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Journal articles on the topic 'Non-isomorphic graph'

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Published: 4 June 2021
Last updated: 4 February 2022

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1

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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2

Lo Faro, Giovanni, Salvatore Milici, and Antoinette Tripodi. "Uniformly Resolvable Decompositions of Kv-I into n-Cycles and n-Stars, for Even n." Mathematics 8, no. 10 (2020): 1755. http://dx.doi.org/10.3390/math8101755.

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If X is a connected graph, then an X-factor of a larger graph is a spanning subgraph in which all of its components are isomorphic to X. Given a set Γ of pairwise non-isomorphic graphs, a uniformly resolvable Γ-decomposition of a graph G is an edge decomposition of G into X-factors for some graph X∈Γ. In this article we completely solve the existence problem for decompositions of Kv-I into Cn-factors and K1,n-factors in the case when n is even.
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3

SRIDHARAN, N., S. AMUTHA, and S. B. RAO. "INDUCED SUBGRAPHS OF GAMMA GRAPHS." Discrete Mathematics, Algorithms and Applications 05, no. 03 (2013): 1350012. http://dx.doi.org/10.1142/s1793830913500122.

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Let G be a graph. The gamma graph of G denoted by γ ⋅ G is the graph with vertex set V(γ ⋅ G) as the set of all γ-sets of G and two vertices D and S of γ ⋅ G are adjacent if and only if |D ∩ S| = γ(G) – 1. A graph H is said to be a γ-graph if there exists a graph G such that γ ⋅ G is isomorphic to H. In this paper, we show that every induced subgraph of a γ-graph is also a γ-graph. Furthermore, if we prove that H is a γ-graph, then there exists a sequence {Gn} of non-isomorphic graphs such that H = γ ⋅ Gn for every n.
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4

Huang, Shaobin, Jiang Zhou, and Changjiang Bu. "Signless Laplacian spectral characterization of graphs with isolated vertices." Filomat 30, no. 14 (2016): 3689–96. http://dx.doi.org/10.2298/fil1614689h.

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A graph is said to be DQS if there is no other non-isomorphic graph with the same signless Laplacian spectrum. For a DQS graph G, we show that G ? rK1 is DQS under certain conditions. Applying these results, some DQS graphs with isolated vertices are obtained.
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5

Jain, Vivek, and Pradeep Kumar. "A note on the power graphs of finite nilpotent groups." Filomat 34, no. 7 (2020): 2451–61. http://dx.doi.org/10.2298/fil2007451j.

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The power graph P(G) of a group G is the graph with vertex set G and two distinct vertices are adjacent if one is a power of the other. Two finite groups are said to be conformal, if they contain the same number of elements of each order. Let Y be a family of all non-isomorphic odd order finite nilpotent groups of class two or p-groups of class less than p. In this paper, we prove that the power graph of each group in Y is isomorphic to the power graph of an abelian group and two groups in Y have isomorphic power graphs if they are conformal. We determine the number of maximal cyclic subgroups
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6

Costalonga, João Paulo, Robert J. Kingan, and Sandra R. Kingan. "Constructing Minimally 3-Connected Graphs." Algorithms 14, no. 1 (2021): 9. http://dx.doi.org/10.3390/a14010009.

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A 3-connected graph is minimally 3-connected if removal of any edge destroys 3-connectivity. We present an algorithm for constructing minimally 3-connected graphs based on the results in (Dawes, JCTB 40, 159-168, 1986) using two operations: adding an edge between non-adjacent vertices and splitting a vertex. To test sets of vertices and edges for 3-compatibility, which depends on the cycles of the graph, we develop a method for obtaining the cycles of G′ from the cycles of G, where G′ is obtained from G by one of the two operations above. We eliminate isomorphic duplicates using certificates g
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7

Shiau, S. Y., R. Joynt, and S. N. Coppersmith. "Physically-motivated dynamical algorithms for the graph isomorphism problem." Quantum Information and Computation 5, no. 6 (2005): 492–506. http://dx.doi.org/10.26421/qic5.6-7.

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The graph isomorphism problem (GI) plays a central role in the theory of computational complexity and has importance in physics and chemistry as well \cite{kobler93,fortin96}. No polynomial-time algorithm for solving GI is known. We investigate classical and quantum physics-based polynomial-time algorithms for solving the graph isomorphism problem in which the graph structure is reflected in the behavior of a dynamical system. We show that a classical dynamical algorithm proposed by Gudkov and Nussinov \cite{gudkov02} as well as its simplest quantum generalization fail to distinguish pairs of
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8

Zhao, Jinxing, and Guixin Deng. "Remark on subgroup intersection graph of finite abelian groups." Open Mathematics 18, no. 1 (2020): 1025–29. http://dx.doi.org/10.1515/math-2020-0066.

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Abstract Let G be a finite group. The subgroup intersection graph \text{Γ}(G) of G is a graph whose vertices are non-identity elements of G and two distinct vertices x and y are adjacent if and only if |\langle x\rangle \cap \langle y\rangle |\gt 1 , where \langle x\rangle is the cyclic subgroup of G generated by x. In this paper, we show that two finite abelian groups are isomorphic if and only if their subgroup intersection graphs are isomorphic.
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9

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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10

Ahmadidelir, Karim. "On the non-commuting graph in finite Moufang loops." Journal of Algebra and Its Applications 17, no. 04 (2018): 1850070. http://dx.doi.org/10.1142/s0219498818500706.

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The non-commuting graph associated to a non-abelian group [Formula: see text], [Formula: see text], is a graph with vertex set [Formula: see text] where distinct non-central elements [Formula: see text] and [Formula: see text] of [Formula: see text] are joined by an edge if and only if [Formula: see text]. The non-commuting graph of a non-abelian finite group has received some attention in existing literature. Recently, many authors have studied the non-commuting graph associated to a non-abelian group. In particular, the authors put forward the following conjectures: Conjecture 1. Let [Formul
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11

Hammer, P. L., and A. K. Kelmans. "On Universal Threshold Graphs." Combinatorics, Probability and Computing 3, no. 3 (1994): 327–44. http://dx.doi.org/10.1017/s096354830000122x.

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A graph G is threshold if there exists a ‘weight’ function w: V(G) → R such that the total weight of any stable set of G is less than the total weight of any non-stable set of G. Let n denote the set of threshold graphs with n vertices. A graph is called n-universal if it contains every threshold graph with n vertices as an induced subgraph. n-universal threshold graphs are of special interest, since they are precisely those n-universal graphs that do not contain any non-threshold induced subgraph.In this paper we shall study minimumn-universal (threshold) graphs, i.e.n-universal (threshold) g
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12

Vatandoost, E., and Y. Golkhandypour. "On the non-cyclic graph of a group." Discrete Mathematics, Algorithms and Applications 11, no. 01 (2019): 1950006. http://dx.doi.org/10.1142/s179383091950006x.

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Here we study some algebraic properties of non-cyclic graphs. In this paper, we show that [Formula: see text] is isomorphic to [Formula: see text] or [Formula: see text] if and only if [Formula: see text] is isomorphic to [Formula: see text] or [Formula: see text], respectively. We characterize all groups of order [Formula: see text] like [Formula: see text] in which [Formula: see text], where [Formula: see text].
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13

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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14

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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15

Azam, Naveed Ahmed, Aleksandar Shurbevski, and Hiroshi Nagamochi. "Enumerating Tree-Like Graphs and Polymer Topologies with a Given Cycle Rank." Entropy 22, no. 11 (2020): 1295. http://dx.doi.org/10.3390/e22111295.

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Cycle rank is an important notion that is widely used to classify, understand, and discover new chemical compounds. We propose a method to enumerate all non-isomorphic tree-like graphs of a given cycle rank with self-loops and no multiple edges. To achieve this, we develop an algorithm to enumerate all non-isomorphic rooted graphs with the required constraints. The idea of our method is to define a canonical representation of rooted graphs and enumerate all non-isomorphic graphs by generating the canonical representation of rooted graphs. An important feature of our method is that for an integ
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16

VAIDYA, SAMIR K., and KALPESH POPAT. "On Equienergetic, Hyperenergetic and Hypoenergetic Graphs." Kragujevac Journal of Mathematics 44, no. 4 (2020): 523–32. http://dx.doi.org/10.46793/kgjmat2004.523v.

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The eigenvalue of a graph G is the eigenvalue of its adjacency matrix and the energy E(G) is the sum of absolute values of eigenvalues of graph G. Two non-isomorphic graphs G1 and G2 of the same order are said to be equienergetic if E(G1) = E(G2). The graphs whose energy is greater than that of complete graph are called hyperenergetic and the graphs whose energy is less than that of its order are called hypoenergetic graphs. The natural question arises: Are there any pairs of equienergetic graphs which are also hyperenergetic (hypoenergetic)? We have found an affirmative answer of this question
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17

REDMOND, SHANE P. "RECOVERING RINGS FROM ZERO-DIVISOR GRAPHS." Journal of Algebra and Its Applications 12, no. 08 (2013): 1350047. http://dx.doi.org/10.1142/s0219498813500473.

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Suppose G is the zero-divisor graph of some commutative ring with 1. When G has four or more vertices, a method is presented to find a specific commutative ring R with 1 such that Γ(R) ≅ G. Furthermore, this ring R can be written as R ≅ R1 × R2 × ⋯ × Rn, where each Ri is local and this representation of R is unique up to factors Ri with isomorphic zero-divisor graphs. It is also shown that for graphs on four or more vertices, no local ring has the same zero-divisor graph as a non-local ring and no reduced ring has the same zero-divisor graph as a non-reduced ring.
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18

Abrosimov, M. B., and P. V. Razumovsky. "About non-isomorphic graph colouring generating by read - faradzhev method." Prikladnaya diskretnaya matematika. Prilozhenie, no. 12 (September 1, 2019): 173–76. http://dx.doi.org/10.17223/2226308x/12/48.

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19

Huang, Xueyi, Qiongxiang Huang, and Lu Lu. "On the Second Least Distance Eigenvalue of a Graph." Electronic Journal of Linear Algebra 32 (February 6, 2017): 531–38. http://dx.doi.org/10.13001/1081-3810.3607.

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Let $G$ be a connected graph on $n$ vertices, and let $D(G)$ be the distance matrix of $G$. Let $\partial_1(G)\ge\partial_2(G)\ge\cdots\ge\partial_n(G)$ denote the eigenvalues of $D(G)$. In this paper, the connected graphs with @n􀀀1(G) at least the smallest root of $x^3=3x^2-11x-6 = 0$ are determined. Additionally, some non-isomorphic distance cospectral graphs are given.
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20

Ali, Akbar. "Tetracyclic graphs with maximum second Zagreb index: A simple approach." Asian-European Journal of Mathematics 11, no. 05 (2018): 1850064. http://dx.doi.org/10.1142/s179355711850064x.

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In the chemical graph theory, graph invariants are usually referred to as topological indices. The second Zagreb index (denoted by [Formula: see text]) is one of the most studied topological indices. For [Formula: see text], let [Formula: see text] be the collection of all non-isomorphic connected graphs with [Formula: see text] vertices and [Formula: see text] edges (such graphs are known as tetracyclic graphs). Recently, Habibi et al. [Extremal tetracyclic graphs with respect to the first and second Zagreb indices, Trans. on Combin. 5(4) (2016) 35–55.] characterized the graph having maximum
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21

Dutta, Jutirekha, Dhiren K. Basnet, and Rajat K. Nath. "On Generalized Non-commuting Graph of a Finite Ring." Algebra Colloquium 25, no. 01 (2018): 149–60. http://dx.doi.org/10.1142/s100538671800010x.

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Let S and K be two subrings of a finite ring R. Then the generalized non-commuting graph of subrings S, K of R, denoted by ГS,K, is a simple graph whose vertex set is [Formula: see text], and where two distinct vertices a, b are adjacent if and only if [Formula: see text] or [Formula: see text] and [Formula: see text]. We determine the diameter, girth and some dominating sets for ГS,K. Some connections between ГS,K and Pr(S, K) are also obtained. Further, ℤ-isoclinism between two pairs of finite rings is defined, and we show that the generalized non-commuting graphs of two ℤ-isoclinic pairs ar
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22

Bapat, Mukund. "WHEEL RELATED ONE POINT UNION OF VERTEX PRIME GRAPHS AND INVARIANCE." International Journal of Engineering Technologies and Management Research 5, no. 3 (2020): 145–50. http://dx.doi.org/10.29121/ijetmr.v5.i3.2018.186.

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We investigate one point unions of W4, and graphs obtained from W4 such as gear graph G4, each cycle edge of W4 replaced with Pm, each pokes of W4 replaced with Pm for vertex prime labeling. All different non isomorphic structures of these graphs obtained by taking one point union graphs are shown to be vertex prime. This property of graphs is called as invariance under vertex prime labeling.
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23

Sangeetha, R., and A. Muthusamy. "Hamilton cycle rich 2-factorization of complete bipartite graphs." Discrete Mathematics, Algorithms and Applications 07, no. 03 (2015): 1550026. http://dx.doi.org/10.1142/s1793830915500263.

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A 2-factorization {F1, F2,…,Fd} of a 2d-regular graph G such that each [Formula: see text] and the remaining Fi's are all Hamilton cycles is called Hamilton cycle rich 2-factorization of G, where Gi's are the given non-isomorphic 2-factors of G. In this paper, we prove that there exists a 2-factorization {F1, F2,…,Fn} of K2n,2n such that F1 ≅ G1, F2 ≅ G2 and the remaining Fi's are Hamilton cycles of K2n,2n, where G1 and G2 are the given two non-isomorphic 2-factors of K2n,2n. In fact our result together with the earlier results settles the existence of Hamilton cycle rich 2-factorizations of K
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24

Jahandideh, M., R. Modabernia, and S. Shokrolahi. "Non-commuting graphs of certain almost simple groups." Asian-European Journal of Mathematics 12, no. 05 (2019): 1950081. http://dx.doi.org/10.1142/s1793557119500815.

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Let [Formula: see text] be a non-abelian finite group and [Formula: see text] be the center of [Formula: see text]. The non-commuting graph, [Formula: see text], associated to [Formula: see text] is the graph whose vertex set is [Formula: see text] and two distinct vertices [Formula: see text] are adjacent if and only if [Formula: see text]. We conjecture that if [Formula: see text] is an almost simple group and [Formula: see text] is a non-abelian finite group such that [Formula: see text], then [Formula: see text]. Among other results, we prove that if [Formula: see text] is a certain almost
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25

Nath, Rajat Kanti, Walaa Nabil Taha Fasfous, Kinkar Chandra Das, and Yilun Shang. "Common Neighborhood Energy of Commuting Graphs of Finite Groups." Symmetry 13, no. 9 (2021): 1651. http://dx.doi.org/10.3390/sym13091651.

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The commuting graph of a finite non-abelian group G with center Z(G), denoted by Γc(G), is a simple undirected graph whose vertex set is G∖Z(G), and two distinct vertices x and y are adjacent if and only if xy=yx. Alwardi et al. (Bulletin, 2011, 36, 49-59) defined the common neighborhood matrix CN(G) and the common neighborhood energy Ecn(G) of a simple graph G. A graph G is called CN-hyperenergetic if Ecn(G)>Ecn(Kn), where n=|V(G)| and Kn denotes the complete graph on n vertices. Two graphs G and H with equal number of vertices are called CN-equienergetic if Ecn(G)=Ecn(H). In this paper we
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26

Xu, Zhipeng, Xiaolong Huang, Fabian Jimenez, and Yuefan Deng. "A New Record of Graph Enumeration Enabled by Parallel Processing." Mathematics 7, no. 12 (2019): 1214. http://dx.doi.org/10.3390/math7121214.

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Using three supercomputers, we broke a record set in 2011, in the enumeration of non-isomorphic regular graphs by expanding the sequence of A006820 in the Online Encyclopedia of Integer Sequences (OEIS), to achieve the number for 4-regular graphs of order 23 as 429,668,180,677,439, while discovering several regular graphs with minimum average shortest path lengths (ASPL) that can be used as interconnection networks for parallel computers. The enumeration of 4-regular graphs and the discovery of minimal-ASPL graphs are extremely time consuming. We accomplish them by adapting GENREG, a classical
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27

Devhare, Sarika, and Vinayak Joshi. "Perfect non-commuting graphs of matrices over chains." Discrete Mathematics, Algorithms and Applications 09, no. 04 (2017): 1750049. http://dx.doi.org/10.1142/s1793830917500495.

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In this paper, we study the non-commuting graph [Formula: see text] of strictly upper triangular [Formula: see text] matrices over an [Formula: see text]-element chain [Formula: see text]. We prove that [Formula: see text] is a compact graph. From [Formula: see text], we construct a poset [Formula: see text]. We further prove that [Formula: see text] is a dismantlable lattice and its zero-divisor graph is isomorphic to [Formula: see text]. Lastly, we prove that [Formula: see text] is a perfect graph.
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28

Lou, Zhenzhen, Qiongxiang Huang, and Xueyi Huang. "On the construction of Q-controllable graphs." Electronic Journal of Linear Algebra 32 (February 6, 2017): 365–79. http://dx.doi.org/10.13001/1081-3810.3298.

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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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Sun, Zequn, Chengming Wang, Wei Hu, et al. "Knowledge Graph Alignment Network with Gated Multi-Hop Neighborhood Aggregation." Proceedings of the AAAI Conference on Artificial Intelligence 34, no. 01 (2020): 222–29. http://dx.doi.org/10.1609/aaai.v34i01.5354.

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Graph neural networks (GNNs) have emerged as a powerful paradigm for embedding-based entity alignment due to their capability of identifying isomorphic subgraphs. However, in real knowledge graphs (KGs), the counterpart entities usually have non-isomorphic neighborhood structures, which easily causes GNNs to yield different representations for them. To tackle this problem, we propose a new KG alignment network, namely AliNet, aiming at mitigating the non-isomorphism of neighborhood structures in an end-to-end manner. As the direct neighbors of counterpart entities are usually dissimilar due to
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Morris, Christopher, Martin Ritzert, Matthias Fey, et al. "Weisfeiler and Leman Go Neural: Higher-Order Graph Neural Networks." Proceedings of the AAAI Conference on Artificial Intelligence 33 (July 17, 2019): 4602–9. http://dx.doi.org/10.1609/aaai.v33i01.33014602.

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In recent years, graph neural networks (GNNs) have emerged as a powerful neural architecture to learn vector representations of nodes and graphs in a supervised, end-to-end fashion. Up to now, GNNs have only been evaluated empirically—showing promising results. The following work investigates GNNs from a theoretical point of view and relates them to the 1-dimensional Weisfeiler-Leman graph isomorphism heuristic (1-WL). We show that GNNs have the same expressiveness as the 1-WL in terms of distinguishing non-isomorphic (sub-)graphs. Hence, both algorithms also have the same shortcomings. Based
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31

Li, Yantao, and Yan-Quan Feng. "Pentavalent One-regular Graphs of Square-free Order." Algebra Colloquium 17, no. 03 (2010): 515–24. http://dx.doi.org/10.1142/s1005386710000490.

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A graph is one-regular if its automorphism group acts regularly on the set of its arcs. Let n be a square-free integer. It is shown in this paper that a pentavalent one-regular graph of order n exists if and only if n = 2 · 5tp1p2 … ps ≥ 62, where t ≤ 1, s ≥ 1, and pi's are distinct primes such that 5|(pi-1). For such an integer n, there are exactly 4s-1 non-isomorphic pentavalent one-regular graphs of order n, which are Cayley graphs on dihedral groups constructed by Kwak et al. This work is a continuation of the classification of cubic one-regular graphs of order twice a square-free integer
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Rouzbehani, Mohammad, Mahmood Pourgholamhossein, and Massoud Amini. "Chain conditions for graph C*-algebras." Forum Mathematicum 32, no. 2 (2020): 491–500. http://dx.doi.org/10.1515/forum-2019-0170.

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AbstractIn this article, we study chain conditions for graph C*-algebras. We show that there are infinitely many mutually non isomorphic Noetherian (and Artinian) purely infinite graph C*-algebras with infinitely many ideals. We prove that if E is a graph, then {C^{*}(E)} is a Noetherian (resp. Artinian) C*-algebra if and only if E satisfies condition (K) and each ascending (resp. descending) sequence of admissible pairs of E stabilizes.
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33

Alikhani, Saeid, Jason Brown, and Somayeh Jahari. "On the domination polynomials of friendship graphs." Filomat 30, no. 1 (2016): 169–78. http://dx.doi.org/10.2298/fil1601169a.

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Let G be a simple graph of order n. The domination polynomial of G is the polynomial D(G,x)= n?i=0 d(G,i)xi, where d(G,i) is the number of dominating sets of G of size i. Let n be any positive integer and Fn be the Friendship graph with 2n + 1 vertices and 3n edges, formed by the join of K1 with nK2. We study the domination polynomials of this family of graphs, and in particular examine the domination roots of the family, and find the limiting curve for the roots. We also show that for every n > 2, Fn is not D-unique, that is, there is another non-isomorphic graph with the same domination p
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Zhou, Jin-Xin, and Mohsen Ghasemi. "Automorphisms of a Family of Cubic Graphs." Algebra Colloquium 20, no. 03 (2013): 495–506. http://dx.doi.org/10.1142/s1005386713000461.

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A Cayley graph Cay (G,S) on a group G with respect to a Cayley subset S is said to be normal if the right regular representation R(G) of G is normal in the full automorphism group of Cay (G,S). For a positive integer n, let Γn be a graph with vertex set {xi,yi|i ∈ ℤ2n} and edge set {{xi,xi+1}, {yi,yi+1}, {x2i,y2i+1}, {y2i,x2i+1}|i ∈ ℤ2n}. In this paper, it is shown that Γn is a Cayley graph and its full automorphism group is isomorphic to [Formula: see text] for n=2, and to [Formula: see text] for n > 2. Furthermore, we determine all pairs of G and S such that Γn= Cay (G,S) is non-normal fo
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Nath, Rajat Kanti, Monalisha Sharma, Parama Dutta, and Yilun Shang. "On r-Noncommuting Graph of Finite Rings." Axioms 10, no. 3 (2021): 233. http://dx.doi.org/10.3390/axioms10030233.

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Let R be a finite ring and r∈R. The r-noncommuting graph of R, denoted by ΓRr, is a simple undirected graph whose vertex set is R and two vertices x and y are adjacent if and only if [x,y]≠r and [x,y]≠−r. In this paper, we obtain expressions for vertex degrees and show that ΓRr is neither a regular graph nor a lollipop graph if R is noncommutative. We characterize finite noncommutative rings such that ΓRr is a tree, in particular a star graph. It is also shown that ΓR1r and ΓR2ψ(r) are isomorphic if R1 and R2 are two isoclinic rings with isoclinism (ϕ,ψ). Further, we consider the induced subgr
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36

Mosavi, Somayeh, and Neda Ahanjideh. "Quasirecognition by prime graph of $C_{n}(4) $, where $ n\geq17 $ is odd." Boletim da Sociedade Paranaense de Matemática 33, no. 1 (2014): 57. http://dx.doi.org/10.5269/bspm.v33i1.21969.

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Let $G$ be a finite group and let $\Gamma(G) $ be the prime graph of $ G$. We assume that $ n\geq 17$ is an odd number. In this paper, we show that if $ \Gamma(G) = \Gamma(C_{n}(4))$, then $ G$ has a unique non-abelian composition factor isomorphic to $C_{n}(4)$. As consequences of our result, $C_{n}(4)$ is quasirecognizable by its spectrum and by prime graph.
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37

Kambites, Mark. "On commuting elements and embeddings of graph groups and monoids." Proceedings of the Edinburgh Mathematical Society 52, no. 1 (2009): 155–70. http://dx.doi.org/10.1017/s0013091507000119.

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AbstractWe study commutation properties of subsets of right-angled Artin groups and trace monoids. We show that if Γ is any graph not containing a four-cycle without chords, then the group G(Γ) does not contain four elements whose commutation graph is a four-cycle; a consequence is that G(Γ) does not have a subgroup isomorphic to a direct product of non-abelian groups. We also obtain corresponding and more general results in the monoid case.
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38

Gelbukh, Irina. "A finite graph is homeomorphic to the Reeb graph of a Morse–Bott function." Mathematica Slovaca 71, no. 3 (2021): 757–72. http://dx.doi.org/10.1515/ms-2021-0018.

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Abstract We prove that a finite graph (allowing loops and multiple edges) is homeomorphic (isomorphic up to vertices of degree two) to the Reeb graph of a Morse–Bott function on a smooth closed n-manifold, for any dimension n ≥ 2. The manifold can be chosen orientable or non-orientable; we estimate the co-rank of its fundamental group (or the genus in the case of surfaces) from below in terms of the cycle rank of the graph. The function can be chosen with any number k ≥ 3 of critical values, and in a few special cases with k < 3. In the case of surfaces, the function can be chosen, except f
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39

Bächle, Andreas, and Leo Margolis. "On the prime graph question for integral group rings of 4-primary groups I." International Journal of Algebra and Computation 27, no. 06 (2017): 731–67. http://dx.doi.org/10.1142/s0218196717500357.

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We study the Prime Graph Question for integral group rings. This question can be reduced to almost simple groups by a result of Kimmerle and Konovalov. We prove that the Prime Graph Question has an affirmative answer for all almost simple groups having a socle isomorphic to [Formula: see text] for [Formula: see text], establishing the Prime Graph Question for all groups where the only non-abelian composition factors are of the aforementioned form. Using this, we determine exactly how far the so-called HeLP method can take us for (almost simple) groups having an order divisible by at most four
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40

Abdollahi, Alireza, and Mojtaba Jazaeri. "On groups admitting no integral Cayley graphs besides complete multipartite graphs." Applicable Analysis and Discrete Mathematics 7, no. 1 (2013): 119–28. http://dx.doi.org/10.2298/aadm121212027a.

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Let G be a non-trivial finite group, S ? G \ {e} be a set such that if a 2 S, then a-1 ? S and e be the identity element of G. Suppose that Cay(G, S) is the Cayley graph with the vertex set G such that two vertices a and b are adjacent whenever a-1 ? S. An arbitrary graph is called integral whenever all eigenvalues of the adjacency matrix are integers. We say that a group G is Cayley integral simple whenever every connected integral Cayley graph on G is isomorphic to a complete multipartite graph. In this paper we prove that if G is a non-simple group, then G is Cayley integral simple if and o
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41

Rowlinson, Peter. "Certain 3-decompositions of complete graphs, with an application to finite fields." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 99, no. 3-4 (1985): 277–81. http://dx.doi.org/10.1017/s0308210500014293.

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SynopsisA necessary condition is obtained for a complete graph to have a decomposition as the line-disjoint union of three isomorphic strongly regular subgraphs. The condition is used to determine the number of non-trivial solutions of the equation x3+y3 = z3 in a finite field of characteristic p ≡ 2 mod 3.
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42

Azam, Naveed Ahmed, Aleksandar Shurbevski, and Hiroshi Nagamochi. "An Efficient Algorithm to Count Tree-Like Graphs with a Given Number of Vertices and Self-Loops." Entropy 22, no. 9 (2020): 923. http://dx.doi.org/10.3390/e22090923.

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Graph enumeration with given constraints is an interesting problem considered to be one of the fundamental problems in graph theory, with many applications in natural sciences and engineering such as bio-informatics and computational chemistry. For any two integers n≥1 and Δ≥0, we propose a method to count all non-isomorphic trees with n vertices, Δ self-loops, and no multi-edges based on dynamic programming. To achieve this goal, we count the number of non-isomorphic rooted trees with n vertices, Δ self-loops and no multi-edges, in O(n2(n+Δ(n+Δ·min{n,Δ}))) time and O(n2(Δ2+1)) space, since ev
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43

SUFIANI, R., S. NAMI, M. GOLMOHAMMADI, and M. A. JAFARIZADEH. "CONTINUOUS TIME QUANTUM WALKS AND QUOTIENT GRAPHS." International Journal of Quantum Information 09, no. 03 (2011): 1005–17. http://dx.doi.org/10.1142/s0219749911007435.

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Continuous-time quantum walks (CTQW) over finite group schemes is investigated, where it is shown that some properties of a CTQW over a group scheme defined on a finite group G induces a CTQW over group scheme defined on G/H, where H is a normal subgroup of G with prime index. This reduction can be helpful in analyzing CTQW on underlying graphs of group schemes. Even though this claim is proved for normal subgroups with prime index (using the Clifford's theorem from representation theory), it is checked in some examples that for other normal subgroups or even non-normal subgroups, the result i
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44

Kwak, Jin Ho, Jaeun Lee, and Alexander Mednykh. "Enumerating Branched Surface Coverings from Unbranched Ones." LMS Journal of Computation and Mathematics 6 (2003): 89–104. http://dx.doi.org/10.1112/s1461157000000395.

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AbstractThe number of non-isomorphic n-fold branched coverings of a given closed surface can be determined by the number of nonisomorphic n-fold unbranched coverings of the surface and the number of nonisomorphic connected n-fold graph coverings of a suitable bouquet of circles. A similar enumeration can also be done for regular branched coverings. Some explicit enumerations are also possible.
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45

Livesey, G. E., and H. A. Donegan. "Using classical graph theory to generate non-isomorphic floorplan distributions in the measurement of egress complexity." Mathematical and Computer Modelling 48, no. 1-2 (2008): 1–10. http://dx.doi.org/10.1016/j.mcm.2007.08.002.

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46

Shaveisi, Farzad. "Some Results on Annihilating-ideal Graphs." Canadian Mathematical Bulletin 59, no. 3 (2016): 641–51. http://dx.doi.org/10.4153/cmb-2016-016-3.

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AbstractThe annihilating-ideal graph of a commutative ring R, denoted by 𝔸𝔾(R), is a graph whose vertex set consists of all non-zero annihilating ideals and two distinct vertices I and J are adjacent if and only if IJ = (0). Here we show that if R is a reduced ring and the independence number of 𝔸𝔾(R) is finite, then the edge chromatic number of 𝔸𝔾(R) equals its maximum degree and this number equals 2|Min(R)|−1 also, it is proved that the independence number of 𝔸𝔾(R) equals 2|Min(R)|−1, where Min(R) denotes the set of minimal prime ideals of R. Then we give some criteria for a graph to be isom
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47

Froncek, Dalibor, and O'Neill Kingston. "Decomposition of complete graphs into connected unicyclic graphs with eight edges and pentagon." Indonesian Journal of Combinatorics 3, no. 1 (2019): 24. http://dx.doi.org/10.19184/ijc.2019.3.1.3.

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<p>A <span class="math"><em>G</em></span>-decomposition of the complete graph <span class="math"><em>K</em><sub><em>n</em></sub></span> is a family of pairwise edge disjoint subgraphs of <span class="math"><em>K</em><sub><em>n</em></sub></span>, all isomorphic to <span class="math"><em>G</em></span>, such that every edge of <span class="math"><em>K</em><sub><em>n</em></sub></span> belongs to exactly
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48

KANG, SOORAN, and AIDAN SIMS. "CO-UNIVERSAL C*-ALGEBRAS ASSOCIATED TO APERIODIC k-GRAPHS." Glasgow Mathematical Journal 56, no. 3 (2013): 537–50. http://dx.doi.org/10.1017/s001708951300044x.

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AbstractWe construct a representation of each finitely aligned aperiodic k-graph Λ on the Hilbert space $\mathcal{H}^{\rm ap}$ with basis indexed by aperiodic boundary paths in Λ. We show that the canonical expectation on $\mathcal{B}(\mathcal{H}^{\rm ap})$ restricts to an expectation of the image of this representation onto the subalgebra spanned by the final projections of the generating partial isometries. We then show that every quotient of the Toeplitz algebra of the k-graph admits an expectation compatible with this one. Using this, we prove that the image of our representation, which is
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49

Zabierowski, M. "The G3 non-isomorphic clustering graph and the problem of the physical reality of the clustering tendency." Astrophysics and Space Science 201, no. 1 (1993): 125–30. http://dx.doi.org/10.1007/bf00626981.

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50

Huang, Da, Jian Zhu, Zhiyong Yu, and Haijun Jiang. "On Consensus Index of Triplex Star-like Networks: A Graph Spectra Approach." Symmetry 13, no. 7 (2021): 1248. http://dx.doi.org/10.3390/sym13071248.

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In this article, the consensus-related performances of the triplex multi-agent systems with star-related structures, which can be measured by the algebraic connectivity and network coherence, have been studied by the characterization of Laplacian spectra. Some notions of graph operations are introduced to construct several triplex networks with star substructures. The methods of graph spectra are applied to derive the network coherence, and some asymptotic behaviors of the indices have been derived. It is found that the operations of adhering star topologies will make the first-order coherence
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