The generalized distance matrix

Cui, Shu-Yu; He, Jing-Xiang; Tian, Gui-Xian

doi:10.1016/j.laa.2018.10.014

Cited by 49 publications

(37 citation statements)

References 15 publications

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“…≥ ∂ n be the generalized distance eigenvalues of G. Note that by Corollary 1, we have ∂ 1 ≥ Tr min = n − 1. On the other hand, by Proposition 5 in [7] we get…”

Section: Proof Sincementioning

confidence: 85%

“…Nikiforov [6] investigated the integration of adjacency spectrum and signless Laplacian spectrum via cunning convex combinations between diagonal degree and adjacency matrices. Recently in [7], properties of generalized distance matrix has its counterpart for each of these particular graph matrices, and these counterparts follow immediately from a straightforward proof. In fact, this matrix leads to merging the distance spectral, distance Laplacian spectral and distance signless Laplacian spectral theories.…”

Section: Introductionmentioning

confidence: 97%

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On Generalized Distance Gaussian Estrada Index of Graphs

Alhevaz

Shang

2019

Symmetry

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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.

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“…≥ ∂ n be the generalized distance eigenvalues of G. Note that by Corollary 1, we have ∂ 1 ≥ Tr min = n − 1. On the other hand, by Proposition 5 in [7] we get…”

Section: Proof Sincementioning

confidence: 85%

Section: Introductionmentioning

confidence: 97%

On Generalized Distance Gaussian Estrada Index of Graphs

Alhevaz

Shang

2019

Symmetry

View full text Add to dashboard Cite

show abstract

“…Recently, in [21], Cui et al introduced the generalized distance matrix D α (G) as a convex combinations of Tr(G) and D(G). Namely, D α (G) = αTr(G)…”

Section: Introductionmentioning

confidence: 99%

On the Generalized Distance Energy of Graphs

2019

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is the distance matrix and Tr(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.

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“…Motivated by [6], Cui et al [7] introduced the generalized distance matrix D α (G) , any result regarding the spectral properties of generalized distance matrix has its counterpart for each of these particular graph matrices, and these counterparts following immediately from a single proof. In fact, this matrix reduces to merging the distance Laplacian spectral, distance spectral, as well as distance signless Laplacian spectral theories.…”

Section: Introductionmentioning

confidence: 99%

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

Alhevaz

Shang

2019

Mathematics

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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.

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The generalized distance matrix

Cited by 49 publications

References 15 publications

On Generalized Distance Gaussian Estrada Index of Graphs

On Generalized Distance Gaussian Estrada Index of Graphs

On the Generalized Distance Energy of Graphs

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

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