Spectra, signless Laplacian and Laplacian spectra of complementary prisms of graphs

Cardoso, Domingos M.; Carvalho, Paula; Freitas, Maria Aguieiras A. de; Vinagre, Cybele T. M.

doi:10.1016/j.laa.2018.01.020

Cited by 13 publications

(9 citation statements)

References 10 publications

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“…while the edge set E(Γ Γ) is the union of the sets [5,Theorem 3.6]). For a general graph Γ the core of its complementary prism Γ Γ was recently studied in [22] (see also the arXiv version [26]).…”

Section: Preliminariesmentioning

confidence: 99%

The core of a vertex transitive complementary prism of a lexicographic product

Orel

2023

Art Discrete Appl. Math.

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The complementary prism of a graph Γ is the graph Γ Γ, which is formed from the union of Γ and its complement Γ by adding an edge between each pair of identical vertices in Γ and Γ. Vertex-transitive self-complementary graphs provide vertex-transitive complementary prisms. It was recently proved by the author that Γ Γ is a core, i.e. all its endomorphisms are automorphisms, whenever Γ is vertex-transitive, self-complementary, and either Γ is a core or its core is a complete graph. In this paper the same conclusion is obtained for some other classes of vertex-transitive self-complementary graphs that can be decomposed as a lexicographic product Γ = Γ 1 [Γ 2 ]. In the process some new results about the homomorphisms of a lexicographic product are obtained.

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Section: Preliminariesmentioning

confidence: 99%

The core of a vertex transitive complementary prism of a lexicographic product

Orel

2023

Art Discrete Appl. Math.

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“…Clearly, if Γ has n vertices, then Γ Γ is regular if and only if Γ is ( n−1 2 )-regular (see also [14,Theorem 3.6]). In this case Γ Γ is ( n+12 )-regular.…”

Section: Sketch Of the Proofmentioning

confidence: 99%

“…The complementary prism was introduced in [49]. Despite it was studied in several papers (see for example [1,8,13,14,15,16,17,19,23,24,28,45,49,50,87,93,118]) it turns out that many important graph invariants of a complementary prism are not determined yet. One such invariant is the automorphism group Aut(Γ Γ), which we describe for arbitrary finite simple graph Γ.…”

Section: Introductionmentioning

confidence: 99%

“…Later during 8ECM he attended the talk of Paula Carvalho, where he learned that graph Γ Γ was introduced in [49] and is known as the 'complementary prism'. He would like to thank Paula Carvalho also for letting him know that Lemmas 2.33 and 2.36 were already proved in [14] and [49], respectively. This work is supported in part by the Slovenian Research Agency (research program P1-0285 and research projects N1-0140, N1-0208 and N1-0210).…”

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confidence: 99%

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A family of non-Cayley cores based on vertex-transitive or strongly regular self-complementary graphs

Orel¹

2021

Preprint

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Given a finite simple graph Γ on n vertices its complementary prism is the graph Γ Γ that is obtained from Γ and its complement Γ by adding a perfect matching, where each its edge connects two copies of the same vertex in Γ and Γ. It generalizes the Petersen graph, which is obtained if Γ is the pentagon. The automorphism group of Γ Γ is described for arbitrary graph Γ. In particular, it is shown that the ratio between the cardinalities of the automorphism groups of Γ Γ and Γ can attain only values 1, 2, 4, and 12. It is shown that the Cheeger number of Γ Γ equals either 1 or 1 − 1 n , and the two corresponding classes of graphs are fully determined. It is proved that Γ Γ is vertex-transitive if and only if Γ is vertex-transitive and self-complementary. In this case the complementary prism is Hamiltonian-connected whenever n > 5, and is not a Cayley graph whenever n > 1. The main results involve endomorphisms of graph Γ Γ. It is shown that Γ Γ is a core, i.e. all its endomorphisms are automorphisms, whenever Γ is strongly regular and self-complementary. The same conclusion is obtained for many vertex-transitive self-complementary graphs. In particular, it is shown that if there exists a vertex-transitive self-complementary graph Γ such that Γ Γ is not a core, then Γ is neither a core nor its core is a complete graph.

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“…In particular, the Petersen graph is the complementary prism 𝐶 5 𝐶 5 and 𝐾 𝑛 •𝐾 1 is the complementary prism 𝐾 𝑛 𝐾 𝑛 . From a structural geometrical point of view, complementary prism networks have been studied through their properties and indices (see chromatic index [2], domination number [3], cycle structure [4], complexity properties [5], spectral properties [6], convexity number [7], chromatic number [8], etc. ).…”

Section: Introductionmentioning

confidence: 99%