggested integrating energy, Laplacian energy, and signless Laplacian energy with single-valued neutrosophic graphs (SVNGs). This integration is used to solve problems that are characterized by indeterminate and inconsistent information. However, when the information is endowed with both positive and negative uncertainty, then bipolar single-valued neutrosophic sets (BSVNs) constitute an appropriate knowledge representation of this framework. A BSVNs is a generalized bipolar fuzzy structure that deals with positive and negative uncertainty in real-life problems with a larger domain. In contrast to the previous study, which directly used truth and indeterminate and false membership, this paper proposes integrating energy, Laplacian energy, and signless Laplacian energy with BSVNs to graph structure considering the positive and negative membership degree to greatly improve decisions in certain problems. Moreover, this paper intends to elaborate on characteristics of eigenvalues, upper and lower bound of energy, Laplacian energy, and signless Laplacian energy. We introduced the concept of a bipolar single-valued neutrosophic graph (BSVNG) for an energy graph and discussed its relevant ideas with the help of examples. Furthermore, the significance of using bipolar concepts over non-bipolar concepts is compared numerically. Finally, the application of energy, Laplacian energy, and signless Laplacian energy in BSVNG are demonstrated in selecting renewable energy sources, while optimal selection is suggested to illustrate the proposed method. This indicates the usefulness and practicality of this proposed approach in real life.

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Relation between signless Laplacian energy, energy of graph and its line graph

Cited by 19 publications

References 15 publications

On the borderenergeticity of line graphs

On the borderenergeticity of line graphs

Correlations Between Network Vulnerability and Laplacian Energies

Novel Concept of Energy in Bipolar Single-Valued Neutrosophic Graphs with Applications

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