On the Second-Largest Reciprocal Distance Signless Laplacian Eigenvalue

Ghorbani, Modjtaba; Ganie, Hilal A.; Shang, Yilun

doi:10.3390/math9050512

Cited by 6 publications

(4 citation statements)

References 23 publications

(31 reference statements)

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“…To this extent these matrices RD(G), RT(G), and RQ(G) may be understood from a completely new perspective, and some interesting topics arise. For the these matrices RD(G), RT(G), and RQ(G), some spectral extremal graphs with fixed structure parameters have been characterized in [8,9]. It is natural to ask whether these results can be generalized to RD α (G).…”

Section: Definitionmentioning

confidence: 99%

Upper and Lower Bounds for the Spectral Radius of Generalized Reciprocal Distance Matrix of a Graph

Gao

Shao

2022

Mathematics

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For a connected graph G on n vertices, recall that the reciprocal distance signless Laplacian matrix of G is defined to be RQ(G)=RT(G)+RD(G), where RD(G) is the reciprocal distance matrix, RT(G)=diag(RT1,RT2,⋯,RTn) and RTi is the reciprocal distance degree of vertex vi. In 2022, generalized reciprocal distance matrix, which is defined by RDα(G)=αRT(G)+(1−α)RD(G),α∈[0,1], was introduced. In this paper, we give some bounds on the spectral radius of RDα(G) and characterize its extremal graph. In addition, we also give the generalized reciprocal distance spectral radius of line graph L(G).

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Section: Definitionmentioning

confidence: 99%

Upper and Lower Bounds for the Spectral Radius of Generalized Reciprocal Distance Matrix of a Graph

Gao

Shao

2022

Mathematics

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“…Several authors [1,4,8,10,11,14,15,20] have determined the distance spectra of graphs that are obtained by applying different graph operations, as well as the distance spectra that characterize the graphs from an application perspective. Recently, in [3,16,17], the authors have determined the upper bounds for the extremal graphs related to reciprocal distance Laplacian spectral radius. The books [5,9] are excellent resources on spectra of graphs for interested readers.…”

Section: Introductionmentioning

confidence: 99%

Universal Distance Spectra of Join of Graphs

Kaliyaperumal,

Desikan

2024

Eur. J. Pure Appl. Math.

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Consider G a simple connected graph. In this paper, we introduce the Universal distance matrix UD (G). For α, β, γ, δ ∈ R and β ̸= 0, the universal distance matrix UD (G) is defined asUD (G) = αTr (G) + βD(G) + γJ + δI,where Tr (G) is the diagonal matrix whose elements are the vertex transmissions, and D(G) is the distance matrix of G. Here J is the all-ones matrix, and I is the identity matrix. In this paper, we obtain the universal distance spectra of regular graph, join of two regular graphs, joined union of three regular graphs, generalized joined union of n disjoint graphs with one arbitrary graph H. As a consequence, we obtain the eigenvalues of distance matrix, distance Laplacian matrix, distance signless Laplacian matrix, generalized distance matrix, distance Seidal matrix and distance matrices of complementary graphs.

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“…Since RL(G) is a positive semidefinite matrix, we will denote the spectral radius of RD L (G) by λ(G) = λ 1 (RD L (G)), called the reciprocal distance Laplacian spectral radius. More work on the matrix RD L (G) can be seen in [8][9][10][11].…”

Section: Introductionmentioning

confidence: 99%

On the Sum and Spread of Reciprocal Distance Laplacian Eigenvalues of Graphs in Terms of Harary Index

2022

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The reciprocal distance Laplacian matrix of a connected graph G is defined as RDL(G)=RT(G)−RD(G), where RT(G) is the diagonal matrix of reciprocal distance degrees and RD(G) is the Harary matrix. Clearly, RDL(G) is a real symmetric matrix, and we denote its eigenvalues as λ1(RDL(G))≥λ2(RDL(G))≥…≥λn(RDL(G)). The largest eigenvalue λ1(RDL(G)) of RDL(G), denoted by λ(G), is called the reciprocal distance Laplacian spectral radius. In this paper, we obtain several upper bounds for the sum of k largest reciprocal distance Laplacian eigenvalues of G in terms of various graph parameters, such as order n, maximum reciprocal distance degree RTmax, minimum reciprocal distance degree RTmin, and Harary index H(G) of G. We determine the extremal cases corresponding to these bounds. As a consequence, we obtain the upper bounds for reciprocal distance Laplacian spectral radius λ(G) in terms of the parameters as mentioned above and characterize the extremal cases. Moreover, we attain several upper and lower bounds for reciprocal distance Laplacian spread RDLS(G)=λ1(RDL(G))−λn−1(RDL(G)) in terms of various graph parameters. We determine the extremal graphs in many cases.

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On the Second-Largest Reciprocal Distance Signless Laplacian Eigenvalue

Cited by 6 publications

References 23 publications

Upper and Lower Bounds for the Spectral Radius of Generalized Reciprocal Distance Matrix of a Graph

Upper and Lower Bounds for the Spectral Radius of Generalized Reciprocal Distance Matrix of a Graph

Universal Distance Spectra of Join of Graphs

On the Sum and Spread of Reciprocal Distance Laplacian Eigenvalues of Graphs in Terms of Harary Index

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