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Journal articles on the topic 'Non-isomorphic graph'

Author: Grafiati

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 (August 22, 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 is solvable, and every non-cyclic Sylow subgroup of a finite DS group is of order [Formula: see text], [Formula: see text], [Formula: see text] or [Formula: see text]. We also give several infinite families of non-Cay-DS solvable groups. In particular we prove that there exist two cospectral non-isomorphic [Formula: see text]-regular Cayley graphs on the dihedral group of order [Formula: see text] for any prime [Formula: see text].

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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 (October 13, 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 (September 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 of a generalized extraspecial p-group (p odd) by determining the power graph of this group. We also determine the power graph of a p-group of order p4 (p odd).

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6

Costalonga, João Paulo, Robert J. Kingan, and Sandra R. Kingan. "Constructing Minimally 3-Connected Graphs." Algorithms 14, no. 1 (January 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 generated by McKay’s isomorphism checker nauty. The algorithm consecutively constructs the non-isomorphic minimally 3-connected graphs with n vertices and m edges from the non-isomorphic minimally 3-connected graphs with n−1 vertices and m−2 edges, n−1 vertices and m−3 edges, and n−2 vertices and m−3 edges.

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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 (September 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 non-isomorphic strongly regular graphs. However, by combining the algorithm of Gudkov and Nussinov with a construction proposed by Rudolph \cite{rudolph02} in which one examines a graph describing the dynamics of two particles on the original graph, we find an algorithm that successfully distinguishes all pairs of non-isomorphic strongly regular graphs that we tested with up to 29 vertices.

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8

Zhao, Jinxing, and Guixin Deng. "Remark on subgroup intersection graph of finite abelian groups." Open Mathematics 18, no. 1 (September 18, 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 (September 20, 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 (April 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 [Formula: see text] and [Formula: see text] be two non-abelian finite groups such that [Formula: see text]. Then [Formula: see text]. Conjecture 2 (AAM’s Conjecture). Let [Formula: see text] be a finite non-abelian simple group and [Formula: see text] be a group such that [Formula: see text]. Then [Formula: see text]. Some authors have proved the first conjecture for some classes of groups (specially for all finite simple groups and non-abelian nilpotent groups with irregular isomorphic non-commuting graphs) but in [Moghaddamfar, About noncommuting graphs, Sib. Math. J. 47(5) (2006) 911–914], Moghaddamfar has shown that it is not true in general with some counterexamples to this conjecture. On the other hand, Solomon and Woldar proved the second conjecture, in [R. Solomon and A. Woldar, Simple groups are characterized by their non-commuting graph, J. Group Theory 16 (2013) 793–824]. In this paper, we will define the same concept for a finite non-commutative Moufang loop [Formula: see text] and try to characterize some finite non-commutative Moufang loops with their non-commuting graph. Particularly, we obtain examples of finite non-associative Moufang loops and finite associative Moufang loops (groups) of the same order which have isomorphic non-commuting graphs. Also, we will obtain some results related to the non-commuting graph of a finite non-commutative Moufang loop. Finally, we give a conjecture stating that the above result is true for all finite simple Moufang loops.

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11

Hammer, P. L., and A. K. Kelmans. "On Universal Threshold Graphs." Combinatorics, Probability and Computing 3, no. 3 (September 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) graphs having the minimum number of vertices.It is shown that for any n ≥ 3 there exist minimum n-universal graphs, which are themselves threshold, and others which are not.Two extremal minimum n-universal graphs having respectively the minimum and the maximum number of edges are described, it is proved that they are unique, and that they are threshold graphs.The set of all minimum n-universal threshold graphs is then described constructively; it is shown that it forms a lattice isomorphic to the n − 1 dimensional Boolean cube, and that the minimum and the maximum elements of this lattice are the two extremal graphs introduced above.The proofs provide a (polynomial) recursive procedure that determines for any threshold graph G with n vertices and for any minimum n-universal threshold graph T, an induced subgraph G' of T isomorphic to 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 (February 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 characterization of the signed ?-graphs containing a triangle.

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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 eigenvalues and eigenvectors of several graph matrices can be quickly computed. Spectral graph parameters contain a lot of information on the graph structure (both global and local) including some information on graph parameters that, in general, are computed by exponential algorithms. Moreover, in some applications in data mining, graph spectra are used to encode graphs themselves. The Euclidean distance between the eigenvalue sequences of two graphs on the same number of vertices is called the spectral distance of graphs. Some other spectral distances (also based on various graph matrices) have been considered as well. Two graphs are considered as similar if their spectral distance is small. If two graphs are at zero distance, they are cospectral. In this sense, cospectral graphs are similar. Other spectrally based measures of similarity between networks (not necessarily having the same number of vertices) have been used in Internet topology analysis, and in other areas. The notion of spectral distance enables the design of various meta-heuristic (e.g., tabu search, variable neighbourhood search) algorithms for constructing graphs with a given spectrum (spectral graph reconstruction). Several spectrally based pattern recognition problems appear in many areas (e.g., image segmentation in computer vision, alignment of protein-protein interaction networks in bio-informatics, recognizing hard instances for combinatorial optimization problems such as the travelling salesman problem). We give a survey of such and other graph spectral recognition techniques used in computer sciences.

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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 (November 13, 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 integer n≥1, it generates all required graphs with n vertices in O(n) time per graph and O(n) space in total, without generating invalid intermediate structures. We performed some experiments to enumerate graphs with a given cycle rank from which it is evident that our method is efficient. As an application of our method, we can generate tree-like polymer topologies of a given cycle rank with self-loops and no multiple edges.

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16

VAIDYA, SAMIR K., and KALPESH POPAT. "On Equienergetic, Hyperenergetic and Hypoenergetic Graphs." Kragujevac Journal of Mathematics 44, no. 4 (December 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 and contribute some new results.

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17

REDMOND, SHANE P. "RECOVERING RINGS FROM ZERO-DIVISOR GRAPHS." Journal of Algebra and Its Applications 12, no. 08 (July 31, 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 @nô1(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 (October 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 [Formula: see text] value among all members of the collection [Formula: see text]. In this short note, an alternative but relatively simple approach is used for characterizing the aforementioned graph.

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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 (January 22, 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 are isomorphic under some conditions.

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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 (February 14, 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 (September 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(m, p), the complete p-partite graph with m vertices in each partite set, except when (m, p) = (2n + 1, 2), in the case that two of the 2-factors are isomorphic to the given two non-isomorphic 2-factors and the remaining are Hamilton cycles.

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Jahandideh, M., R. Modabernia, and S. Shokrolahi. "Non-commuting graphs of certain almost simple groups." Asian-European Journal of Mathematics 12, no. 05 (September 3, 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 simple group and [Formula: see text] is a non-abelian group with isomorphic non-commuting graphs, then [Formula: see text].

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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 (September 8, 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 compute the common neighborhood energy of Γc(G) for several classes of finite non-abelian groups, including the class of groups such that the central quotient is isomorphic to group of symmetries of a regular polygon, and conclude that these graphs are not CN-hyperenergetic. We shall also obtain some pairs of finite non-abelian groups such that their commuting graphs are CN-equienergetic.

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Xu, Zhipeng, Xiaolong Huang, Fabian Jimenez, and Yuefan Deng. "A New Record of Graph Enumeration Enabled by Parallel Processing." Mathematics 7, no. 12 (December 10, 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 regular graph generator, to three supercomputers with thousands of processor cores.

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Devhare, Sarika, and Vinayak Joshi. "Perfect non-commuting graphs of matrices over chains." Discrete Mathematics, Algorithms and Applications 09, no. 04 (August 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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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, Muhao Chen, Jian Dai, Wei Zhang, and Yuzhong Qu. "Knowledge Graph Alignment Network with Gated Multi-Hop Neighborhood Aggregation." Proceedings of the AAAI Conference on Artificial Intelligence 34, no. 01 (April 3, 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 the schema heterogeneity, AliNet introduces distant neighbors to expand the overlap between their neighborhood structures. It employs an attention mechanism to highlight helpful distant neighbors and reduce noises. Then, it controls the aggregation of both direct and distant neighborhood information using a gating mechanism. We further propose a relation loss to refine entity representations. We perform thorough experiments with detailed ablation studies and analyses on five entity alignment datasets, demonstrating the effectiveness of AliNet.

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Morris, Christopher, Martin Ritzert, Matthias Fey, William L. Hamilton, Jan Eric Lenssen, Gaurav Rattan, and Martin Grohe. "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 on this, we propose a generalization of GNNs, so-called k-dimensional GNNs (k-GNNs), which can take higher-order graph structures at multiple scales into account. These higher-order structures play an essential role in the characterization of social networks and molecule graphs. Our experimental evaluation confirms our theoretical findings as well as confirms that higher-order information is useful in the task of graph classification and regression.

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Li, Yantao, and Yan-Quan Feng. "Pentavalent One-regular Graphs of Square-free Order." Algebra Colloquium 17, no. 03 (September 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 given by Zhou and Feng.

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Rouzbehani, Mohammad, Mahmood Pourgholamhossein, and Massoud Amini. "Chain conditions for graph C*-algebras." Forum Mathematicum 32, no. 2 (March 1, 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 polynomial. Also we construct some families of graphs whose real domination roots are only -2 and 0. Finally, we conclude by discussing the domination polynomials of a related family of graphs, the n-book graphs Bn, formed by joining n copies of the cycle graph C4 with a common edge.

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Zhou, Jin-Xin, and Mohsen Ghasemi. "Automorphisms of a Family of Cubic Graphs." Algebra Colloquium 20, no. 03 (July 4, 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 for G. Using this, all connected cubic non-normal Cayley graphs of order 8p are constructed explicitly for each prime p.

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35

Nath, Rajat Kanti, Monalisha Sharma, Parama Dutta, and Yilun Shang. "On r-Noncommuting Graph of Finite Rings." Axioms 10, no. 3 (September 19, 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 subgraph ΔRr of ΓRr (induced by the non-central elements of R) and obtain results on clique number and diameter of ΔRr along with certain characterizations of finite noncommutative rings such that ΔRr is n-regular for some positive integer n. As applications of our results, we characterize certain finite noncommutative rings such that their noncommuting graphs are n-regular for n≤6.

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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 (February 3, 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 (February 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 (June 1, 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 for a few special cases, as the height function associated with an immersion ℝ3.

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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 (September 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 different primes.

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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 only if G ? Zp2 for some prime number p or G ? Z2 x Z2. Moreover, we show that there exist finite non-abelian simple groups which are not Cayley integral simple.

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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 (August 22, 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 every tree can be uniquely viewed as a rooted tree by either regarding its unicentroid as the root, or in the case of bicentroid, by introducing a virtual vertex on the bicentroid and assuming the virtual vertex to be the root. By this result, we get a lower bound and an upper bound on the number of tree-like polymer topologies of chemical compounds with any “cycle rank”.

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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 (April 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 is also true! It means that CTQW over the graph on G, starting from any arbitrary vertex, is isomorphic to the CTQW over the quotient graph on G/H if we take the sum of the amplitudes corresponding to the vertices belonging to the same cosets.

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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 (July 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 (September 1, 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 isomorphic with an annihilating-ideal graph of a ring. For example, it is shown that every bipartite annihilating-ideal graph is a complete bipartite graph with at most two horns. Among other results, it is shown that a ûnite graph 𝔸𝔾(R) is not Eulerian, and that it is Hamiltonian if and only if R contains no Gorenstain ring as its direct summand.

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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 (June 30, 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 one copy of <span class="math"><em>G</em></span>. Using standard decomposition techniques based on <span class="math"><em>ρ</em></span>-labelings, introduced by Rosa in 1967, and their modifications we show that each of the ten non-isomorphic connected unicyclic graphs with eight edges containing the pentagon decomposes the complete graph <span class="math"><em>K</em><sub><em>n</em></sub></span> whenever the necessary conditions are satisfied.</p>

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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 (August 13, 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 canonically isomorphic to the Cuntz–Krieger algebra, is co-universal for Toeplitz–Cuntz–Krieger families consisting of non-zero partial isometries.

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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 (July 12, 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 increase a constant value under the triplex structures as parameters tend to infinity, and the second-order coherence have some equality relations as the node related parameters tend to infinity. Finally, the consensus related indices of the triplex systems with the same number of nodes but non-isomorphic graph structures have been compared and simulated to verify the results.

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