Skew-adjacency matrices of graphs

Cavers, Michael S.; Cioabă, Sebastian M.; Fallat, Shaun M.; Gregory, David A.; Haemers, Willem H.; Kirkland, Steve; McDonald, Judith J.; Tsatsomeros, Michael J.

doi:10.1016/j.laa.2012.01.019

Cited by 69 publications

(51 citation statements)

References 12 publications

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“…We immediately arrive at the result of Cavers et al [7]: [7].) An undirected graph G has no even cycles if and only if all of its oriented graphs are all cospectral.…”

Section: Basic Propertiessupporting

confidence: 55%

Hermitian-adjacency matrices and Hermitian energies of mixed graphs

Liu

2015

Linear Algebra and its Applications

158

135

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A complex adjacency matrix of a mixed graph is introduced in the present paper, which is a Hermitian matrix and called the Hermitian-adjacency matrix. It incorporates both adjacency matrix of an undirected graph and skew-adjacency matrix of an oriented graph. Some of its properties are studied. Furthermore, properties of its characteristic polynomial are studied. Cospectral problems among mixed graphs, including mixed graphs and their underlying graphs, oriented graphs and their underlying graphs, are studied. We give equivalent conditions for a mixed graph (especially oriented graph) that share the same spectrum with its underlying graph. As a consequence, we reconfirm a conjecture which was proposed by Cui and Hou in Ref. [8]. We also show that the spectrum of the Hermitian matrix of a mixed graph is invariant when changing the value of any its cut edge (if any). Correspondingly, an energy of a mixed graph is introduced and called the Hermitian energy. It incorporates both the energy of an undirected graph and the skew energy of an oriented graph. Some of its bounds are given. Especially, the mixed graphs with optimal upper bound of Hermitian energy are characterized. An infinite family of mixed graphs attaining the maximum Hermitian energy is constructed. Moreover, the Hermitian energy of a mixed tree is showed to be equal to ✩ 183 the energy of its underlying tree. Finally, the integral formula for Hermitian energy of a mixed graph is given.

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“…We immediately arrive at the result of Cavers et al [7]: [7].) An undirected graph G has no even cycles if and only if all of its oriented graphs are all cospectral.…”

Section: Basic Propertiessupporting

confidence: 55%

Hermitian-adjacency matrices and Hermitian energies of mixed graphs

Liu

2015

Linear Algebra and its Applications

158

135

View full text Add to dashboard Cite

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“…It was followed by the distance energy (based on the eigenvalues of the distance matrix), [37] normalized Laplacian energy (based on the eigenvalues of the normalized Laplacian matrix [38] which independently was introduced under the name of "Randić energy"), [39] etc. Consonni and Todeschini [40] defined the energy of any real symmetric matrix with eigenvalues 1 2 , , , n x x x  as Nikiforov extended the energy-concept to any matrix.…”

Section: The Graph Energy Delugementioning

confidence: 99%

“…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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“…On the other hand, from Observation 1 we can delete k − 2 outer vertices from P l 1 ,l 2 ,··· ,l k such that the resultant graph is of P l 1 ,l 2 with l 1 + l 2 = n − k. By Lemma 1,…”

Section: Lemma 1 Let G σ Be An Arbitrary Oriented Graph On N Verticementioning

confidence: 99%