Constructing non-isomorphic signless Laplacian cospectral graphs

Let [Formula: see text] and [Formula: see text] be three graphs on disjoint sets of vertices and [Formula: see text] has [Formula: see text] edges. Let [Formula: see text] be the graph obtained from [Formula: see text] and [Formula: see text] in the following way: (1) Delete all the edges of [Formula: see text] and consider [Formula: see text] disjoint copies of [Formula: see text]. (2) Join each vertex of the [Formula: see text]th copy of [Formula: see text] to the end vertices of the [Formula: see text]th edge of [Formula: see text]. Let [Formula: see text] be the graph obtained from [Formula: see text] by joining each vertex of [Formula: see text] with each vertex of [Formula: see text] In this paper, we determine the adjacency (respectively, Laplacian, signless Laplacian) spectrum of [Formula: see text] in terms of those of [Formula: see text] and [Formula: see text] As an application, we construct infinite pairs of cospectral graphs.

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On the path cospectral graphs and path signless Laplacian matrix of graphs

Journal of Mathematical and Computational Science ◽

10.28919/jmcs/4510 ◽

2020 ◽

Keyword(s):

Laplacian Matrix ◽

Cospectral Graphs ◽

Signless Laplacian Matrix ◽

Signless Laplacian

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Signless Laplacian and normalized Laplacian on the H-join operation of graphs

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830914500463 ◽

2014 ◽

Vol 06 (03) ◽

pp. 1450046 ◽

Cited By ~ 5

Author(s):

Bao-Feng Wu ◽

Yuan-Yuan Lou ◽

Chang-Xiang He

Keyword(s):

Spectral Radius ◽

Regular Graphs ◽

Arbitrary Graph ◽

Cospectral Graphs ◽

Signless Laplacian ◽

Integral Graphs

In this paper, we consider a generalized join operation, that is, the H-join on graphs, where H is an arbitrary graph. In terms of the signless Laplacian and the normalized Laplacian, we determine the spectra of the graphs obtained by this operation on regular graphs. Some additional consequences on the spectral radius, integral graphs and cospectral graphs, etc. are also obtained.

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Signless Laplacian spectrum of a class of generalized corona and its application

Discrete Mathematics Algorithms and Applications ◽

10.1142/s179383091850060x ◽

2018 ◽

Vol 10 (05) ◽

pp. 1850060

Author(s):

Pengli Lu ◽

Ke Gao ◽

Yumo Wu

Keyword(s):

Laplacian Spectrum ◽

Cospectral Graphs ◽

Signless Laplacian Spectrum ◽

Subdivision Graph ◽

Signless Laplacian ◽

Corona Graphs

Let [Formula: see text] be a graph with [Formula: see text] edges, [Formula: see text] the subdivision graph of [Formula: see text] with [Formula: see text] the set of inserted vertices of [Formula: see text]. The generalized subdivision-edge corona graph [Formula: see text] of [Formula: see text] and [Formula: see text] is the graph obtained from [Formula: see text] and [Formula: see text] by joining the [Formula: see text]th vertex of [Formula: see text] to every vertex of [Formula: see text]. In this paper, we determine the [Formula: see text]-polynomial of the graph [Formula: see text]. Also, we construct infinitely many pairs of [Formula: see text]-cospectral graphs and compute the incidence energy of subdivision-edge corona graphs.

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Spectra of Indu–Bala product of graphs and some new pairs of cospectral graphs

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830919500563 ◽

2019 ◽

Vol 11 (05) ◽

pp. 1950056

Author(s):

Shreekant Patil ◽

Mallikarjun Mathapati

Keyword(s):

Spanning Trees ◽

Product Formula ◽

Regular Graphs ◽

Kirchhoff Index ◽

Cospectral Graphs ◽

Laplacian Spectra ◽

Signless Laplacian ◽

Graph Operation ◽

Product Of Graphs ◽

Number Of Spanning Trees

Recently Indulal and Balakrishnan [Distance spectrum of Indu–Bala product of graphs, AKCE Int. J. Graph Comb. 13 (2016) 230–234] put forward a new graph operation, namely, the Indu–Bala product [Formula: see text] of graphs [Formula: see text] and [Formula: see text], and it is obtained from two disjoint copies of the join [Formula: see text] of [Formula: see text] and [Formula: see text] by joining the corresponding vertices in the two copies of [Formula: see text]. In this paper, we obtain the adjacency spectra, the Laplacian spectra and the signless Laplacian spectra of [Formula: see text] in terms of the corresponding spectra of [Formula: see text] and [Formula: see text]. As applications, these results enable us to construct infinitely many pairs of respective cospectral graphs. Further, the Laplacian spectra enable us to get the formulas of the number of spanning trees and Kirchhoff index of [Formula: see text] in terms of the Laplacian spectra of regular graphs [Formula: see text] and [Formula: see text].

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On the spectra of a new duplication based corona of graphs

Notes on Number Theory and Discrete Mathematics ◽

10.7546/nntdm.2021.27.1.208-220 ◽

2021 ◽

Vol 27 (1) ◽

pp. 208-220

Author(s):

Renny P. Varghese ◽

◽

D. Susha ◽

Keyword(s):

Spanning Trees ◽

Graph Product ◽

Kirchhoff Index ◽

Cospectral Graphs ◽

Laplacian Spectra ◽

Laplacian Energy ◽

Signless Laplacian ◽

Product Of Graphs ◽

Number Of Spanning Trees

In this paper we introduce a new corona-type product of graphs namely duplication corresponding corona. Here we mainly determine the adjacency, Laplacian and signless Laplacian spectra of the new graph product. In addition to that, we find out the incidence energy, the number of spanning trees, Kirchhoff index and Laplacian-energy-like invariant of the new graph. Also we discuss some new classes of cospectral graphs.

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On the construction of Q-controllable graphs

Electronic Journal of Linear Algebra ◽

10.13001/1081-3810.3298 ◽

2017 ◽

Vol 32 ◽

pp. 365-379 ◽

Cited By ~ 1

Author(s):

Zhenzhen Lou ◽

Qiongxiang Huang ◽

Xueyi Huang

Keyword(s):

Connected Graph ◽

Laplacian Spectrum ◽

Laplacian Eigenvalues ◽

Cospectral Graphs ◽

Signless Laplacian Spectrum ◽

Signless Laplacian

A connected graph is called Q-controllable if its signless Laplacian eigenvalues are mutually distinct and main. Two graphs G and H are said to be Q-cospectral if they share the same signless Laplacian spectrum. In this paper, infinite families of Q-controllable graphs are constructed, by using the operator of rooted product introduced by Godsil and McKay. In the process, innitely many non-isomorphic Q-cospectral graphs are also constructed, especially, including those graphs whose signless Laplacian eigenvalues are mutually distinct.

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Cospectral constructions for several graph matrices using cousin vertices

Special Matrices ◽

10.1515/spma-2020-0143 ◽

2021 ◽

Vol 10 (1) ◽

pp. 9-22

Author(s):

Kate Lorenzen

Keyword(s):

Adjacency Matrix ◽

Structural Information ◽

Distance Matrix ◽

Laplacian Matrix ◽

Cospectral Graphs ◽

Combinatorial Laplacian ◽

Signless Laplacian Matrix ◽

Signless Laplacian

Abstract Graphs can be associated with a matrix according to some rule and we can find the spectrum of a graph with respect to that matrix. Two graphs are cospectral if they have the same spectrum. Constructions of cospectral graphs help us establish patterns about structural information not preserved by the spectrum. We generalize a construction for cospectral graphs previously given for the distance Laplacian matrix to a larger family of graphs. In addition, we show that with appropriate assumptions this generalized construction extends to the adjacency matrix, combinatorial Laplacian matrix, signless Laplacian matrix, normalized Laplacian matrix, and distance matrix. We conclude by enumerating the prevelance of this construction in small graphs for the adjacency matrix, combinatorial Laplacian matrix, and distance Laplacian matrix.

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