On the eigenvalues of signed complete graphs

Akbari, S.; Dalvandi, S.; Heydari, F.; Maghasedi, M.

doi:10.1080/03081087.2017.1403548

Cited by 11 publications

(4 citation statements)

References 2 publications

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“…For more information about switching and recent work on signed graphs, we refer to [1,3,4,5,6,8,11,12,13,14,15,16,17]. We note that the sign function for the signed subgraph is the restriction of the original function.…”

Section: Introductionmentioning

confidence: 99%

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On the eigenvalues of complete bipartite signed graphs

Pirzada,

Shamsher,

Bhat

2024

Ars Math. Contemp.

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Let Γ = (G, σ) be a signed graph, where σ is the sign function on the edges of G. The adjacency matrix of Γ is defined canonically. Let (K p,q , σ), p ≤ q, be a complete bipartite signed graph with bipartition (U p , V q ), where U p = {u 1 , u 2 , . . . , u p } and V q = {v 1 , v 2 , . . . , v q }. Let (K p,q , σ)[U r ∪ V s ], r ≤ p and s ≤ q, be an induced signed subgraph on minimum vertices r+s, which contains all negative edges of the signed graph (K p,q , σ). In this paper, we show that the nullity of the signed graph (K p,q , σ) is at least p+q −2k −2, where k = min(r, s). The spectrum of a complete bipartite signed graph whose negative edges induce either a disjoint complete bipartite subgraphs or a path is determined. Finally, we obtain the spectrum of a complete bipartite signed graph whose negative edges (positive edges) induce a regular subgraph H. It turns out that there is a relationship between the eigenvalues of this complete bipartite signed graph and the non-negative eigenvalues of H.

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Section: Introductionmentioning

confidence: 99%

“…In the recent years, several researchers have shown interest in signed graphs, including complete bipartite signed graphs and complete signed graphs, which have a variety of applications, see [1,2] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

On the eigenvalues of complete bipartite signed graphs

Pirzada,

Shamsher,

Bhat

2024

Ars Math. Contemp.

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“…Also, the spectrum of A(Γ) is called the spectrum of Γ and the largest eigenvalue is often called the index. The spectrum of signed graphs has been studied by many authors, for instance see [1,3,8,9].…”

Section: Introductionmentioning

confidence: 99%

On the Signed Complete Graphs with Maximum Index

Kafai

Heydari

Rad³

et al. 2021

Preprint

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Let Γ = (K n , H − ) be a signed complete graph whose negative edges induce a subgraph H. The index of Γ is the largest eigenvalue of its adjacency matrix. In this paper we study the index of Γ when H is a unicyclic graph. We show that among all signed complete graphs of order n > 5 whose negative edges induce a unicyclic graph of order k and maximizes the index, the negative edges induce a triangle with all remaining vertices being pendant at the same vertex of the triangle.

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“…If all edges of a signed graph are positive (resp., negative), then we denote it by (G, +) (resp., (G, −)). The signed graphs have been studied by many authors, for instance see [1,2,3].…”

Section: Introductionmentioning

confidence: 99%

On the eigenvalues of some signed graphs

Souri

Heydari

Maghasedi

2019

Preprint

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Let G be a simple graph and A(G) be the adjacency matrix of G. The matrix S(G) = J − I − 2A(G) is called the Seidel matrix of G, where I is an identity matrix and J is a square matrix all of whose entries are equal to 1. Clearly, if G is a graph of order n with no isolated vertex, then the Seidel matrix of G is also the adjacency matrix of a signed complete graph Kn whose negative edges induce G. In this paper, we study the Seidel eigenvalues of the complete multipartite graph Kn 1 ,...,n k and investigate its Seidel characteristic polynomial. We show that if there are at least three parts of size ni, for some i = 1, . . . , k, then Kn 1 ,...,n k is determined, up to switching, by its Seidel spectrum.

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On the eigenvalues of signed complete graphs

Cited by 11 publications

References 2 publications

On the eigenvalues of complete bipartite signed graphs

On the eigenvalues of complete bipartite signed graphs

On the Signed Complete Graphs with Maximum Index

On the eigenvalues of some signed graphs

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