Lower bounds of the Kirchhoff and degree Kirchhoff indices

Milovanovic, Igor Z.; Milovanovic, Emina I.; Glogic, Edin

doi:10.5937/spsunp1501025m

Cited by 5 publications

(5 citation statements)

References 20 publications

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“…and named it Laplacian-energy-like invariant. Details of the theory of these Laplacian-spectrum-based invariants can be found, for example, in [11,17,19,[21][22][23][24][25]30].…”

Section: Introductionmentioning

confidence: 99%

On relations between Kirchhoff index, Laplacian energy, Laplacian-energy-like invariant and degree deviation of graphs

Milošević

Milovanovic

Matejic

et al. 2020

Filomat

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Let G be a simple connected graph of order n and size m, vertex degree sequence d1 ? d2 ?...? dn > 0, and let ?1 ? ? 2 ? ... ? ?n-1 > ?n = 0 be the eigenvalues of its Laplacian matrix. Laplacian energy LE, Laplacian-energy-like invariant LEL and Kirchhoff index Kf, are graph invariants defined in terms of Laplacian eigenvalues. These are, respectively, defined as LE(G) = ?n,i=1 |?i-2m/n|, LEL(G) = ?n-1 i=1 ??i and Kf (G) = n ?n-1,i=1 1/?i. A vertex-degree-based topological index referred to as degree deviation is defined as S(G) = ?n,i=1 |di- 2m/n|. Relations between Kf and LE, Kf and LEL, as well as Kf and S are obtained.

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“…and named it Laplacian-energy-like invariant. Details of the theory of these Laplacian-spectrum-based invariants can be found, for example, in [11,17,19,[21][22][23][24][25]30].…”

Section: Introductionmentioning

confidence: 99%

On relations between Kirchhoff index, Laplacian energy, Laplacian-energy-like invariant and degree deviation of graphs

Milošević

Milovanovic

Matejic

et al. 2020

Filomat

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“…Inequalities (18) and (19) were proven in [22] (see also [23]). By a similar argument as in case of Theorem 3.2, the following results can be proved.…”

Section: On Interplay Between the Kirchhoff And The Narumi-katayama Imentioning

confidence: 95%

On the relationship between the Kirchhoff and the Narumi-Katayama indices

Milovanovic¹,

Glogic²,

Matejic³

et al. 2019

Filomat

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Let G be a simple connected graph with n vertices and m edges, sequence of vertex degrees ∆ = d 1 ≥ d 2 ≥ • • • ≥ d n = δ > 0 and diagonal matrix D = diag(d 1 , d 2 ,. .. , d n) of its vertex degrees. Denote by K f (G) = n n−1 i=1 1 µ i , where µ i are the Laplacian eigenvalues of graph G, the Kirchhoff index of G, and by NK = n i=1 d i the Narumi-Katayama index. In this paper we prove some inequalities that exhibit relationship between the Kirchhoff and Narumi-Katayama indices.

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“…The lower bound (24) is a pretty good bound obtained circa 2014-2015 (see, e.g., [12,25,26,27]), but is superseded by the state-of-the-art result in [25, Theorem 1].…”

Section: Lmentioning

confidence: 99%

Kemeny's constant and the effective graph resistance

Wang

Dubbeldam

Mieghem

2017

Linear Algebra and its Applications

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Kemeny's constant and its relation to the effective graph resistance has been established for regular graphs by Palacios et al. [1]. Based on the Moore-Penrose pseudo-inverse of the Laplacian matrix, we derive a new closed-form formula and deduce upper and lower bounds for the Kemeny constant. Furthermore, we generalize the relation between the Kemeny constant and the effective graph resistance for a general connected, undirected graph.

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Lower bounds of the Kirchhoff and degree Kirchhoff indices

Abstract: Let G be an undirected connected graph with n, n ≥ 3, vertices and m edges. If

Cited by 5 publications

References 20 publications

On relations between Kirchhoff index, Laplacian energy, Laplacian-energy-like invariant and degree deviation of graphs

On relations between Kirchhoff index, Laplacian energy, Laplacian-energy-like invariant and degree deviation of graphs

On the relationship between the Kirchhoff and the Narumi-Katayama indices

Kemeny's constant and the effective graph resistance

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