scholarly journals Merging the Spectral Theories of Distance Estrada and Distance Signless Laplacian Estrada Indices of Graphs

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 995 ◽  
Author(s):  
Abdollah Alhevaz ◽  
Maryam Baghipur ◽  
Yilun Shang
Keyword(s):  
Distance Matrix ◽  
Wiener Index ◽  
Upper Bounds ◽  
Extremal Graphs ◽  
Lower And Upper Bounds ◽  
Graph Spectrum ◽  
Estrada Index ◽  
Signless Laplacian ◽  
Graph Parameters ◽  
Generalized Distance

Suppose that G is a simple undirected connected graph. Denote by D ( G ) the distance matrix of G and by T r ( G ) the diagonal matrix of the vertex transmissions in G, and let α ∈ [ 0 , 1 ] . The generalized distance matrix D α ( G ) is defined as D α ( G ) = α T r ( G ) + ( 1 − α ) D ( G ) , where 0 ≤ α ≤ 1 . If ∂ 1 ≥ ∂ 2 ≥ … ≥ ∂ n are the eigenvalues of D α ( G ) ; we define the generalized distance Estrada index of the graph G as D α E ( G ) = ∑ i = 1 n e ∂ i − 2 α W ( G ) n , where W ( G ) denotes for the Wiener index of G. It is clear from the definition that D 0 E ( G ) = D E E ( G ) and 2 D 1 2 E ( G ) = D Q E E ( G ) , where D E E ( G ) denotes the distance Estrada index of G and D Q E E ( G ) denotes the distance signless Laplacian Estrada index of G. This shows that the concept of generalized distance Estrada index of a graph G merges the theories of distance Estrada index and the distance signless Laplacian Estrada index. In this paper, we obtain some lower and upper bounds for the generalized distance Estrada index, in terms of various graph parameters associated with the structure of the graph G, and characterize the extremal graphs attaining these bounds. We also highlight relationship between the generalized distance Estrada index and the other graph-spectrum-based invariants, including generalized distance energy. Moreover, we have worked out some expressions for D α E ( G ) of some special classes of graphs.

scholarly journals On Generalized Distance Gaussian Estrada Index of Graphs

Symmetry ◽  
2019 ◽  
Vol 11 (10) ◽  
pp. 1276 ◽  
Author(s):  
Abdollah Alhevaz ◽  
Maryam Baghipur ◽  
Yilun Shang
Keyword(s):  
Connected Graph ◽  
Distance Matrix ◽  
Laplacian Matrix ◽  
Wiener Index ◽  
Quantum Information Theory ◽  
Extremal Graphs ◽  
Estrada Index ◽  
Graph Parameters ◽  
Generalized Distance

For a simple undirected connected graph G of order n, let D ( G ) , D L ( G ) , D Q ( G ) and T r ( G ) be, respectively, the distance matrix, the distance Laplacian matrix, the distance signless Laplacian matrix and the diagonal matrix of the vertex transmissions of G. The generalized distance matrix D α ( G ) is signified by D α ( G ) = α T r ( G ) + ( 1 - α ) D ( G ) , where α ∈ [ 0 , 1 ] . Here, we propose a new kind of Estrada index based on the Gaussianization of the generalized distance matrix of a graph. Let ∂ 1 , ∂ 2 , … , ∂ n be the generalized distance eigenvalues of a graph G. We define the generalized distance Gaussian Estrada index P α ( G ) , as P α ( G ) = ∑ i = 1 n e - ∂ i 2 . Since characterization of P α ( G ) is very appealing in quantum information theory, it is interesting to study the quantity P α ( G ) and explore some properties like the bounds, the dependence on the graph topology G and the dependence on the parameter α . In this paper, we establish some bounds for the generalized distance Gaussian Estrada index P α ( G ) of a connected graph G, involving the different graph parameters, including the order n, the Wiener index W ( G ) , the transmission degrees and the parameter α ∈ [ 0 , 1 ] , and characterize the extremal graphs attaining these bounds.

scholarly journals Bounds for Generalized Distance Spectral Radius and the Entries of the Principal Eigenvector

2021 ◽  
Vol 52 (1) ◽  
pp. 69-89
Author(s):  
Hilal Ahmad ◽  
Abdollah Alhevaz ◽  
Maryam Baghipur ◽  
Gui-Xian Tian
Keyword(s):  
Spectral Radius ◽  
Distance Matrix ◽  
Upper Bounds ◽  
Upper And Lower Bounds ◽  
Extremal Graphs ◽  
Lower And Upper Bounds ◽  
Linear Combinations ◽  
Graph Parameters ◽  
Simple Connected Graph ◽  
Generalized Distance

For a simple connected graph $G$, the convex linear combinations $D_{\alpha}(G)$ of \ $Tr(G)$ and $D(G)$ is defined as $D_{\alpha}(G)=\alpha Tr(G)+(1-\alpha)D(G)$, $0\leq \alpha\leq 1$. As $D_{0}(G)=D(G)$, $2D_{\frac{1}{2}}(G)=D^{Q}(G)$, $D_{1}(G)=Tr(G)$ and $D_{\alpha}(G)-D_{\beta}(G)=(\alpha-\beta)D^{L}(G)$, this matrix reduces to merging the distance spectral and distance signless Laplacian spectral theories. In this paper, we study the spectral properties of the generalized distance matrix $D_{\alpha}(G)$. We obtain some lower and upper bounds for the generalized distance spectral radius, involving different graph parameters and characterize the extremal graphs. Further, we obtain upper and lower bounds for the maximal and minimal entries of the $ p $-norm normalized Perron vector corresponding to spectral radius $ \partial(G) $ of the generalized distance matrix $D_{\alpha}(G)$ and characterize the extremal graphs.

scholarly journals Bounds for the Generalized Distance Eigenvalues of a Graph

Symmetry ◽  
2019 ◽  
Vol 11 (12) ◽  
pp. 1529 ◽  
Author(s):  
Abdollah Alhevaz ◽  
Maryam Baghipur ◽  
Hilal Ahmad Ganie ◽  
Yilun Shang
Keyword(s):  
Spectral Radius ◽  
Undirected Graph ◽  
Minimum Degree ◽  
Distance Matrix ◽  
Laplacian Matrix ◽  
Extremal Graphs ◽  
Special Cases ◽  
Signless Laplacian ◽  
Graph Parameters ◽  
Generalized Distance

Let G be a simple undirected graph containing n vertices. Assume G is connected. Let D ( G ) be the distance matrix, D L ( G ) be the distance Laplacian, D Q ( G ) be the distance signless Laplacian, and T r ( G ) be the diagonal matrix of the vertex transmissions, respectively. Furthermore, we denote by D α ( G ) the generalized distance matrix, i.e., D α ( G ) = α T r ( G ) + ( 1 − α ) D ( G ) , where α ∈ [ 0 , 1 ] . In this paper, we establish some new sharp bounds for the generalized distance spectral radius of G, making use of some graph parameters like the order n, the diameter, the minimum degree, the second minimum degree, the transmission degree, the second transmission degree and the parameter α , improving some bounds recently given in the literature. We also characterize the extremal graphs attaining these bounds. As an special cases of our results, we will be able to cover some of the bounds recently given in the literature for the case of distance matrix and distance signless Laplacian matrix. We also obtain new bounds for the k-th generalized distance eigenvalue.

Bounds and extremal graphs for Harary energy

2021 ◽  
pp. 2150149
Author(s):  
A. Alhevaz ◽  
M. Baghipur ◽  
H. A. Ganie ◽  
K. C. Das
Keyword(s):  
Connected Graph ◽  
Distance Matrix ◽  
Upper Bounds ◽  
Extremal Graphs ◽  
Lower And Upper Bounds ◽  
Energy Formula ◽  
Matrix Formula ◽  
The Absolute ◽  
Graph Parameters ◽  
Reciprocal Distance Matrix

Let [Formula: see text] be a connected graph of order [Formula: see text] and let [Formula: see text] be the reciprocal distance matrix (also called Harary matrix) of the graph [Formula: see text]. Let [Formula: see text] be the eigenvalues of the reciprocal distance matrix [Formula: see text] of the connected graph [Formula: see text] called the reciprocal distance eigenvalues of [Formula: see text]. The Harary energy [Formula: see text] of a connected graph [Formula: see text] is defined as sum of the absolute values of the reciprocal distance eigenvalues of [Formula: see text], that is, [Formula: see text] In this paper, we establish some new lower and upper bounds for [Formula: see text] in terms of different graph parameters associated with the structure of the graph [Formula: see text]. We characterize the extremal graphs attaining these bounds. We also obtain a relation between the Harary energy and the sum of [Formula: see text] largest adjacency eigenvalues of a connected graph.

scholarly journals On the Generalized Distance Energy of Graphs

Mathematics ◽  
2019 ◽  
Vol 8 (1) ◽  
pp. 17 ◽  
Author(s):  
Abdollah Alhevaz ◽  
Maryam Baghipur ◽  
Hilal A. Ganie ◽  
Yilun Shang
Keyword(s):  
Lower Bounds ◽  
Complete Graph ◽  
Connected Graph ◽  
Distance Matrix ◽  
Wiener Index ◽  
Diagonal Matrix ◽  
Upper And Lower Bounds ◽  
Star Graph ◽  
Extremal Graphs ◽  
Generalized Distance

The generalized distance matrix D α ( G ) of a connected graph G is defined as D α ( G ) = α T r ( G ) + ( 1 − α ) D ( G ) , where 0 ≤ α ≤ 1 , D ( G ) is the distance matrix and T r ( G ) is the diagonal matrix of the node transmissions. In this paper, we extend the concept of energy to the generalized distance matrix and define the generalized distance energy E D α ( G ) . Some new upper and lower bounds for the generalized distance energy E D α ( G ) of G are established based on parameters including the Wiener index W ( G ) and the transmission degrees. Extremal graphs attaining these bounds are identified. It is found that the complete graph has the minimum generalized distance energy among all connected graphs, while the minimum is attained by the star graph among trees of order n.

scholarly journals Some Extremal Graphs with Respect to Sombor Index

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1202
Author(s):  
Kinkar Chandra Das ◽  
Yilun Shang
Keyword(s):  
Molecular Structure ◽  
Chromatic Number ◽  
Clique Number ◽  
Upper Bounds ◽  
Extremal Graphs ◽  
Lower And Upper Bounds ◽  
Graph Parameters ◽  
Structure Descriptor

Let G be a graph with set of vertices V(G)(|V(G)|=n) and edge set E(G). Very recently, a new degree-based molecular structure descriptor, called Sombor index is denoted by SO(G) and is defined as SO=SO(G)=∑vivj∈E(G)dG(vi)2+dG(vj)2, where dG(vi) is the degree of the vertex vi in G. In this paper we present some lower and upper bounds on the Sombor index of graph G in terms of graph parameters (clique number, chromatic number, number of pendant vertices, etc.) and characterize the extremal graphs.

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.

scholarly journals On the Second-Largest Reciprocal Distance Signless Laplacian Eigenvalue

Mathematics ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 512
Author(s):  
Maryam Baghipur ◽  
Modjtaba Ghorbani ◽  
Hilal A. Ganie ◽  
Yilun Shang
Keyword(s):  
Lower Bounds ◽  
Complete Graph ◽  
Distance Matrix ◽  
Upper And Lower Bounds ◽  
Laplacian Eigenvalue ◽  
Signless Laplacian ◽  
Graph Parameters ◽  
Simple Connected Graph ◽  
Second Largest Eigenvalue ◽  
Reciprocal Distance Matrix

The signless Laplacian reciprocal distance matrix for a simple connected graph G is defined as RQ(G)=diag(RH(G))+RD(G). Here, RD(G) is the Harary matrix (also called reciprocal distance matrix) while diag(RH(G)) represents the diagonal matrix of the total reciprocal distance vertices. In the present work, some upper and lower bounds for the second-largest eigenvalue of the signless Laplacian reciprocal distance matrix of graphs in terms of various graph parameters are investigated. Besides, all graphs attaining these new bounds are characterized. Additionally, it is inferred that among all connected graphs with n vertices, the complete graph Kn and the graph Kn−e obtained from Kn by deleting an edge e have the maximum second-largest signless Laplacian reciprocal distance eigenvalue.

Laplacian Energy of Digraphs and a Minimum Laplacian Energy Algorithm

2015 ◽  
Vol 26 (03) ◽  
pp. 367-380 ◽  
Author(s):  
Xingqin Qi ◽  
Edgar Fuller ◽  
Rong Luo ◽  
Guodong Guo ◽  
Cunquan Zhang
Keyword(s):  
Oriented Graph ◽  
Laplacian Matrix ◽  
Upper Bounds ◽  
Spectral Graph Theory ◽  
Extremal Graphs ◽  
Lower And Upper Bounds ◽  
Spectral Graph ◽  
Laplacian Energy ◽  
Energy Formula ◽  
Matrix Formula

In spectral graph theory, the Laplacian energy of undirected graphs has been studied extensively. However, there has been little work yet for digraphs. Recently, Perera and Mizoguchi (2010) introduced the directed Laplacian matrix [Formula: see text] and directed Laplacian energy [Formula: see text] using the second spectral moment of [Formula: see text] for a digraph [Formula: see text] with [Formula: see text] vertices, where [Formula: see text] is the diagonal out-degree matrix, and [Formula: see text] with [Formula: see text] whenever there is an arc [Formula: see text] from the vertex [Formula: see text] to the vertex [Formula: see text] and 0 otherwise. They studied the directed Laplacian energies of two special families of digraphs (simple digraphs and symmetric digraphs). In this paper, we extend the study of Laplacian energy for digraphs which allow both simple and symmetric arcs. We present lower and upper bounds for the Laplacian energy for such digraphs and also characterize the extremal graphs that attain the lower and upper bounds. We also present a polynomial algorithm to find an optimal orientation of a simple undirected graph such that the resulting oriented graph has the minimum Laplacian energy among all orientations. This solves an open problem proposed by Perera and Mizoguchi at 2010.

scholarly journals On the first Zagreb index and multiplicative Zagreb coindices of graphs

2016 ◽  
Vol 24 (1) ◽  
pp. 153-176 ◽  
Author(s):  
Kinkar Ch. Das ◽  
Nihat Akgunes ◽  
Muge Togan ◽  
Aysun Yurttas ◽  
I. Naci Cangul ◽  
...  
Keyword(s):  
Degree Sequence ◽  
Molecular Graph ◽  
Upper Bounds ◽  
Extremal Graphs ◽  
Lower And Upper Bounds ◽  
First Zagreb Index ◽  
Zagreb Index ◽  
Irregularity Index ◽  
Vertex Set ◽  
The First Zagreb Index

AbstractFor a (molecular) graph G with vertex set V (G) and edge set E(G), the first Zagreb index of G is defined as, where dG(vi) is the degree of vertex vi in G. Recently Xu et al. introduced two graphical invariantsandnamed as first multiplicative Zagreb coindex and second multiplicative Zagreb coindex, respectively. The Narumi-Katayama index of a graph G, denoted by NK(G), is equal to the product of the degrees of the vertices of G, that is, NK(G) =. The irregularity index t(G) of G is defined as the number of distinct terms in the degree sequence of G. In this paper, we give some lower and upper bounds on the first Zagreb index M1(G) of graphs and trees in terms of number of vertices, irregularity index, maxi- mum degree, and characterize the extremal graphs. Moreover, we obtain some lower and upper bounds on the (first and second) multiplicative Zagreb coindices of graphs and characterize the extremal graphs. Finally, we present some relations between first Zagreb index and Narumi-Katayama index, and (first and second) multiplicative Zagreb index and coindices of graphs.

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