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Journal articles on the topic 'Spectrum of graphs'

Author: Grafiati
Published: 4 June 2021
Last updated: 7 February 2022

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1

Alhevaz, Abdollah, Maryam Baghipur, Hilal A. Ganie, and Yilun Shang. "The Generalized Distance Spectrum of the Join of Graphs." Symmetry 12, no. 1 (2020): 169. http://dx.doi.org/10.3390/sym12010169.

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Let G be a simple connected graph. In this paper, we study the spectral properties of the generalized distance matrix of graphs, the convex combination of the symmetric distance matrix D ( G ) and diagonal matrix of the vertex transmissions T r ( G ) . We determine the spectrum of the join of two graphs and of the join of a regular graph with another graph, which is the union of two different regular graphs. Moreover, thanks to the symmetry of the matrices involved, we study the generalized distance spectrum of the graphs obtained by generalization of the join graph operation through their eig
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2

Ghorbani, M., and M. Songhori. "On the spectrum of Cayley graphs." Algebra and Discrete Mathematics 30, no. 2 (2020): 194–206. http://dx.doi.org/10.12958/adm544.

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The set of eigenvalues of the adjacency matrix of a graph is called the spectrum of it. In the present paper, we introduce the spectrum of Cayley graphs of order pqr in terms of character table, where p,q,r are prime numbers. We also, stablish some properties of Cayley graphs of non-abelian groups with a normal symmetric connected subset.
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3

Cvetkovic, Dragos, and Vesna Todorcevic. "Cospectrality graphs of Smith graphs." Filomat 33, no. 11 (2019): 3269–76. http://dx.doi.org/10.2298/fil1911269c.

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Graphs whose spectrum belongs to the interval [-2,2] are called Smith graphs. The structure of a Smith graph with a given spectrum depends on a system of Diophantine linear algebraic equations. We have established in [1] several properties of this system and showed how it can be simplified and effectively applied. In this way a spectral theory of Smith graphs has been outlined. In the present paper we introduce cospectrality graphs for Smith graphs and study their properties through examples and theoretical consideration. The new notion is used in proving theorems on cospectrality of Smith gra
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4

SUNTORNPOCH, BORWORN, and YOTSANAN MEEMARK. "CAYLEY GRAPHS OVER A FINITE CHAIN RING AND GCD-GRAPHS." Bulletin of the Australian Mathematical Society 93, no. 3 (2016): 353–63. http://dx.doi.org/10.1017/s0004972715001380.

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We extend spectral graph theory from the integral circulant graphs with prime power order to a Cayley graph over a finite chain ring and determine the spectrum and energy of such graphs. Moreover, we apply the results to obtain the energy of some gcd-graphs on a quotient ring of a unique factorisation domain.
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5

Cvetkovic, Dragos. "Spectral recognition of graphs." Yugoslav Journal of Operations Research 22, no. 2 (2012): 145–61. http://dx.doi.org/10.2298/yjor120925025c.

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At some time, in the childhood of spectral graph theory, it was conjectured that non-isomorphic graphs have different spectra, i.e. that graphs are characterized by their spectra. Very quickly this conjecture was refuted and numerous examples and families of non-isomorphic graphs with the same spectrum (cospectral graphs) were found. Still some graphs are characterized by their spectra and several mathematical papers are devoted to this topic. In applications to computer sciences, spectral graph theory is considered as very strong. The benefit of using graph spectra in treating graphs is that
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6

Iranmanesh, Mohammad, and Mahboubeh Saheli. "Toward a Laplacian spectral determination of signed ∞-graphs." Filomat 32, no. 6 (2018): 2283–94. http://dx.doi.org/10.2298/fil1806283i.

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A signed graph consists of a (simple) graph G=(V,E) together with a function ? : E ? {+,-} called signature. Matrices can be associated to signed graphs and the question whether a signed graph is determined by the set of its eigenvalues has gathered the attention of several researchers. In this paper we study the spectral determination with respect to the Laplacian spectrum of signed ?-graphs. After computing some spectral invariants and obtain some constraints on the cospectral mates, we obtain some non isomorphic signed graphs cospectral to signed ?-graphs and we study the spectral character
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7

Yu, Guidong, Tao Yu, Xiangwei Xia, and Huan Xu. "Spectral Sufficient Conditions on Pancyclic Graphs." Complexity 2021 (July 15, 2021): 1–8. http://dx.doi.org/10.1155/2021/3630245.

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A pancyclic graph of order n is a graph with cycles of all possible lengths from 3 to n . In fact, it is NP-complete that deciding whether a graph is pancyclic. Because the spectrum of graphs is convenient to be calculated, in this study, we try to use the spectral theory of graphs to study this problem and give some sufficient conditions for a graph to be pancyclic in terms of the spectral radius and the signless Laplacian spectral radius of the graph.
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8

Sciriha, Irene, and Stephanie Farrugia. "On the Spectrum of Threshold Graphs." ISRN Discrete Mathematics 2011 (January 17, 2011): 1–21. http://dx.doi.org/10.5402/2011/108509.

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The antiregular connected graph on r vertices is defined as the connected graph whose vertex degrees take the values of r−1 distinct positive integers. We explore the spectrum of its adjacency matrix and show common properties with those of connected threshold graphs, having an equitable partition with a minimal number r of parts. Structural and combinatorial properties can be deduced for related classes of graphs and in particular for the minimal configurations in the class of singular graphs.
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9

Alhevaz, Abdollah, Maryam Baghipur, and Somnath Paul. "Spectrum of graphs obtained by operations." Asian-European Journal of Mathematics 13, no. 02 (2018): 2050045. http://dx.doi.org/10.1142/s179355712050045x.

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The distance signless Laplacian matrix of a simple connected graph [Formula: see text] is defined as [Formula: see text], where [Formula: see text] is the distance matrix of [Formula: see text] and [Formula: see text] is the diagonal matrix whose main diagonal entries are the vertex transmissions in [Formula: see text]. In this paper, we first determine the distance signless Laplacian spectrum of the graphs obtained by generalization of the join and lexicographic product graph operations (namely joined union) in terms of their adjacency spectrum and the eigenvalues of an auxiliary matrix, dete
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10

Li, Shuchao, and Shujing Wang. "The $A_{\alpha}$- spectrum of graph product." Electronic Journal of Linear Algebra 35 (February 1, 2019): 473–81. http://dx.doi.org/10.13001/1081-3810.3857.

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Let $A(G)$ and $D(G)$ denote the adjacency matrix and the diagonal matrix of vertex degrees of $G$, respectively. Define $$ A_{\alpha}(G)=\alpha D(G)+(1-\alpha)A(G) $$ for any real $\alpha\in [0,1]$. The collection of eigenvalues of $A_{\alpha}(G)$ together with multiplicities is called the $A_{\alpha}$-\emph{spectrum} of $G$. Let $G\square H$, $G[H]$, $G\times H$ and $G\oplus H$ be the Cartesian product, lexicographic product, directed product and strong product of graphs $G$ and $H$, respectively. In this paper, a complete characterization of the $A_{\alpha}$-spectrum of $G\square H$ for arb
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11

Liu, Yu, and Lihua You. "Further Results on the Nullity of Signed Graphs." Journal of Applied Mathematics 2014 (2014): 1–8. http://dx.doi.org/10.1155/2014/483735.

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The nullity of a graph is the multiplicity of the eigenvalue zero in its spectrum. A signed graph is a graph with a sign attached to each of its edges. In this paper, we apply the coefficient theorem on the characteristic polynomial of a signed graph and give two formulae on the nullity of signed graphs with cut-points. As applications of the above results, we investigate the nullity of the bicyclic signed graphΓ∞p,q,l, obtain the nullity set of unbalanced bicyclic signed graphs, and thus determine the nullity set of bicyclic signed graphs.
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12

Adiga, Chandrashekar, Kinkar Das, and B. R. Rakshith. "Some Graphs Determined by their Signless Laplacian (Distance) Spectra." Electronic Journal of Linear Algebra 36, no. 36 (2020): 461–72. http://dx.doi.org/10.13001/ela.2020.4951.

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In literature, there are some results known about spectral determination of graphs with many edges. In [M.~C\'{a}mara and W.H.~Haemers. Spectral characterizations of almost complete graphs. {\em Discrete Appl. Math.}, 176:19--23, 2014.], C\'amara and Haemers studied complete graph with some edges deleted for spectral determination. In fact, they found that if the deleted edges form a matching, a complete graph $K_m$ provided $m \le n-2$, or a complete bipartite graph, then it is determined by its adjacency spectrum. In this paper, the graph $K_{n}\backslash K_{l,m}$ $(n>l+m)$ which is obtai
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13

RAJASEKARAN, SANGUTHEVAR, and VAMSI KUNDETI. "SPECTRUM BASED TECHNIQUES FOR GRAPH ISOMORPHISM." International Journal of Foundations of Computer Science 20, no. 03 (2009): 479–99. http://dx.doi.org/10.1142/s0129054109006693.

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The graph isomorphism problem is to check if two given graphs are isomorphic. Graph isomorphism is a well studied problem and numerous algorithms are available for its solution. In this paper we present algorithms for graph isomorphism that employ the spectra of graphs. An open problem that has fascinated many a scientist is if there exists a polynomial time algorithm for graph isomorphism. Though we do not solve this problem in this paper, the algorithms we present take polynomial time. These algorithms have been tested on a good collection of instances. However, we have not been able to prov
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14

Palathingal, Jeepamol J., Gopalapillai Indulal, and S. Aparna Lakshmanan. "Spectrum of Gallai Graph of Some Graphs." Indian Journal of Pure and Applied Mathematics 51, no. 4 (2020): 1829–41. http://dx.doi.org/10.1007/s13226-020-0499-0.

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15

Ganie, Hilal A. "On distance Laplacian spectrum (energy) of graphs." Discrete Mathematics, Algorithms and Applications 12, no. 05 (2020): 2050061. http://dx.doi.org/10.1142/s1793830920500615.

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For a simple connected graph [Formula: see text] of order [Formula: see text] having distance Laplacian eigenvalues [Formula: see text], the distance Laplacian energy [Formula: see text] is defined as [Formula: see text], where [Formula: see text] is the Wiener index of [Formula: see text]. We obtain the distance Laplacian spectrum of the joined union of graphs [Formula: see text] in terms of their distance Laplacian spectrum and the spectrum of an auxiliary matrix. As application, we obtain the distance Laplacian spectrum of the lexicographic product of graphs. We study the distance Laplacian
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16

Farooq, Rashid, Mehar Ali Malik, Qudsia Naureen, and Shariefuddin Pirzada. "On the nullity of a family of tripartite graphs." Acta Universitatis Sapientiae, Informatica 8, no. 1 (2016): 96–107. http://dx.doi.org/10.1515/ausi-2016-0006.

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Abstract The eigenvalues of the adjacency matrix of a graph form the spectrum of the graph. The multiplicity of the eigenvalue zero in the spectrum of a graph is called nullity of the graph. Fan and Qian (2009) obtained the nullity set of n-vertex bipartite graphs and characterized the bipartite graphs with nullity n − 4 and the regular n-vertex bipartite graphs with nullity n − 6. In this paper, we study similar problem for a class of tripartite graphs. As observed the nullity problem in tripartite graphs does not follow as an extension to that of the nullity of bipartite graphs, this makes t
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17

Atik, Fouzul, and Pratima Panigrahi. "Graphs with few distinct distance eigenvalues irrespective of the diameters." Electronic Journal of Linear Algebra 29 (September 20, 2015): 194–205. http://dx.doi.org/10.13001/1081-3810.2947.

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The distance matrix of a simple connected graph $G$ is $D(G)=(d_{ij})$, where $d_{ij}$ is the distance between $i$th and $j$th vertices of $G$. The multiset of all eigenvalues of $D(G)$ is known as the distance spectrum of $G$. Lin et al.(On the distance spectrum of graphs. \newblock {\em Linear Algebra Appl.}, 439:1662-1669, 2013) asked for existence of graphs other than strongly regular graphs and some complete $k$-partite graphs having exactly three distinct distance eigenvalues. In this paper some classes of graphs with arbitrary diameter and satisfying this property is constructed. For ea
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18

Thomas, Ann Susa, Sunny Joseph Kalayathankal, and Joseph Varghese Kureethara. "An Introductory Note on the Spectrum and Energy of Molecular Graphs." Mapana - Journal of Sciences 16, no. 2 (2017): 17–27. http://dx.doi.org/10.12723/mjs.41.3.

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Graph Theory is one branch of Mathematics that laid the foundations of the structural studies in Chemistry. The fact that every molecule or compound can be represented as a network of vertices (elements) and edges (bonds) provoked the question of the predictability of the physical and chemical properties of molecules and compounds. Spectrum, π-electron energy, Spectral Radius etc. are predictable using graph theoretical methods. This is an introductory paper about spectrum and energy of molecular graphs.
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19

Paul, Somnath. "On distance and distance Laplacian spectra of corona of two graphs." Discrete Mathematics, Algorithms and Applications 08, no. 01 (2016): 1650007. http://dx.doi.org/10.1142/s1793830916500075.

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Corona of two graphs has been defined in [F. Harary, Graph Theory (Addison-Wesley, 1969)]. In this paper, we study the distance and the distance Laplacian spectra of corona of two graphs and describe the complete distance (distance Laplacian) spectrum for some particular cases. As an application, we show that the corona operation can be used to create distance singular graphs. We also show that these results enable us to construct infinitely many pairs of distance (respectively, distance Laplacian) cospectral graphs. Last, we give a graph transformation and discuss its effect on the distance L
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20

Hamzeh, Asma, and Ali Ashrafi. "Spectrum and L-spectrum of the power graph and its main supergraph for certain finite groups." Filomat 31, no. 16 (2017): 5323–34. http://dx.doi.org/10.2298/fil1716323h.

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Let G be a finite group. The power graph P(G) and its main supergraph S(G) are two simple graphs with the same vertex set G. Two elements x,y ? G are adjacent in the power graph if and only if one is a power of the other. They are joined in S(G) if and only if o(x)|o(y) or o(y)|o(x). The aim of this paper is to compute the characteristic polynomial of these graph for certain finite groups. As a consequence, the spectrum and Laplacian spectrum of these graphs for dihedral, semi-dihedral, cyclic and dicyclic groups were computed.
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21

Cvetkovic, Dragos, and Mirko Lepovic. "Sets of cospectral graphs with least eigenvalue at least -2 and some related results." Bulletin: Classe des sciences mathematiques et natturalles 129, no. 29 (2004): 85–102. http://dx.doi.org/10.2298/bmat0429085c.

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In this paper we study the phenomenon of cospectrality in generalized line graphs and in exceptional graphs. The paper contains a table of sets of Co spectral graphs with least eigenvalue at least ?2 and at most 8 vertices together with some comments and theoretical explanations of the phenomena suggested by the table. In particular, we prove that the multiplicity of the number 0 in the spectrum of a generalized line graph L(G) is at least the number of petals of the corresponding root graph G. .
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22

Palathingal, Jeepamol J., Aparna S. Lakshmanan, and Gopalapillai Indulal. "Spectrum of anti-gallai graph of some graphs." Indian Journal of Pure and Applied Mathematics 52, no. 1 (2021): 304–11. http://dx.doi.org/10.1007/s13226-021-00066-z.

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23

Imran, Muhammad, Yasir Ali, Mehar Ali Malik, and Kiran Hasnat. "Chromatic spectrum of some classes of 2-regular bipartite colored graphs." Journal of Intelligent & Fuzzy Systems 41, no. 1 (2021): 1125–33. http://dx.doi.org/10.3233/jifs-210066.

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Chromatic spectrum of a colored graph G is a multiset of eigenvalues of colored adjacency matrix of G. The nullity of a disconnected graph is equal to sum of nullities of its components but we show that this result does not hold for colored graphs. In this paper, we investigate the chromatic spectrum of three different classes of 2-regular bipartite colored graphs. In these classes of graphs, it is proved that the nullity of G is not sum of nullities of components of G. We also highlight some important properties and conjectures to extend this problem to general graphs.
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24

Abudayah, Mohammad, Omar Alomari, and Torsten Sander. "On the N-spectrum of oriented graphs." Open Mathematics 18, no. 1 (2020): 486–95. http://dx.doi.org/10.1515/math-2020-0167.

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Abstract Given any digraph D, its non-negative spectrum (or N-spectrum, shortly) consists of the eigenvalues of the matrix AA T , where A is the adjacency matrix of D. In this study, we relate the classical spectrum of undirected graphs to the N-spectrum of their oriented counterparts, permitting us to derive spectral bounds. Moreover, we study the spectral effects caused by certain modifications of a given digraph.
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Abudayah, Mohammad, Omar Alomari, and Torsten Sander. "Spectrum of free-form Sudoku graphs." Open Mathematics 16, no. 1 (2018): 1445–54. http://dx.doi.org/10.1515/math-2018-0125.

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AbstractA free-form Sudoku puzzle is a square arrangement ofm×mcells such that the cells are partitioned intomsubsets (called blocks) of equal cardinality. The goal of the puzzle is to place integers 1, . . ,min the cells such that the numbers in every row, column and block are distinct. Represent each cell by a vertex and add edges between two vertices exactly when the corresponding cells, according to the rules, must contain different numbers. This yields the associated free-form Sudoku graph. This article studies the eigenvalues of free-form Sudoku graphs, most notably integrality. Further,
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26

Simic, Slobodan, and Zoran Stanic. "On Q-integral (3,s)-semiregular bipartite graphs." Applicable Analysis and Discrete Mathematics 4, no. 1 (2010): 167–74. http://dx.doi.org/10.2298/aadm1000002s.

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A graph is called Q-integral if its signless Laplacian spectrum consists entirely of integers. We establish some general results regarding signless Laplacians of semiregular bipartite graphs. Especially, we consider those semiregular bipartite graphs with integral signless Laplacian spectrum. In some particular cases we determine the possible Q-spectra and consider the corresponding graphs.
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27

Zakharov, A. A., A. E. Barinov, and A. L. Zhiznyakov. "RECOGNITION OF HUMAN POSE FROM IMAGES BASED ON GRAPH SPECTRA." ISPRS - International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences XL-5/W6 (May 18, 2015): 9–12. http://dx.doi.org/10.5194/isprsarchives-xl-5-w6-9-2015.

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Recognition of human pose is an actual problem in computer vision. To increase the reliability of the recognition it is proposed to use structured information in the form of graphs. The spectrum of graphs is applied for the comparison of the structures. Image skeletonization is used to construct graphs. Line segments are the nodes of the graph. The end point of line segments are the edges of the graph. The angles between adjacent segments are used to set the weights of the adjacency matrix. The Laplacian matrix is used to generate the spectrum graph. The algorithm consists of the following ste
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28

Andjelic, Milica, Tamara Koledin, and Zoran Stanic. "Distance spectrum and energy of graphs with small diameter." Applicable Analysis and Discrete Mathematics 11, no. 1 (2017): 108–22. http://dx.doi.org/10.2298/aadm1701108a.

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In this paper we express the distance spectrum of graphs with small diameter in terms of the eigenvalues of their adjacency matrix. We also compute the distance energy of particular types of graph and determine a sequence of infinite families of distance equienergetic graphs.
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29

Nath, Rajat Kanti, and Jutirekha Dutta. "Spectrum of commuting graphs of some classes of finite groups." MATEMATIKA 33, no. 1 (2017): 87. http://dx.doi.org/10.11113/matematika.v33.n1.812.

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In this paper, we initiate the study of spectrum of the commuting graphs of finite non-abelian groups. We first compute the spectrum of this graph for several classes of finite groups, in particular AC-groups. We show that the commuting graphs of finite non-abelian AC-groups are integral. We also show that the commuting graph of a finite non-abelian group G is integral if G is not isomorphic to the symmetric group of degree 4 and the commuting graph of G is planar. Further, it is shown that the commuting graph of G is integral if its commuting graph is toroidal.
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30

Ahmadi, Omran, Noga Alon, Ian F. Blake, and Igor E. Shparlinski. "Graphs with integral spectrum." Linear Algebra and its Applications 430, no. 1 (2009): 547–52. http://dx.doi.org/10.1016/j.laa.2008.08.020.

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31

Hamidi, Mohammad, and Arsham Saeid. "Accessible spectrum of graphs." Applicable Analysis and Discrete Mathematics 15, no. 1 (2021): 1–26. http://dx.doi.org/10.2298/aadm180319007h.

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This paper computes eigenvalues of discrete complete hypergraphs and partitioned hypergraphs. We define positive equivalence relation on hypergraphs that establishes a connection between hypergraphs and graphs. With this regards it makes a connection between spectrum of graphs and spectrum of quotient of any hypergraphs. Finally, this study tries to construct spectrum of path trees via quotient of partitioned hypergraphs.
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32

Abdian, Ali Zeydi, and S. Morteza Mirafzal. "The spectral characterizations of the connected multicone graphs Kw ▽ LHS and Kw ▽ LGQ(3,9)." Discrete Mathematics, Algorithms and Applications 10, no. 02 (2018): 1850019. http://dx.doi.org/10.1142/s1793830918500192.

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In the past decades, graphs that are determined by their spectrum have received much more and more attention, since they have been applied to several fields, such as randomized algorithms, combinatorial optimization problems and machine learning. An important part of spectral graph theory is devoted to determining whether given graphs or classes of graphs are determined by their spectra or not. So, finding and introducing any class of graphs which are determined by their spectra can be an interesting and important problem. The main aim of this study is to characterize two classes of multicone
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33

Balinska, Krystyna, Dragos Cvetkovic, Zoran Radosavljevic, Slobodan Simic, and Dragan Stevanovic. "A survey on integral graphs." Publikacije Elektrotehnickog fakulteta - serija: matematika, no. 13 (2002): 42–65. http://dx.doi.org/10.2298/petf0213042b.

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34

Lorenzen, Kate. "Cospectral constructions for several graph matrices using cousin vertices." Special Matrices 10, no. 1 (2021): 9–22. http://dx.doi.org/10.1515/spma-2020-0143.

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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
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Cardoso, Domingos, Paula Carvalho, Paula Rama, Slobodan Simic, and Zoran Stanic. "Lexicographic polynomials of graphs and their spectra." Applicable Analysis and Discrete Mathematics 11, no. 2 (2017): 258–72. http://dx.doi.org/10.2298/aadm1702258c.

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For a (simple) graph H and non-negative integers c0, c1,..., cd (cd ? 0), p(H) = ?dk=0 ck?Hk is the lexicographic polynomial in H of degree d, where the sum of two graphs is their join and ck?Hk is the join of ck copies of Hk. The graph Hk is the kth power of H with respect to the lexicographic product (H0 = K1). The spectrum (if H is connected and regular) and the Laplacian spectrum (in general case) of p(H) are determined in terms of the spectrum of H and ck's. Constructions of infinite families of cospectral or integral graphs are announced.
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36

Urakawa, Hajime. "The Spectrum of an Infinite Graph." Canadian Journal of Mathematics 52, no. 5 (2000): 1057–84. http://dx.doi.org/10.4153/cjm-2000-044-2.

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AbstractIn this paper, we consider the (essential) spectrum of the discrete Laplacian of an infinite graph. We introduce a new quantity for an infinite graph, in terms of which we give new lower bound estimates of the (essential) spectrum and give also upper bound estimates when the infinite graph is bipartite. We give sharp estimates of the (essential) spectrum for several examples of infinite graphs.
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37

Alhevaz, Abdollah, Maryam Baghipur, and Ebrahim Hashemi. "Further results on the distance signless Laplacian spectrum of graphs." Asian-European Journal of Mathematics 11, no. 05 (2018): 1850066. http://dx.doi.org/10.1142/s1793557118500663.

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The distance signless Laplacian matrix [Formula: see text] of a connected graph [Formula: see text] is defined as [Formula: see text], where [Formula: see text] is the distance matrix of [Formula: see text] and [Formula: see text] is the diagonal matrix whose main entries are the vertex transmissions of [Formula: see text], and the spectral radius of a connected graph [Formula: see text] is the largest eigenvalue of [Formula: see text]. In this paper, first we obtain the [Formula: see text]-eigenvalues of the join of certain regular graphs. Next, we give some new bounds on the distance signles
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38

Barik, S., R. B. Bapat, and S. Pati. "On the Laplacian spectra of product graphs." Applicable Analysis and Discrete Mathematics 9, no. 1 (2015): 39–58. http://dx.doi.org/10.2298/aadm150218006b.

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Graph products and their structural properties have been studied extensively by many researchers. We investigate the Laplacian eigenvalues and eigenvectors of the product graphs for the four standard products, namely, the Cartesian product, the direct product, the strong product and the lexicographic product. A complete characterization of Laplacian spectrum of the Cartesian product of two graphs has been done by Merris. We give an explicit complete characterization of the Laplacian spectrum of the lexicographic product of two graphs using the Laplacian spectra of the factors. For the other tw
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39

Jordan, Jonathan. "SPECTRUM OF THE LAPLACIAN OF AN ASYMMETRIC FRACTAL GRAPH." Proceedings of the Edinburgh Mathematical Society 49, no. 1 (2006): 101–13. http://dx.doi.org/10.1017/s001309150400063x.

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AbstractWe consider a simple self-similar sequence of graphs that does not satisfy the symmetry conditions that imply the existence of a spectral decimation property for the eigenvalues of the graph Laplacians. We show that, for this particular sequence, a very similar property to spectral decimation exists, and we obtain a complete description of the spectra of the graphs in the sequence.
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40

Song, Haizhou, and Lulu Tian. "On the Maximal-Adjacency-Spectrum Unicyclic Graphs with Given Maximum Degree." Mathematical Problems in Engineering 2020 (July 6, 2020): 1–23. http://dx.doi.org/10.1155/2020/9861834.

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In this paper, we study the properties and structure of the maximal-adjacency-spectrum unicyclic graphs with given maximum degree. We obtain some necessary conditions on the maximal-adjacency-spectrum unicyclic graphs in the set of unicyclic graphs with n vertices and maximum degree Δ and describe the structure of the maximal-adjacency-spectrum unicyclic graphs in the set. Besides, we also give a new upper bound on the adjacency spectral radius of unicyclic graphs, and this new upper bound is the best upper bound expressed by vertices n and maximum degree Δ from now on.
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41

Abdollahi, Alireza, Shahrooz Janbaz, and Mojtaba Jazaeri. "Groups all of whose undirected Cayley graphs are determined by their spectra." Journal of Algebra and Its Applications 15, no. 09 (2016): 1650175. http://dx.doi.org/10.1142/s0219498816501759.

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The adjacency spectrum [Formula: see text] of a graph [Formula: see text] is the multiset of eigenvalues of its adjacency matrix. Two graphs with the same spectrum are called cospectral. A graph [Formula: see text] is “determined by its spectrum” (DS for short) if every graph cospectral to it is in fact isomorphic to it. A group is DS if all of its Cayley graphs are DS. A group [Formula: see text] is Cay-DS if every two cospectral Cayley graphs of [Formula: see text] are isomorphic. In this paper, we study finite DS groups and finite Cay-DS groups. In particular we prove that a finite DS group
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42

Dutta, Jutirekha, Walaa Nabil Taha Fasfous, and Rajat Kanti Nath. "Spectrum and genus of commuting graphs of some classes of finite rings." Acta et Commentationes Universitatis Tartuensis de Mathematica 23, no. 1 (2019): 5–12. http://dx.doi.org/10.12697/acutm.2019.23.01.

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We consider commuting graphs of some classes of finite rings and compute their spectrum and genus. We show that the commuting graph of a finite CC-ring is integral. We also characterize some finite rings whose commuting graphs are planar.
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43

Jog, S. R., and Raju Kotambari. "On the Adjacency, Laplacian, and Signless Laplacian Spectrum of Coalescence of Complete Graphs." Journal of Mathematics 2016 (2016): 1–11. http://dx.doi.org/10.1155/2016/5906801.

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Coalescence as one of the operations on a pair of graphs is significant due to its simple form of chromatic polynomial. The adjacency matrix, Laplacian matrix, and signless Laplacian matrix are common matrices usually considered for discussion under spectral graph theory. In this paper, we compute adjacency, Laplacian, and signless Laplacian energy (Qenergy) of coalescence of pair of complete graphs. Also, as an application, we obtain the adjacency energy of subdivision graph and line graph of coalescence from itsQenergy.
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44

Coutino, Mario, Sundeep Prabhakar Chepuri, Takanori Maehara, and Geert Leus. "Fast Spectral Approximation of Structured Graphs with Applications to Graph Filtering." Algorithms 13, no. 9 (2020): 214. http://dx.doi.org/10.3390/a13090214.

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To analyze and synthesize signals on networks or graphs, Fourier theory has been extended to irregular domains, leading to a so-called graph Fourier transform. Unfortunately, different from the traditional Fourier transform, each graph exhibits a different graph Fourier transform. Therefore to analyze the graph-frequency domain properties of a graph signal, the graph Fourier modes and graph frequencies must be computed for the graph under study. Although to find these graph frequencies and modes, a computationally expensive, or even prohibitive, eigendecomposition of the graph is required, the
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45

FIOL, M. A., E. GARRIGA, and J. L. A. YEBRA. "On Twisted Odd Graphs." Combinatorics, Probability and Computing 9, no. 3 (2000): 227–40. http://dx.doi.org/10.1017/s0963548300004181.

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The twisted odd graphs are obtained from the well-known odd graphs through an involutive automorphism. As expected, the twisted odd graphs share some of the interesting properties of the odd graphs but, in general, they seem to have a more involved structure. Here we study some of their basic properties, such as their automorphism group, diameter, and spectrum. They turn out to be examples of the so-called boundary graphs, which are graphs satisfying an extremal property that arises from a bound for the diameter of a graph in terms of its distinct eigenvalues.
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46

Taheri, R., M. Behboodi, and A. Tehranian. "The spectrum subgraph of the annihilating-ideal graph of a commutative ring." Journal of Algebra and Its Applications 14, no. 08 (2015): 1550130. http://dx.doi.org/10.1142/s0219498815501303.

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In this paper we introduce and study the spectrum graph of a commutative ring R, denoted by 𝔸𝔾s(R), that is, the graph whose vertices are all non-zero prime ideals of R with non-zero annihilator and two distinct vertices P1, P2 are adjacent if and only if P1P2 = (0). This is an induced subgraph of the annihilating-ideal graph 𝔸𝔾(R) of R. Among other results, we present the structures of all graphs which can be realized as the spectrum graph of a commutative ring. Then we show that for a non-domain Noetherian ring R, 𝔸𝔾s(R), is a connected graph if and only if 𝔸𝔾s(R) is a star graph if and only
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47

Pushpalatha, A. P., G. Jothilakshmi, S. Suganthi, and V. Swaminathan. "Forcing Independent Spectrum in Graphs." International Journal of Computer Applications 21, no. 2 (2011): 1–6. http://dx.doi.org/10.5120/2487-3355.

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48

Silva, Ana. "Graphs with small fall-spectrum." Discrete Applied Mathematics 254 (February 2019): 183–88. http://dx.doi.org/10.1016/j.dam.2018.06.037.

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49

Deng, Bo, Caibing Chang, Haixing Zhao, and Kinkar Chandra Das. "Construction for the Sequences of Q-Borderenergetic Graphs." Mathematical Problems in Engineering 2020 (July 18, 2020): 1–5. http://dx.doi.org/10.1155/2020/6176849.

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This research intends to construct a signless Laplacian spectrum of the complement of any k-regular graph G with order n. Through application of the join of two arbitrary graphs, a new class of Q-borderenergetic graphs is determined with proof. As indicated in the research, with a regular Q-borderenergetic graph, sequences of regular Q-borderenergetic graphs can be constructed. The procedures for such a construction are determined and demonstrated. Significantly, all the possible regular Q-borderenergetic graphs of order 7<n≤10 are determined.
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50

Lu, Pengli, Ke Gao, and Yumo Wu. "Signless Laplacian spectrum of a class of generalized corona and its application." Discrete Mathematics, Algorithms and Applications 10, no. 05 (2018): 1850060. http://dx.doi.org/10.1142/s179383091850060x.

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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]. Al
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