On the spectral characterization of pineapple graphs

Topcu, Hatice; Sorgun, Sezer; Haemers, Willem H.

doi:10.1016/j.laa.2016.06.018

Cited by 16 publications

(4 citation statements)

References 6 publications

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“…Notice that this proposition does not give any graph which is cospectral and nonisomorphic with a mixed extension of P 3 , because in all three cases the second graph has more vertices than the first one. For the pineapple graph K q p , which is a mixed extension of P 3 of type (p − 1, 1, −q), we find (see [7]).…”

Section: Spectral Characterizationsmentioning

confidence: 86%

On the Spectral Characterization of Mixed Extensions of $P_{3}$

Haemers

Sorgun

Topcu

2019

Electron. J. Combin.

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A mixed extension of a graph G is a graph H obtained from G by replacing each vertex of G by a clique or a coclique, whilst two vertices in H corresponding to distinct vertices x and y of G are adjacent whenever x and y are adjacent in G. If G is the path P 3 , then H has at most three adjacency eigenvalues unequal to 0 and −1. Recently, the first author classified the graphs with the mentioned eigenvalue property. Using this classification we investigate mixed extension of P 3 on being determined by the adjacency spectrum. We present several cospectral families, and with the help of a computer we find all graphs on at most 25 vertices that are cospectral with a mixed extension of P 3 . present several infinite families of cospectral graphs with all but three eigenvalues equal to 0 or −1.2 Mixed extensions of P 3 Let G be a graph with vertex set V (G) = {1, . . . , m} and let V 1 , . . . , V m be mutually disjoint nonempty finite sets. A graph H with vertex set V (H) = V 1 ∪ . . . ∪V m is defined as follows. For each i ∈ {1, . . . , m}, all vertices of V i are either mutually adjacent (form a clique), or mutually nonadjacent (form a coclique). For any u ∈ V i and v ∈ V j (i = j) {u, v} in an edge in H if and only if {i, j} is an edge in G. The graph H is called a mixed extension of G. A mixed extension is represented by an m-tuple (t 1 , . . . , t m ) of nonzero integers, where t i > 0 indicates that V i is a clique of order t i and t i < 0 means that V i is a coclique of order −t i . A mixed extension of G is a special case of a generalized composition or G-join, introduced in [3] and [6], respectively. We refer to [5], [3] and [6] for basic results on mixed extensions, and to [1] or [4] for graph spectra.Suppose H is a mixed extension of the path P 3 of type (t 1 , t 2 , t 3 ). Then the adjacency matrix of H admits the following structure. (As usual, J is an all-ones matrix, J n is the n × n all-ones matrix, and I n is the identity matrix of order n.)

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Section: Spectral Characterizationsmentioning

confidence: 86%

On the Spectral Characterization of Mixed Extensions of $P_{3}$

Haemers

Sorgun

Topcu

2019

Electron. J. Combin.

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“…and The pineapple graph K q p is obtained by appending q pendant edges to a vertex of a complete graph K p ( q ≥ 1, p ≥ 3) [14,15]. From (3.4) and Figure 2, we can see easily that JB n could also be obtained from a pineapple graph K n n+1 by adding loop to the all of vertices of its maximum clique K n+1 .…”

Section: Proofmentioning

confidence: 99%

On the spectra of generalized Fibonomial and Jacobsthal-binomial graphs

Topcu¹,

Yücel²

2021

Turk J Math

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In this work, we first give a more general form of the binomial, Fibonomial, and balance-binomial graphs that is called generalized Fibonomial graph. We also argue the spectra of generalized Fibonomial graph. Next, we introduce a new type of graph on Jacobsthal numbers that is called Jacobsthal-binomial graph and denoted by JBn . We obtain the adjacency, Laplacian and signless Laplacian characteristic polynomials of JBn , respectively. We lastly give inequalities for the adjacency, Laplacian and signless Laplacian energies of JBn .

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“…Up to now, very few classes of graphs with very special structures have been proved to be DS. Usually it is case that the graphs shown to be DS have very few edges, such as the T-shape trees [5], the ∞-graphs [6], the lollipop graphs [7], the θ-graphs [8], the graphs with index at most 2 + √ 5 [9], and the pineapple graphs [10], to just name a few. For dense graphs, it is usually quite difficult to show them to be DS, for example, the complement of the path Pn was shown to be DS in [4], but the proof is much more involved than the proof that the path P n is DS.…”

Section: Introductionmentioning

confidence: 99%

Spectral characterization of the complete graph removing a path of small length

Mao

Cioabă

Wang

2019

Discrete Applied Mathematics

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A graph G is said to be determined by its spectrum if any graph having the same spectrum as G is isomorphic to G. Let K n \ P ℓ be the graph obtained from K n by removing edges of P ℓ , where P ℓ is a path of length ℓ − 1 which is a subgraph of a complete graph K n . Cámara and Haemers [11] conjectured that K n \P ℓ is determined by its adjacency spectrum for every 2 ≤ ℓ ≤ n. In this paper we show that the conjecture is true for 7 ≤ ℓ ≤ 9.

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On the spectral characterization of pineapple graphs

Cited by 16 publications

References 6 publications

On the Spectral Characterization of Mixed Extensions of $P_{3}$

On the Spectral Characterization of Mixed Extensions of $P_{3}$

On the spectra of generalized Fibonomial and Jacobsthal-binomial graphs

Spectral characterization of the complete graph removing a path of small length

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