Energy of m-Polar Fuzzy Digraphs

2020 ◽  
pp. 469-491 ◽  
Author(s):  
Muhammad Akram ◽  
Danish Saleem ◽  
Ganesh Ghorai
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Adjacency Matrix ◽  
Fuzzy Graph ◽  
Laplacian Energy ◽  
Signless Laplacian ◽  
Different Types ◽  
Matrix Eigenvalues

In this chapter, firstly some basic definitions like fuzzy graph, its adjacency matrix, eigenvalues, and its different types of energies are presented. Some upper bound and lower bound for the energy of this graph are also obtained. Then certain notions, including energy of m-polar fuzzy digraphs, Laplacian energy of m-polar fuzzy digraphs and signless Laplacian energy of m-polar fuzzy digraphs are presented. These concepts are illustrated with several example, and some of their properties are investigated.

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 New Bounds for the α-Indices of Graphs

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1668
Author(s):  
Eber Lenes ◽  
Exequiel Mallea-Zepeda ◽  
Jonnathan Rodríguez
Keyword(s):  
Lower Bound ◽  
Spectral Radius ◽  
Upper Bound ◽  
Adjacency Matrix ◽  
Independence Number ◽  
Minimal Degree ◽  
Diagonal Matrix ◽  
The Matrix

Let G be a graph, for any real 0≤α≤1, Nikiforov defines the matrix Aα(G) as Aα(G)=αD(G)+(1−α)A(G), where A(G) and D(G) are the adjacency matrix and diagonal matrix of degrees of the vertices of G. This paper presents some extremal results about the spectral radius ρα(G) of the matrix Aα(G). In particular, we give a lower bound on the spectral radius ρα(G) in terms of order and independence number. In addition, we obtain an upper bound for the spectral radius ρα(G) in terms of order and minimal degree. Furthermore, for n>l>0 and 1≤p≤⌊n−l2⌋, let Gp≅Kl∨(Kp∪Kn−p−l) be the graph obtained from the graphs Kl and Kp∪Kn−p−l and edges connecting each vertex of Kl with every vertex of Kp∪Kn−p−l. We prove that ρα(Gp+1)<ρα(Gp) for 1≤p≤⌊n−l2⌋−1.

TWO SHARP UPPER BOUNDS FOR THE SIGNLESS LAPLACIAN SPECTRAL RADIUS OF GRAPHS

2011 ◽  
Vol 03 (02) ◽  
pp. 185-191 ◽  
Author(s):  
YA-HONG CHEN ◽  
RONG-YING PAN ◽  
XIAO-DONG ZHANG
Keyword(s):  
Spectral Radius ◽  
Upper Bound ◽  
Adjacency Matrix ◽  
Laplacian Matrix ◽  
Upper Bounds ◽  
Extremal Graphs ◽  
Laplacian Spectral Radius ◽  
Signless Laplacian Matrix ◽  
Signless Laplacian Spectral Radius ◽  
Signless Laplacian

The signless Laplacian matrix of a graph is the sum of its degree diagonal and adjacency matrices. In this paper, we present a sharp upper bound for the spectral radius of the adjacency matrix of a graph. Then this result and other known results are used to obtain two new sharp upper bounds for the signless Laplacian spectral radius. Moreover, the extremal graphs which attain an upper bound are characterized.

scholarly journals On Harary energy and Reciprocal distance Laplacian energies1

2021 ◽  
Vol 2090 (1) ◽  
pp. 012102
Author(s):  
Macarena Trigo
Keyword(s):  
Lower Bound ◽  
Laplacian Energy ◽  
Signless Laplacian ◽  
The Absolute

Abstract Let G be an graph simple, undirected, connected and unweighted graphs. The Reciprocal distance energy of a graph G is equal to the sum of the absolute values of the reciprocal distance eigenvalues. In this work, we find a lower bound for the Harary energy, reciprocal distance Laplacian energy and reciprocal distance signless Laplacian energy of a graph. Moreover, we find relationship between the Harary energy and Reciprocal distance Laplacian energies.

scholarly journals On Ensuring Multilayer Wirability by Stretching Layouts

VLSI Design ◽  
1998 ◽  
Vol 7 (4) ◽  
pp. 365-383
Author(s):  
Teofilo F. Gonzalez ◽  
Si-Qing Zheng
Keyword(s):  
Dynamic Programming ◽  
Lower Bound ◽  
Upper Bound ◽  
Area Increase ◽  
Different Types ◽  
Np Complete ◽  
Layout Area ◽  
The One

Every knock-knee layout is four-layer wirable. However, there are knock-knee layouts that cannot be wired in less than four layers. While it is easy to determine whether a knock-knee layout is one-layer wirable or two-layer wirable, the problem of determining three-layer wirability of knock-knee layouts is NP-complete. A knock-knee layout may be stretched vertically (horizontally) by introducing empty rows (columns) so that it can be wired in fewer than four layers. In this paper we discuss two different types of stretching schemes. It is known that under these two stretching schemes, any knock-knee layout is three-layer wirable by stretching it up to (4/3) of the knock-knee layout area (upper bound). We show that there are knock-knee layouts that when stretched and wired in three layers under scheme I (II) require at least 1.2 (1.07563) of the original layout area. Our lower bound for the area increase factor can be used to guide the search for effective stretching-based dynamic programming three-layer wiring algorithms similar to the one presented in [8].

scholarly journals Skew-rank of an oriented graph in terms of the rank and dimension of cycle space of its underlying graph

Filomat ◽  
2018 ◽  
Vol 32 (4) ◽  
pp. 1303-1312 ◽  
Author(s):  
Yong Lu ◽  
Ligong Wang ◽  
Qiannan Zhou
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Adjacency Matrix ◽  
Oriented Graph ◽  
Connected Components ◽  
Cycle Space ◽  
Vertex Number ◽  
Underlying Graph ◽  
Skew Adjacency Matrix

Let G? be an oriented graph and S(G?) be its skew-adjacency matrix, where G is called the underlying graph of G?. The skew-rank of G?, denoted by sr(G?), is the rank of S(G?). Denote by d(G) = |E(G)|-|V(G)| + ?(G) the dimension of cycle spaces of G, where |E(G)|, |V(G)| and ?(G) are the edge number, vertex number and the number of connected components of G, respectively. Recently, Wong, Ma and Tian [European J. Combin. 54 (2016) 76-86] proved that sr(G?) ? r(G) + 2d(G) for an oriented graph G?, where r(G) is the rank of the adjacency matrix of G, and characterized the graphs whose skew-rank attain the upper bound. However, the problem of the lower bound of sr(G?) of an oriented graph G? in terms of r(G) and d(G) of its underlying graph G is left open till now. In this paper, we prove that sr(G?) ? r(G)-2d(G) for an oriented graph G? and characterize the graphs whose skew-rank attain the lower bound.

scholarly journals Signless Laplacian Energy in Products of Intuitionistic Fuzzy Graphs

2019 ◽  
Vol 8 (3) ◽  
pp. 8536-8545
Keyword(s):  
Tensor Product ◽  
Cartesian Product ◽  
Fuzzy Graph ◽  
Lexicographic Product ◽  
Intuitionistic Fuzzy ◽  
Strong Product ◽  
Fuzzy Graphs ◽  
Laplacian Energy ◽  
Signless Laplacian ◽  
Intuitionistic Fuzzy Graphs

The observation of an Intuitionistic Fuzzy Graph’s signless laplacian energy is expanded innumerous products in Intuitionistic Fuzzy Graph. During this paper, we have got the value of signless laplacian Energy in unrelated products such as Cartesian product, Lexicographic Product, Tensor product and Strong Product, product, product and product amongst 2 intuitionistic Fuzzy graphs. Additionally we tend to study the relation between the Signless laplacian Energy within the varied products in 2 Intuitionistic Fuzzy Graphs

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.

Sign in / Sign up

Export Citation Format

Share Document