The graphs cospectral with the pineapple graph

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

doi:10.1016/j.dam.2018.10.002

Cited by 11 publications

(6 citation statements)

References 3 publications

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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 [4,6,7]. A mixed extension of star S k is represented by an k-tuple (±r 1 , .…”

Section: Resultsmentioning

confidence: 99%

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On the graphs having at most one positive eccentricity eigenvalue

Sorgun¹,

Küçük²

2020

Preprint

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Section: Resultsmentioning

confidence: 99%

“…+ r k , k) as an induced subgraph. Also, from the partition of the matrix, it can contain CS (7,3) or CS (7,4) or CS(n+2, n) as induced subgraph as well as the eccentricity matrices of them is principal submatrix of it. That's, there can be three cases.…”

Section: Theorem 211 Let H Denote the Class Of Connected Graphs Which...mentioning

confidence: 99%

On the graphs having at most one positive eccentricity eigenvalue

Sorgun¹,

Küçük²

2020

Preprint

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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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“…For the pineapple graphs it is known for which p and q the graphs are determined by its spectrum. For the precise conditions on p and q we refer to [8]. It follows that among the connected graphs the pineapple graph is determined by its spectrum.…”

Section: Spectral Characterizationsmentioning

confidence: 99%

“…See [8] for more examples of graphs non-isomorphic but cospectral with K q p . As remarked before, there exists no connected example.…”

Section: Spectral Characterizationsmentioning

confidence: 99%

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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The graphs cospectral with the pineapple graph

Cited by 11 publications

References 3 publications

On the graphs having at most one positive eccentricity eigenvalue

On the graphs having at most one positive eccentricity eigenvalue

On the spectra of generalized Fibonomial and Jacobsthal-binomial graphs

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

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