Bounds for the signless Laplacian energy

The energy of a graph is equal to the sum of the absolute values of its eigenvalues. Line graphs play an important role in the study of graph theory. Generalized line graphs extend the ideas of both line graphs and cocktail party graphs. In this paper, we establish relations between the energy of the generalized line graph of a graph G and the Laplacian and signless Laplacian energies of G. We give upper and lower bounds for the energy of generalized line graphs. Finally, we present upper and lower bounds for some special graphs.

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“…, a n )) < E(G) + 4 n i=1 a i . (2) When r i > 2, i = 1, 2, we have m > n, q 1 > q 2 . Then E(L (G; a 1 , a 2 , . .…”

Section: Semiregular Graphsmentioning

confidence: 99%

“…Many results on graph energy have appeared in the mathematical literature, see surveys [9,11], and the recent papers [7,13,16,19,21]. The relations between energy, Laplacian energy and signless Laplacian energy of a graph G can be found in [2,8,12,22,24]. Line graph plays an important role in the study of graph theory.…”

Section: Introductionmentioning

confidence: 99%

Energy of generalized line graphs

Lang

Wang

2012

Linear Algebra and its Applications

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“…In reality, the number of existing graph energies may be still greater, and more such will for sure appear in the future. 1) (ordinary) graph energy [12] 2) extended adjacency energy [30] 3) Laplacian energy [36] 4) energy of matrix [41] 5) minimum robust domination energy [49] 6) energy of set of vertices [50] 7) distance energy [37] 8) Laplacian-energy-like invariant [51] 9) Consonni-Todeschini energies [40] 10) energy of (0,1)-matrix [52] 11) incidence energy [53] 12) maximum-degree energy [54] 13) skew Laplacian energy [55] 14) oriented incidence energy [56] 15) skew energy [57] 16) Randić energy [39] 17) normalized Laplacian energy [38] 18) energy of matroid [58] 19) energy of polynomial [42] 20) Harary energy [59] 21) sum-connectivity energy [60] 22) second-stage energy [61] 23) signless Laplacian energy [62] 24) PI energy [63] 25) Szeged energy [64] 26) He energy [65] 27) energy of orthogonal matrix [66] 28) common-neighborhood energy [67] 29) matching energy [43] 30) Seidel energy [68] 31) ultimate energy [69] 32) minimum-covering energy [70] 33) resistance-distance energy [71] 34) Kirchhoff energy [72] 35) color energy [73] 36) normalized incidence energy [74] 37) Laplacian distance energy [75] 38) Laplacian incidence energy [76] 39) Laplacian minimum dominating energy …”

Section: The Graph Energy Delugementioning

confidence: 99%

The Total π-Electron Energy Saga

Gutman¹,

Furtula²

2017

Croat. Chem. Acta

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Abstract:The total π-electron energy, as calculated within the Hückel tight-binding molecular orbital approximation, is a quantum-theoretical characteristic of conjugated molecules that has been conceived as early as in the 1930s. In 1978, a minor modification of the definition of total π-electron energy was put forward, that made this quantity interesting and attractive to mathematical investigations. The concept of graph energy, introduced in 1978, became an extensively studied graph-theoretical topic, with many hundreds of published papers. A great variety of graph energies is being considered in the current mathematical-chemistry and mathematical literature. Recently, some unexpected applications of these graph energies were discovered, in biology, medicine, and image processing.We provide historic, bibliographic, and statistical data on the research on total π-electron energy and graph energies, and outline its present state of art. The goal of this survey is to provide, for the first time, an as-complete-as-possible list of various existing variants of graph energy, and thus help the readers to avoid getting lost in the jungle of references on this topic.

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“…The work by Gutman [22] signposted a fruitful relationship between graph energy and various eigenvalues of matrices associated to graphs. This leads to a wealth of work on energy-like graph spectral invariant in regards to, e.g., (signless) Laplacian matrix [27,28], Randić matrix [29], and distance matrix [30]. More recent results can be found in [17,18,24,[29][30][31][32][33].…”

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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Bounds for the signless Laplacian energy

Cited by 78 publications

References 12 publications

Energy of generalized line graphs

Energy of generalized line graphs

The Total π-Electron Energy Saga

On the Generalized Distance Energy of Graphs

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