Energy of spherical fuzzy graphs

2020 ◽  
pp. 321-332
Author(s):  
S. Yahya Mohamed ◽  
A. Mohamed Ali
Keyword(s):  
Adjacency Matrix ◽  
Upper Bounds ◽  
Fuzzy Graph ◽  
Lower And Upper Bounds ◽  
Fuzzy Graphs

In this paper, the notion of energy extended to spherical fuzzy graph. The adjacency matrix of a spherical fuzzy graph is defined and we compute the energy of a spherical fuzzy graph as the sum of absolute values of eigenvalues of the adjacency matrix of the spherical fuzzy graph. Also, the lower and upper bounds for the energy of spherical fuzzy graphs are obtained.

On Laplacian energy of picture fuzzy graphs in site selection problem

2021 ◽  
pp. 1-18
Author(s):  
Mahima Poonia ◽  
Rakesh Kumar Bajaj
Keyword(s):  
Decision Making ◽  
Site Selection ◽  
Upper Bounds ◽  
Selection Problem ◽  
Hydropower Plant ◽  
Fuzzy Graph ◽  
Fuzzy Information ◽  
Fuzzy Graphs ◽  
Laplacian Energy ◽  
Selection For

In the present work, the adjacency matrix, the energy and the Laplacian energy for a picture fuzzy graph/directed graph have been introduced along with their lower and the upper bounds. Further, in the selection problem of decision making, a methodology for the ranking of the available alternatives has been presented by utilizing the picture fuzzy graph and its energy/Laplacian energy. For the shake of demonstrating the implementation of the introduced methodology, the task of site selection for the hydropower plant has been carried out as an application. The originality of the introduced approach, comparative remarks, advantageous features and limitations have also been studied in contrast with intuitionistic fuzzy and Pythagorean fuzzy information.

scholarly journals An Intuitionistic Fuzzy Graph’s Signless Laplacian Energy

2018 ◽  
Vol 7 (4.10) ◽  
pp. 892
Author(s):  
Obbu Ramesh ◽  
S. Sharief Basha
Keyword(s):  
Adjacency Matrix ◽  
Fuzzy Graph ◽  
Intuitionistic Fuzzy ◽  
Fuzzy Graphs ◽  
Laplacian Energy ◽  
Signless Laplacian ◽  
Intuitionistic Fuzzy Graphs

We are extending concept into the Intuitionistic fuzzy graph’ Signless Laplacian energy  instead of the Signless Laplacian energy of fuzzy graph. Now we demarcated an Intuitionistic fuzzy graph’s Signless adjacency matrix and also  an Intuitionistic fuzzy graph’s Signless Laplacian energy. Here we find the Signless Laplacian energy ‘s Intuitionistic fuzzy graphs above and below   boundaries of   an with suitable examples.   

scholarly journals Energy of Pythagorean Fuzzy Graphs with Applications

Mathematics ◽  
2018 ◽  
Vol 6 (8) ◽  
pp. 136 ◽  
Author(s):  
Muhammad Akram ◽  
Sumera Naz
Keyword(s):  
Decision Making ◽  
Fuzzy Sets ◽  
Satellite Communication ◽  
Research Study ◽  
Upper Bounds ◽  
Intuitionistic Fuzzy Sets ◽  
Lower And Upper Bounds ◽  
Fuzzy Graphs ◽  
Laplacian Energy ◽  
Duality Property

Pythagorean fuzzy sets (PFSs), an extension of intuitionistic fuzzy sets (IFSs), inherit the duality property of IFSs and have a more powerful ability than IFSs to model the obscurity in practical decision-making problems. In this research study, we compute the energy and Laplacian energy of Pythagorean fuzzy graphs (PFGs) and Pythagorean fuzzy digraphs (PFDGs). Moreover, we derive the lower and upper bounds for the energy and Laplacian energy of PFGs. Finally, we present numerical examples, including the design of a satellite communication system and the evaluation of the schemes of reservoir operation to illustrate the applications of our proposed concepts in decision making.

scholarly journals On the Aα-Eigenvalues of Signed Graphs

Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 1990
Author(s):  
Germain Pastén ◽  
Oscar Rojo ◽  
Luis Medina
Keyword(s):  
Adjacency Matrix ◽  
Undirected Graph ◽  
Upper Bounds ◽  
Signed Graph ◽  
Lower And Upper Bounds ◽  
Signed Graphs ◽  
Basic Properties ◽  
Underlying Graph ◽  
Positive Semidefiniteness ◽  
Vertex Degrees

For α∈[0,1], let Aα(Gσ)=αD(G)+(1−α)A(Gσ), where G is a simple undirected graph, D(G) is the diagonal matrix of its vertex degrees and A(Gσ) is the adjacency matrix of the signed graph Gσ whose underlying graph is G. In this paper, basic properties of Aα(Gσ) are obtained, its positive semidefiniteness is studied and some bounds on its eigenvalues are derived—in particular, lower and upper bounds on its largest eigenvalue are obtained.

scholarly journals Some Properties on Estrada Index of Folded Hypercubes Networks

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Jia-Bao Liu ◽  
Xiang-Feng Pan ◽  
Jinde Cao
Keyword(s):  
Adjacency Matrix ◽  
Upper Bounds ◽  
Simple Graph ◽  
Spectral Graph Theory ◽  
Lower And Upper Bounds ◽  
Estrada Index ◽  
Spectral Graph ◽  
Explicit Formulae ◽  
Folded Hypercube ◽  
Hypercube Networks

LetGbe a simple graph withnvertices and letλ1,λ2,…,λnbe the eigenvalues of its adjacency matrix; the Estrada indexEEGof the graphGis defined as the sum of the termseλi,  i=1,2,…,n. Then-dimensional folded hypercube networksFQnare an important and attractive variant of then-dimensional hypercube networksQn, which are obtained fromQnby adding an edge between any pair of vertices complementary edges. In this paper, we establish the explicit formulae for calculating the Estrada index of the folded hypercubes networksFQnby deducing the characteristic polynomial of the adjacency matrix in spectral graph theory. Moreover, some lower and upper bounds for the Estrada index of the folded hypercubes networksFQnare proposed.

Results on t-Fuzzy Graphs

2020 ◽  
Vol 16 (01) ◽  
pp. 143-161
Author(s):  
R. A. Borzooei ◽  
B. Sheikh Hoseini ◽  
M. Mohseni Takallo
Keyword(s):  
Adjacency Matrix ◽  
Boolean Matrix ◽  
Matrix Product ◽  
Fuzzy Graph ◽  
Natural Computation ◽  
Fuzzy Graphs

It is much more practical to use the [Formula: see text]-norm than the minimum in defining the fuzzy graph. Hence, the concept of [Formula: see text]-fuzzy graph was introduced by Mordeson et al. in [J. N. Mordeson and S. Mathew, [Formula: see text]-norm Fuzzy Graphs, New Mathematics and Natural Computation 14(1) (2018) 129–143]. In this paper, we concentrate on the relatively new concept of fuzzy adjacency matrix of that [Formula: see text]-fuzzy graphs. The purpose of this paper is to further develop a product of this fuzzy adjacency matrix based on [Formula: see text]-fuzzy graphs. Moreover, two kinds of Boolean matrix are defined and relation between Normal Boolean matrix, product of adjacency matrix and 1-level cut Boolean matrix is obtained.

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 Bounds for the Energy of Graphs

Mathematics ◽  
2021 ◽  
Vol 9 (14) ◽  
pp. 1687
Author(s):  
Slobodan Filipovski ◽  
Robert Jajcay
Keyword(s):  
Adjacency Matrix ◽  
Maximum Degree ◽  
Minimum Degree ◽  
Upper Bounds ◽  
Lower And Upper Bounds ◽  
The Absolute ◽  
Minimum Degrees

Let G be a graph on n vertices and m edges, with maximum degree Δ(G) and minimum degree δ(G). Let A be the adjacency matrix of G, and let λ1≥λ2≥…≥λn be the eigenvalues of G. The energy of G, denoted by E(G), is defined as the sum of the absolute values of the eigenvalues of G, that is E(G)=|λ1|+…+|λn|. The energy of G is known to be at least twice the minimum degree of G, E(G)≥2δ(G). Akbari and Hosseinzadeh conjectured that the energy of a graph G whose adjacency matrix is nonsingular is in fact greater than or equal to the sum of the maximum and the minimum degrees of G, i.e., E(G)≥Δ(G)+δ(G). In this paper, we present a proof of this conjecture for hyperenergetic graphs, and we prove an inequality that appears to support the conjectured inequality. Additionally, we derive various lower and upper bounds for E(G). The results rely on elementary inequalities and their application.

Extension of the Spatial PDI Model to Three Dimensions

1997 ◽  
Vol 84 (1) ◽  
pp. 176-178
Author(s):  
Frank O'Brien
Keyword(s):  
Population Density ◽  
Dimensional Space ◽  
Finite Lattice ◽  
Three Dimensional ◽  
Upper Bounds ◽  
Three Dimensions ◽  
Lower And Upper Bounds ◽  
Density Index ◽  
Three Dimensional Space

The author's population density index ( PDI) model is extended to three-dimensional distributions. A derived formula is presented that allows for the calculation of the lower and upper bounds of density in three-dimensional space for any finite lattice.

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