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Journal articles on the topic 'Circulant graphs'

Author: Grafiati

Published: 4 June 2021

Last updated: 25 January 2023

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1

Duong, Linh, Brenda K. Kroschel, Michael Riddell, Kevin N. Vander Meulen, and Adam Van Tuyl. "Maximum nullity and zero forcing of circulant graphs." Special Matrices 8, no. 1 (December 6, 2020): 221–34. http://dx.doi.org/10.1515/spma-2020-0106.

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AbstractThe zero forcing number of a graph has been applied to communication complexity, electrical power grid monitoring, and some inverse eigenvalue problems. It is well-known that the zero forcing number of a graph provides a lower bound on the minimum rank of a graph. In this paper we bound and characterize the zero forcing number of various circulant graphs, including families of bipartite circulants, as well as all cubic circulants. We extend the definition of the Möbius ladder to a type of torus product to obtain bounds on the minimum rank and the maximum nullity on these products. We obtain equality for torus products by employing orthogonal Hankel matrices. In fact, in every circulant graph for which we have determined these numbers, the maximum nullity equals the zero forcing number. It is an open question whether this holds for all circulant graphs.

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2

Vilfred, V. "A Theory of Cartesian Product and Factorization of Circulant Graphs." Journal of Discrete Mathematics 2013 (December 26, 2013): 1–10. http://dx.doi.org/10.1155/2013/163740.

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3

Chia, Gek-Ling, and Chong-Keang Lim. "Counting 2-circulant graphs." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 39, no. 2 (October 1985): 270–81. http://dx.doi.org/10.1017/s1446788700022527.

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AbstractAlspach and Sutcliffe call a graph X(S, q, F) 2-circulant if it consists of two isomorphic copies of circulant graphs X(p, S) and X(p, qS) on p vertices with “cross-edges” joining one another in a prescribed manner. In this paper, we enumerate the nonisomorphic classes of 2-circulant graphs X(S, q, F) such that |S| = m and |F| = k. We also determine a necessary and sufficient condition for a 2-circulant graph to be a GRR. The nonisomorphic classes of GRR on 2p vertices are also enumerated.

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4

THOMSON, ALISON, and SANMING ZHOU. "FROBENIUS CIRCULANT GRAPHS OF VALENCY FOUR." Journal of the Australian Mathematical Society 85, no. 2 (October 2008): 269–82. http://dx.doi.org/10.1017/s1446788708000979.

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AbstractA first kind Frobenius graph is a Cayley graph Cay(K,S) on the Frobenius kernel of a Frobenius group $K \rtimes H$ such that S=aH for some a∈K with 〈aH〉=K, where H is of even order or a is an involution. It is known that such graphs admit ‘perfect’ routing and gossiping schemes. A circulant graph is a Cayley graph on a cyclic group of order at least three. Since circulant graphs are widely used as models for interconnection networks, it is thus highly desirable to characterize those which are Frobenius of the first kind. In this paper we first give such a characterization for connected 4-valent circulant graphs, and then describe optimal routing and gossiping schemes for those which are first kind Frobenius graphs. Examples of such graphs include the 4-valent circulant graph with a given diameter and maximum possible order.

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5

SAXENA, NITIN, SIMONE SEVERINI, and IGOR E. SHPARLINSKI. "PARAMETERS OF INTEGRAL CIRCULANT GRAPHS AND PERIODIC QUANTUM DYNAMICS." International Journal of Quantum Information 05, no. 03 (June 2007): 417–30. http://dx.doi.org/10.1142/s0219749907002918.

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The intention of the paper is to move a step towards a classification of network topologies that exhibit periodic quantum dynamics. We show that the evolution of a quantum system whose hamiltonian is identical to the adjacency matrix of a circulant graph is periodic if and only if all eigenvalues of the graph are integers (that is, the graph is integral). Motivated by this observation, we focus on relevant properties of integral circulant graphs. Specifically, we bound the number of vertices of integral circulant graphs in terms of their degree, characterize bipartiteness and give exact bounds for their diameter. Additionally, we prove that circulant graphs with odd order do not allow perfect state transfer.

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6

Rodríguez, José M., and José M. Sigarreta. "The hyperbolicity constant of infinite circulant graphs." Open Mathematics 15, no. 1 (June 9, 2017): 800–814. http://dx.doi.org/10.1515/math-2017-0061.

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Abstract If X is a geodesic metric space and x1, x2, x3 ∈ X, a geodesic triangle T = {x1, x2, x3} is the union of the three geodesics [x1x2], [x2x3] and [x3x1] in X. The space X is δ-hyperbolic (in the Gromov sense) if any side of T is contained in a δ-neighborhood of the union of the two other sides, for every geodesic triangle T in X. Deciding whether or not a graph is hyperbolic is usually very difficult; therefore, it is interesting to find classes of graphs which are hyperbolic. A graph is circulant if it has a cyclic group of automorphisms that includes an automorphism taking any vertex to any other vertex. In this paper we prove that infinite circulant graphs and their complements are hyperbolic. Furthermore, we obtain several sharp inequalities for the hyperbolicity constant of a large class of infinite circulant graphs and the precise value of the hyperbolicity constant of many circulant graphs. Besides, we give sharp bounds for the hyperbolicity constant of the complement of every infinite circulant graph.

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7

Angeles-Canul, R. J., R. M. Norton, M. C. Opperman, C. C. Paribello, M. C. Russell, and C. Tamon. "Perfect state transfer, integral circulants, and join of graphs." Quantum Information and Computation 10, no. 3&4 (March 2010): 325–42. http://dx.doi.org/10.26421/qic10.3-4-10.

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We propose new families of graphs which exhibit quantum perfect state transfer. Our constructions are based on the join operator on graphs, its circulant generalizations, and the Cartesian product of graphs. We build upon the results of Ba\v{s}i\'{c} and Petkovi\'{c} ({\em Applied Mathematics Letters} {\bf 22}(10):1609-1615, 2009) and construct new integral circulants and regular graphs with perfect state transfer. More specifically, we show that the integral circulant $\textsc{ICG}_{n}(\{2,n/2^{b}\} \cup Q)$ has perfect state transfer, where $b \in \{1,2\}$, $n$ is a multiple of $16$ and $Q$ is a subset of the odd divisors of $n$. Using the standard join of graphs, we also show a family of double-cone graphs which are non-periodic but exhibit perfect state transfer. This class of graphs is constructed by simply taking the join of the empty two-vertex graph with a specific class of regular graphs. This answers a question posed by Godsil (arxiv.org math/08062074).

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8

El-Mesady, Ahmed, Aleksandr Y. Romanov, Aleksandr A. Amerikanov, and Alexander D. Ivannikov. "On Bipartite Circulant Graph Decompositions Based on Cartesian and Tensor Products with Novel Topologies and Deadlock-Free Routing." Algorithms 16, no. 1 (December 22, 2022): 10. http://dx.doi.org/10.3390/a16010010.

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Recent developments in commutative algebra, linear algebra, and graph theory allow us to approach various issues in several fields. Circulant graphs now have a wider range of practical uses, including as the foundation for optical networks, discrete cellular neural networks, small-world networks, models of chemical reactions, supercomputing and multiprocessor systems. Herein, we are concerned with the decompositions of the bipartite circulant graphs. We propose the Cartesian and tensor product approaches as helping tools for the decompositions. The proposed approaches enable us to decompose the bipartite circulant graphs into many categories of graphs. We consider the use cases of applying the described theory of bipartite circulant graph decomposition to the problems of finding new topologies and deadlock-free routing in them when building supercomputers and networks-on-chip.

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9

Triyani, Triyani, Mashuri Mashuri, Bunga Tirai Anarkis, and Slamet Riyadi. "The spectrum on prism graph using circulant matrix." Bulletin of Applied Mathematics and Mathematics Education 2, no. 1 (May 18, 2022): 1–10. http://dx.doi.org/10.12928/bamme.v2i1.5129.

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Spectral graph theory discusses about the algebraic properties of graphs based on the spectrum of a graph. This article investigated the spectrum of prism graph. The method used in this research is the circulant matrix. The results showed that prism graph P2,s is a regular graph of degree 3, for s odd and s ≥ 3, P2,s is a circulantt graph with regular spectrum.

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10

Zhou, Houqing. "The Wiener Index of Circulant Graphs." Journal of Chemistry 2014 (2014): 1–4. http://dx.doi.org/10.1155/2014/742121.

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Circulant graphs are an important class of interconnection networks in parallel and distributed computing. In this paper, we discuss the relation of the Wiener index and the Harary index of circulant graphs and the largest eigenvalues of distance matrix and reciprocal distance matrix of circulants. We obtain the following consequence:W/λ=H/μ;2W/n=λ;2H/n=μ, whereW,Hdenote the Wiener index and the Harary index andλ,μdenote the largest eigenvalues of distance matrix and reciprocal distance matrix of circulant graphs, respectively. Moreover we also discuss the Wiener index of nonregular graphs with cut edges.

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11

MONAKHOVA, E. A. "A SURVEY ON UNDIRECTED CIRCULANT GRAPHS." Discrete Mathematics, Algorithms and Applications 04, no. 01 (March 2012): 1250002. http://dx.doi.org/10.1142/s1793830912500024.

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Circulant graphs have been extensively investigated over the past 30 years because of their broad application to different fields of theory and practice. Two known surveys on circulant networks including a survey on undirected circulants have been published: by Bermond et al. [Distributed loop computer networks: A survey, J. Parallel Distributed Comput.24 (1995) 2–10] and by Hwang [A survey on multi-loop networks, Theoret. Comput. Sci.299 (2003) 107–121]. The present paper includes the results which have not been presented there, in particular the works of Russian researchers, and also a lot of new results obtained in the area of research of circulant networks. We focus on the survey connected with study of structural and communicative properties of circulant networks.

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12

So, Wasin. "Integral circulant graphs." Discrete Mathematics 306, no. 1 (January 2006): 153–58. http://dx.doi.org/10.1016/j.disc.2005.11.006.

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13

Thomson, Alison, and Sanming Zhou. "Rotational circulant graphs." Discrete Applied Mathematics 162 (January 2014): 296–305. http://dx.doi.org/10.1016/j.dam.2013.08.044.

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14

Kwon, Young Soo, A. D. Mednykh, and I. A. Mednykh. "Complexity of discrete Seifert foliations over a graph." Доклады Академии наук 486, no. 4 (June 10, 2019): 411–15. http://dx.doi.org/10.31857/s0869-56524864411-415.

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In the present paper, we study the complexity of an infinite family of graphs Hn = Hn(G1, G2, ..., Gm) that are discrete Seifert foliations over a graph H on m vertices with fibers G1, G2, ..., Gm. Each fiber Gi = Cn(si,1, si,2, ..., si,ki) of this foliation is the circulant graph on n vertices with jumps si,1, si,2, ..., si,ki. The family of discrete Seifert foliations is sufficiently large. It includes the generalized Petersen graphs, I-graphs, Y-graphs, H-graphs, sandwiches of circulant graphs, discrete torus graph and others. We obtain a closed formula for the number t(n) of spanning trees in Hn in terms of Chebyshev polynomials, investigate some arithmetical properties of this function and find its asymptotics as n → ∞.

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15

Song, Shu Jiao, Weiqian Zhang, and Can Xu. "Locating and Identifying Codes in Circulant Graphs." Discrete Dynamics in Nature and Society 2021 (October 28, 2021): 1–9. http://dx.doi.org/10.1155/2021/4056910.

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Identifying and locating-dominating codes have been studied widely in circulant graphs. Recently, Ville Junnila et al. (Optimal bounds on codes for location in circulant graphs, Cryptography and Communications; 2019) studied identifying and locating-dominating codes in circulants C n 1 , d , C n 1 , d − 1 , d , and C n 1 , d − 1 , d , d + 1 . In this paper, identifying, locating, and self-identifying codes in the circulant graphs C n k , d , C n k , d − k , d , and C n k , d − k , d , d + k are studied, and this extends Junnila et al.’s results to general cases.

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16

Brooks, Josephine, Alvaro Carbonero, Joseph Vargas, Rigoberto Flórez, Brendan Rooney, and Darren Narayan. "Removing Symmetry in Circulant Graphs and Point-Block Incidence Graphs." Mathematics 9, no. 2 (January 14, 2021): 166. http://dx.doi.org/10.3390/math9020166.

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An automorphism of a graph is a mapping of the vertices onto themselves such that connections between respective edges are preserved. A vertex v in a graph G is fixed if it is mapped to itself under every automorphism of G. The fixing number of a graph G is the minimum number of vertices, when fixed, fixes all of the vertices in G. The determination of fixing numbers is important as it can be useful in determining the group of automorphisms of a graph-a famous and difficult problem. Fixing numbers were introduced and initially studied by Gibbons and Laison, Erwin and Harary and Boutin. In this paper, we investigate fixing numbers for graphs with an underlying cyclic structure, which provides an inherent presence of symmetry. We first determine fixing numbers for circulant graphs, showing in many cases the fixing number is 2. However, we also show that circulant graphs with twins, which are pairs of vertices with the same neighbourhoods, have considerably higher fixing numbers. This is the first paper that investigates fixing numbers of point-block incidence graphs, which lie at the intersection of graph theory and combinatorial design theory. We also present a surprising result-identifying infinite families of graphs in which fixing any vertex fixes every vertex, thus removing all symmetries from the graph.

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17

Talebi, A. A., M.Zameni, and Hossein Rashmanlou. "Vertex Domination Critical in Circulant Graphs." International Journal of Fuzzy Mathematical Archive 12, no. 02 (2017): 55–66. http://dx.doi.org/10.22457/ijfma.v12n2a2.

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A graph G is vertex domination critical if for any vertex v of G, the domination number of G – v is less than the domination number of G. We call these graphs γ-critical if domination number of G is γ. In this paper, we determine the domination and the total domination number of Cir(n,A) for two particular generating sets A of Zn, and then study γ-critical in these graphs.

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18

SUNTORNPOCH, BORWORN, and YOTSANAN MEEMARK. "CAYLEY GRAPHS OVER A FINITE CHAIN RING AND GCD-GRAPHS." Bulletin of the Australian Mathematical Society 93, no. 3 (January 22, 2016): 353–63. http://dx.doi.org/10.1017/s0004972715001380.

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

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19

Lo, P., S. Rajaram, D. Schepens, D. Sullivan, C. Tamon, and J. Ward. "Mixing of quantum walk on circulant bunkbeds." Quantum Information and Computation 6, no. 4&5 (July 2006): 370–81. http://dx.doi.org/10.26421/qic6.4-5-5.

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This paper gives new observations on the mixing dynamics of a continuous-time quantum walk on circulants and their bunkbeds. These bunkbeds are defined through two standard graph operators: the join G + H and the Cartesian product G \cprod H of graphs G and H. Our results include the following: (i) The quantum walk is average uniform mixing on circulants with bounded eigenvalue multiplicity; this extends a known fact about the cycles C_{n}. (ii) Explicit analysis of the probability distribution of the quantum walk on the join of circulants; this explains why complete multipartite graphs are not average uniform mixing, using the fact K_{n} = K_{1} + K_{n-1} and K_{n,\ldots,n} = \overline{K}_{n} + \ldots + \overline{K}_{n}. (iii) The quantum walk on the Cartesian product of a $m$-vertex path P_{m} and a circulant G, namely, P_{m} \cprod G, is average uniform mixing if G is; this highlights a difference between circulants and the hypercubes Q_{n} = P_{2} \cprod Q_{n-1}. Our proofs employ purely elementary arguments based on the spectra of the graphs.

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20

Monakhova, E. A., and O. G. Monakhov. "The diameter vulnerability of two-dimensional optimal circulant networks." Journal of Physics: Conference Series 2099, no. 1 (November 1, 2021): 012054. http://dx.doi.org/10.1088/1742-6596/2099/1/012054.

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Abstract This paper studies the effect of changing the diameter of the network in circulant networks of dimension two with unreliable elements (nodes, links). The well-known (Δ,D,D′, s)-problem is to find (Δ, D)-graphs with maximum degree Δ and diameter D such that the subgraphs obtained from the original graph by deleting any set of up to s vertices (edges) have diameter at most D′. For a family of optimal circulants of degree four we found the ranges of the orders of the graphs that preserve the diameter of the graph for one (two) vertex or edge failures. It is proved that in the investigated circulant networks in case of failure of one or two edges (vertices), the diameter can increase by no more than one, and in case of failure of three elements by no more than two (edge failures) or three (vertex failures). It is shown that two-dimensional optimal circulants, in comparison with two-dimensional tori, have a better diameter in case of element failures.

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21

Paulraja, P., and Kumar Sampath. "On hamiltonian decompositions of tensor products of graphs." Applicable Analysis and Discrete Mathematics 13, no. 1 (2019): 178–202. http://dx.doi.org/10.2298/aadm170803003p.

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Finding a hamiltonian decomposition of G is one of the challenging problems in graph theory. We do not know for what classes of graphs G and H, their tensor product G x H is hamiltonian decomposable. In this paper, we have proved that, if G is a hamiltonian decomposable circulant graph with certain properties and H is a hamiltonian decomposable multigraph, then G x H is hamiltonian decomposable. In particular, tensor products of certain sparse hamiltonian decomposable circulant graphs are hamiltonian decomposable.

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22

Sander, J. W., and T. Sander. "On So's conjecture for integral circulant graphs." Applicable Analysis and Discrete Mathematics 9, no. 1 (2015): 59–72. http://dx.doi.org/10.2298/aadm150226009s.

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Each integral circulant graph ICG(n,D) is characterised by its order n and a set D of positive divisors of n in such a way that it has vertex set Z=nZ and edge set {(a,b) : a, b ? Z=nZ, gcd(a - b,n) ? D}. According to a conjecture of So two integral circulant graphs are isomorphic if and only if they are isospectral, i.e. they have the same eigenvalues (counted with multiplicities). We prove a weaker form of this conjecture, namely, that two integral circulant graphs with multiplicative divisor sets are isomorphic if and only if their spectral vectors coincide.

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23

Ali, Niran Abbas, Adem Kilicman, and Hazim Michman Trao. "Restricted triangulation on circulant graphs." Open Mathematics 16, no. 1 (April 18, 2018): 358–69. http://dx.doi.org/10.1515/math-2018-0033.

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AbstractThe restricted triangulation existence problem on a given graph decides whether there exists a triangulation on the graph’s vertex set that is restricted with respect to its edge set. Let G = C(n, S) be a circulant graph on n vertices with jump value set S. We consider the restricted triangulation existence problem for G. We determine necessary and sufficient conditions on S for which G admitting a restricted triangulation. We characterize a set of jump values S(n) that has the smallest cardinality with C(n, S(n)) admits a restricted triangulation. We present the measure of non-triangulability of Kn − G for a given G.

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24

Kadyan, Monu, and Bikash Bhattacharjya. "Integral mixed circulant graphs." Discrete Mathematics 346, no. 1 (January 2023): 113142. http://dx.doi.org/10.1016/j.disc.2022.113142.

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25

Damnjanović, Ivan, and Dragan Stevanović. "On circulant nut graphs." Linear Algebra and its Applications 633 (January 2022): 127–51. http://dx.doi.org/10.1016/j.laa.2021.10.006.

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26

Broere, Izak, and Johannes H. Hattingh. "PRODUCTS OF CIRCULANT GRAPHS." Quaestiones Mathematicae 13, no. 2 (January 1990): 191–216. http://dx.doi.org/10.1080/16073606.1990.9631612.

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27

Vander Meulen, Kevin N., Adam Van Tuyl, and Catriona Watt. "Cohen–Macaulay Circulant Graphs." Communications in Algebra 42, no. 5 (January 16, 2014): 1896–910. http://dx.doi.org/10.1080/00927872.2012.749886.

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28

Shyue-Ming Tang, Yue-Li Wang, and Chien-Yi Li. "Generalized Recursive Circulant Graphs." IEEE Transactions on Parallel and Distributed Systems 23, no. 1 (January 2012): 87–93. http://dx.doi.org/10.1109/tpds.2011.109.

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29

Cichacz, Sylwia, and Dalibor Froncek. "Distance magic circulant graphs." Discrete Mathematics 339, no. 1 (January 2016): 84–94. http://dx.doi.org/10.1016/j.disc.2015.07.002.

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30

Lih, Ko-Wei, Daphne Der-Fen Liu, and Xuding Zhu. "Star Extremal Circulant Graphs." SIAM Journal on Discrete Mathematics 12, no. 4 (January 1999): 491–99. http://dx.doi.org/10.1137/s0895480198342838.

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31

Brown, Jason, and Richard Hoshino. "Well-covered circulant graphs." Discrete Mathematics 311, no. 4 (February 2011): 244–51. http://dx.doi.org/10.1016/j.disc.2010.11.007.

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32

Qin, Yan-Li, Binzhou Xia, and Sanming Zhou. "Stability of circulant graphs." Journal of Combinatorial Theory, Series B 136 (May 2019): 154–69. http://dx.doi.org/10.1016/j.jctb.2018.10.004.

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33

Salman, Muhammad, Imran Javaid, and Muhammad Anwar Chaudhry. "Resolvability in circulant graphs." Acta Mathematica Sinica, English Series 28, no. 9 (March 19, 2012): 1851–64. http://dx.doi.org/10.1007/s10114-012-0417-4.

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34

Basic, Milan. "Hamiltonian properties on a class of circulant interconnection networks." Filomat 32, no. 1 (2018): 71–85. http://dx.doi.org/10.2298/fil1801071b.

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Classes of circulant graphs play an important role in modeling interconnection networks in parallel and distributed computing. They also find applications in modeling quantum spin networks supporting the perfect state transfer. It has been noticed that unitary Cayley graphs as a class of circulant graphs possess many good properties such as small diameter, mirror symmetry, recursive structure, regularity, etc. and therefore can serve as a model for efficient interconnection networks. In this paper we go a step further and analyze some other characteristics of unitary Cayley graphs important for the modeling of a good interconnection network. We show that all unitary Cayley graphs are hamiltonian. More precisely, every unitary Cayley graph is hamiltonian-laceable (up to one exception for X6) if it is bipartite, and hamiltonianconnected if it is not. We prove this by presenting an explicit construction of hamiltonian paths on Xnm using the hamiltonian paths on Xn and Xm for gcd(n,m) = 1. Moreover, we also prove that every unitary Cayley graph is bipancyclic and every nonbipartite unitary Cayley graph is pancyclic.

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35

LEUNG, KA HIN, VINH NGUYEN, and WASIN SO. "NONEXISTENCE OF A CIRCULANT EXPANDER FAMILY." Bulletin of the Australian Mathematical Society 83, no. 1 (November 17, 2010): 87–95. http://dx.doi.org/10.1017/s0004972710001644.

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AbstractThe expansion constant of a simple graph G of order n is defined as where $E(S, \overline {S})$ denotes the set of edges in G between the vertex subset S and its complement $\overline {S}$. An expander family is a sequence {Gi} of d-regular graphs of increasing order such that h(Gi)>ϵ for some fixed ϵ>0. Existence of such families is known in the literature, but explicit construction is nontrivial. A folklore theorem states that there is no expander family of circulant graphs only. In this note, we provide an elementary proof of this fact by first estimating the second largest eigenvalue of a circulant graph, and then employing Cheeger’s inequalities where G is a d-regular graph and λ2(G) denotes the second largest eigenvalue of G. Moreover, the associated equality cases are discussed.

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36

Tamizh Chelvam, T., and S. Raja. "Integral circulant graphs with four distinct eigenvalues." Discrete Mathematics, Algorithms and Applications 10, no. 05 (October 2018): 1850057. http://dx.doi.org/10.1142/s179383091850057x.

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Let [Formula: see text], the finite cyclic group of order [Formula: see text]. Assume that [Formula: see text] and [Formula: see text]. The circulant graph [Formula: see text] is the undirected graph having the vertex set [Formula: see text] and edge set [Formula: see text]. Let [Formula: see text] be a set of positive, proper divisors of the integer [Formula: see text]. In this paper, by using [Formula: see text] we characterize certain connected integral circulant graphs with four distinct eigenvalues.

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37

Jagannathan, M., Vernold Vivin.J, and Veninstine Vivik.J. "The Equitable Chromatic Bounds on Splitting of Block Circulant Graphs." Mathematical Problems in Engineering 2022 (November 8, 2022): 1–17. http://dx.doi.org/10.1155/2022/7603023.

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An equitable vertex coloring for the splitting of block circulant graphs is investigated. The block circulant graphs comprises block circulant matrices, where each block is itself a matrix. These blocks in each row are cyclically shifted one place to the right from those of the previous row. We approached such block circulant graphs in matrix representation and derived their independent sets using the neighbourhoods of each vertex. This classification makes the vertex coloring process to be simpler and equitable in most cases. In this framework, the equitable chromatic numbers are obtained for splitting on block circulant graphs, namely prism, antiprism, crossed and closed sun graphs.

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38

Antalan, John Rafael Macalisang, and Francis Joseph Campena. "A Breadth-first Search Tree Construction for Multiplicative Circulant Graphs." European Journal of Pure and Applied Mathematics 14, no. 1 (January 31, 2021): 248–64. http://dx.doi.org/10.29020/nybg.ejpam.v14i1.3884.

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In this paper, we give a recursive method in constructing a breadth-first search tree for multiplicative circulant graphs of order power of odd. We then use the proposed construction in reproving some results concerning multiplicative circulant graph's diameter, average distance and distance spectral radius. We also determine the graph's Wiener index, vertex-forwarding index, and a bound for its edge-forwarding index. Finally, we discuss some possible research works in which the proposed construction can be applied.

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39

Sander, J. W., and T. Sander. "The energy of integral circulant graphs with prime power order." Applicable Analysis and Discrete Mathematics 5, no. 1 (2011): 22–36. http://dx.doi.org/10.2298/aadm110131003s.

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The energy of a graph is the sum of the moduli of the eigenvalues of its adjacency matrix. We study the energy of integral circulant graphs, also called gcd graphs. Such a graph can be characterized by its vertex count n and a set D of divisors of n such that its vertex set is Zn and its edge set is {{a,b} : a, b ? Zn; gcd(a-b, n)? D}. For an integral circulant graph on ps vertices, where p is a prime, we derive a closed formula for its energy in terms of n and D. Moreover, we study minimal and maximal energies for fixed ps and varying divisor sets D.

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Ahmadi, A., R. Belk, C. Tamon, and C. Wendler. "On mixing in continuous-time quantum walks on some circulant graphs." Quantum Information and Computation 3, no. 6 (November 2003): 611–18. http://dx.doi.org/10.26421/qic3.6-4.

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Classical random walks on well-behaved graphs are rapidly mixing towards the uniform distribution. Moore and Russell showed that the continuous-time quantum walk on the hypercube is instantaneously uniform mixing. We show that the continuous-time quantum walks on other well-behaved graphs do not exhibit this uniform mixing. We prove that the only graphs amongst balanced complete multipartite graphs that have the instantaneous exactly uniform mixing property are the complete graphs on two, three and four vertices, and the cycle graph on four vertices. Our proof exploits the circulant structure of these graphs. Furthermore, we conjecture that most complete cycles and Cayley graphs of the symmetric group lack this mixing property as well.

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Mönius, Katja. "Constructions of isospectral circulant graphs." Elemente der Mathematik 75, no. 2 (April 8, 2020): 45–57. http://dx.doi.org/10.4171/em/404.

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Munir, Mobeen, Waqas Nazeer, Zakia Shahzadi, and Shin Kang. "Some Invariants of Circulant Graphs." Symmetry 8, no. 11 (November 18, 2016): 134. http://dx.doi.org/10.3390/sym8110134.

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ZHANG, Hong. "On Edge Transitive Circulant Graphs." Tokyo Journal of Mathematics 19, no. 1 (June 1996): 51–55. http://dx.doi.org/10.3836/tjm/1270043218.

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Sjogren, Jon A. "Circulant blocks and rotational graphs." Linear and Multilinear Algebra 34, no. 2 (March 1993): 141–76. http://dx.doi.org/10.1080/03081089308818218.

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Richter, Sebastian, and Israel Rocha. "Layout of random circulant graphs." Linear Algebra and its Applications 559 (December 2018): 95–113. http://dx.doi.org/10.1016/j.laa.2018.09.003.

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Stevanović, Dragan, and Ivan Stanković. "Remarks on hyperenergetic circulant graphs." Linear Algebra and its Applications 400 (May 2005): 345–48. http://dx.doi.org/10.1016/j.laa.2005.01.001.

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Lin, Wensong. "Some star extremal circulant graphs." Discrete Mathematics 271, no. 1-3 (September 2003): 169–77. http://dx.doi.org/10.1016/s0012-365x(02)00872-5.

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Marklof, Jens, and Andreas Strömbergsson. "Diameters of random circulant graphs." Combinatorica 33, no. 4 (August 2013): 429–66. http://dx.doi.org/10.1007/s00493-013-2820-6.

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Marušič, Dragan. "Strong regularity and circulant graphs." Discrete Mathematics 78, no. 1-2 (1989): 119–25. http://dx.doi.org/10.1016/0012-365x(89)90168-4.

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Poljak, Svatopluk, and Daniel Turzík. "Max-cut in circulant graphs." Discrete Mathematics 108, no. 1-3 (October 1992): 379–92. http://dx.doi.org/10.1016/0012-365x(92)90690-h.

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