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 Hyper-Wiener indices of polyphenyl chains and polyphenyl spiders

2019 ◽  
Vol 17 (1) ◽  
pp. 668-676
Author(s):  
Tingzeng Wu ◽  
Huazhong Lü
Keyword(s):  
Lower Bounds ◽  
Connected Graph ◽  
Wiener Index ◽  
Upper And Lower Bounds ◽  
Extremal Graphs ◽  
Recurrence Formulae

Abstract Let G be a connected graph and u and v two vertices of G. The hyper-Wiener index of graph G is $\begin{array}{} WW(G)=\frac{1}{2}\sum\limits_{u,v\in V(G)}(d_{G}(u,v)+d^{2}_{G}(u,v)) \end{array}$, where dG(u, v) is the distance between u and v. In this paper, we first give the recurrence formulae for computing the hyper-Wiener indices of polyphenyl chains and polyphenyl spiders. We then obtain the sharp upper and lower bounds for the hyper-Wiener index among polyphenyl chains and polyphenyl spiders, respectively. Moreover, the corresponding extremal graphs are determined.

scholarly journals On the Sum of Distance Laplacian Eigenvalues of Graphs

2021 ◽  
Vol 54 ◽  
Author(s):  
Shariefuddin Pirzada ◽  
Saleem Khan
Keyword(s):  
Lower Bounds ◽  
Connected Graph ◽  
Distance Matrix ◽  
Laplacian Matrix ◽  
Wiener Index ◽  
Upper Bounds ◽  
Diagonal Matrix ◽  
Laplacian Eigenvalues ◽  
Eigenvalues Of Graphs

Let $G$ be a connected graph with $n$ vertices, $m$ edges and having diameter $d$. The distance Laplacian matrix $D^{L}$ is defined as $D^L=$Diag$(Tr)-D$, where Diag$(Tr)$ is the diagonal matrix of vertex transmissions and $D$ is the distance matrix of $G$. The distance Laplacian eigenvalues of $G$ are the eigenvalues of $D^{L}$ and are denoted by $\delta_{1}, ~\delta_{1},~\dots,\delta_{n}$. In this paper, we obtain (a) the upper bounds for the sum of $k$ largest and (b) the lower bounds for the sum of $k$ smallest non-zero, distance Laplacian eigenvalues of $G$ in terms of order $n$, diameter $d$ and Wiener index $W$ of $G$. We characterize the extremal cases of these bounds. As a consequence, we also obtain the bounds for the sum of the powers of the distance Laplacian eigenvalues of $G$.

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 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.

scholarly journals Multiplicative Zagreb indices of cacti

2016 ◽  
Vol 08 (03) ◽  
pp. 1650040 ◽  
Author(s):  
Shaohui Wang ◽  
Bing Wei
Keyword(s):  
Graph Theory ◽  
Lower Bounds ◽  
Connected Graph ◽  
Upper And Lower Bounds ◽  
Extremal Graphs ◽  
Zagreb Index ◽  
Zagreb Indices ◽  
Material Chemistry ◽  
Cactus Graph ◽  
Cactus Graphs

Let [Formula: see text] be multiplicative Zagreb index of a graph [Formula: see text]. A connected graph is a cactus graph if and only if any two of its cycles have at most one vertex in common, which is a generalization of trees and has been the interest of researchers in the field of material chemistry and graph theory. In this paper, we use a new tool to obtain the upper and lower bounds of [Formula: see text] for all cactus graphs and characterize the corresponding extremal graphs.

On the sum of the generalized distance eigenvalues of graphs

2020 ◽  
pp. 2050100
Author(s):  
Hilal A. Ganie ◽  
Abdollah Alhevaz ◽  
Maryam Baghipur
Keyword(s):  
Connected Graph ◽  
Independence Number ◽  
Distance Matrix ◽  
Wiener Index ◽  
Index Formula ◽  
Graph Invariants ◽  
Matrix Formula ◽  
Simple Connected Graph ◽  
Eigenvalues Of Graphs ◽  
Generalized Distance

In this paper, we study the generalized distance matrix [Formula: see text] assigned to simple connected graph [Formula: see text], which is the convex combinations of Tr[Formula: see text] and [Formula: see text] and defined as [Formula: see text] where [Formula: see text] and Tr[Formula: see text] denote the distance matrix and diagonal matrix of the vertex transmissions of a simple connected graph [Formula: see text], respectively. Denote with [Formula: see text], the generalized distance eigenvalues of [Formula: see text]. For [Formula: see text], let [Formula: see text] and [Formula: see text] be, respectively, the sum of [Formula: see text]-largest generalized distance eigenvalues and the sum of [Formula: see text]-smallest generalized distance eigenvalues of [Formula: see text]. We obtain bounds for [Formula: see text] and [Formula: see text] in terms of the order [Formula: see text], the Wiener index [Formula: see text] and parameter [Formula: see text]. For a graph [Formula: see text] of diameter 2, we establish a relationship between the [Formula: see text] and the sum of [Formula: see text]-largest generalized adjacency eigenvalues of the complement [Formula: see text]. We characterize the connected bipartite graph and the connected graphs with given independence number that attains the minimum value for [Formula: see text]. We also obtain some bounds for the graph invariants [Formula: see text] and [Formula: see text].

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 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 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.

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