A Note On Quadrangular Embedding Of Abelian Cayley Graphs

The genus graphs have been studied by many authors, but just a few results concerning in special cases: Planar, Toroidal, Complete, Bipartite and Cartesian Product of Bipartite. We present here a general lower bound for the genus of a abelian Cayley graph and construct a family of circulant graphs which reach this bound.17333134

Counting circulant graphs of prime-power order by decomposing into orbit enumeration problems

AbstractThe set of circulant graphs with pk vertices (k⩾1,p an odd prime) is decomposed into a collection of well-specified subsets. The number of these subsets is equal to the kth Catalan number, and they are in one-to-one correspondence with the monotone underdiagonal walks on the plane integer (k+1)×(k+1) lattice. The counting of non-isomorphic circulant graphs in each of the subsets is presented as an orbit enumeration problem of Pólya type with respect to a certain Abelian group of multipliers. The descriptions are given in terms of equalities and congruences between multipliers in accordance with an isomorphism theorem for such circulant graphs. In this way, explicit uniform counting formulae have been obtained for various types of circulant graphs with p2 vertices

Circulant Graphs And Spherical Codes

Circulant graphs are homogeneous graphs with special properties which have been used to build interconnection networks for parallel computing. The association of circulant graphs to a spherical codes in dimension 2 k is presented here via the construction of an isomorphic graph supported by a lattice in ℝk. © 2006 SBrT.4245Adam, A., Research problem 2-10 (1967) J. Combinatorial Theory, 2, p. 393Biglieri, E., Elia, M., Cyclic-Group Codes for the Gaussian Channel (1976) IEEE Trans. Inform. Theory, Correspondence, pp. 624-629F. Boesch, R. Tindell, Circulant and Their Connectivities, J. of Graph Theory, 8 (1984) 487-499Costa, S.I.R., Muniz, M., Agustini, E., Palazzo, R., Graphs, Tessellations, and Perfect Codes on Flat Tori (2004) IEEE Trans. Inform. Theory, 50 (10), pp. 2363-2378Costa, S.I.R., Strapasson, J.E., Muniz, M., Carlos, T.B., Siqueira, R.M., Circulant Graphs Viewed as Graphs on Flat Tori SubmittedElpas, B., Turner, J., Graphs with circulant adjacency matrices (1970) J. Combin. Theory, 9, pp. 229-240Liskovets, V., Pöschel, R., Counting circulant graphs of prime-power order by decomposing into orbit enumeration problems (2000) Discrete Mathematics, 214, pp. 173-191Martínez, C., Beivide, R., Gutierrez, J., Gabidulin, E., On the Perfect t-Dominanting Set Problem in Circulant Graphs and Codes over Gaussian Integers (2005) Proceedings ISIT - IEEE International Symposium on Information Theory, pp. 1-5Slepian, D., Group codes for the Gaussian Channel (1968) The Bell System Technical Journal, 47, pp. 575-602Siqueira, R., Costa, S.I.R., Minimum Distance Upper Bounds for Commutative Group Codes (2006) Proceedings of IEEE Information Theory Workshop (ITW, , Uruguay, March 13-1

Circulant Graphs And Tessellations On Flat Tori

Circulant graphs are characterized here as quotient lattices, which are realized as vertices connected by a knot on a k-dimensional flat torus tessellated by hypercubes or hyperparallelotopes. Via this approach we present geometric interpretations for a bound on the diameter of a circulant graph, derive new bounds for the genus of a class of circulant graphs and establish connections with spherical codes and perfect codes in Lee spaces. © 2010 Elsevier Inc. All rights reserved.434818111823Heuberger, C., On planarity and colorability of circulant graphs (2003) Discrete Math., 268, pp. 153-169Muga II, F.P., Undirected circulant graphs (1994) International Symposium on Parallel Architectures, Algorithms and Networks, pp. 113-118Dougherty, R., Faber, V., The degree-diameter problem for several varieties of Cayley graphs. I: The Abelian Case (2004) SIAM J. Discrete Math., 17 (3), pp. 478-519C. Martínez, On the perfect t-dominating set problem in circulant graphs and codes over Gaussian integers (2005) Proceedings ISIT - IEEE International Symposium on Information Theory, pp. 1-5Parkson, T.D., Circulant graph imbeddings (1980) J. Combin. Theory (B), 29, pp. 310-320Golomb, S.W., Welch, L.R., Algebraic coding and the Lee metric (1968) Proc. Sympos. Math. Res. Center, pp. 175-194. , Madison, Wis., John Wiley, New YorkHorak, P., On perfect Lee codes (2008) Discrete Mathematics, , doi:10.1016/j.disc.2008.03.019Ádám, A., Research problem 2-10 (1967) J. Combin. Theory, 2, p. 393Elspas, B., Turner, J., Graphs with circulant adjacency matrices (1970) J. Combin. Theory, 9, pp. 229-240Liskovets, V., Pöschel, R., Counting circulant graphs of prime-power order by decomposing into orbit enumeration problems (2000) Discrete Math., 214, pp. 173-191Muzychuk, M., Ádám's conjecture is true in the square-free case (1995) J. Combin. Theory A, 72 (1), pp. 118-134Boesch, F., Tindell, R., Circulants and their connectivities (1984) J. Graph Theory, 8, pp. 487-499Stillwell, J., (1992) Geometry of Surfaces, , Springer-Verlag New YorkCosta, S.I.R., Muniz, M., Agustini, E., Palazzo, R., Graphs, tessellations, and perfect codes on flat tori (2004) IEEE Trans. Inform. Theory, 50 (10), pp. 2363-2378Conway, J.H., Sloane, N.J.A., (1999) Sphere Packings, Lattices and Groups, , Springer-Verlag New YorkCosta, S.I.R., Strapasson, J.E., Muniz, M., Siqueira, R.M., Circulant graphs, lattices and spherical codes (2007) Internat. J. Appl. Math., 20 (5), pp. 581-594Kirschenhofer, P., Pethõ, A., Tichy, R., On analytical and Diophantine properties of a family of counting functions (1999) Acta Sci. Math. (Szeged), 65 (12), pp. 47-59Albdaiwi, B.F., Bose, B., Quasi-perfect Lee distance codes (2003) IEEE Trans. Inform. Theory, 49, pp. 1535-1539Gross, J.L., Tucker, T.W., (2001) Topological Graph Theory, , Dover NYTrudeau, R.J., (1976) Introduction to Graph Theory, , Dover New YorkMolitierno, J.J., On the algebraic connectivity of graphs as a function of genus (2006) Linear Algebra and Its Applications, 419 (2-3), pp. 519-531. , DOI 10.1016/j.laa.2006.05.014, PII S002437950600266