Folded Cube Graph


The folded n-cube graph, perhaps better termed "folded hypercube graph," is a graph obtained by merging vertices of the n-hypercube graph Q_n that are antipodal, i.e., lie at a distance n (the graph diameter of Q_n). Brouwer et al. 1989 (p. 222) use the notation  square _k for the folded k-cube graph.

For n>2, the folded n-cube graph is regular of degree n. It has 2^(n-1) vertices, 2^(n-2)n edges, and diameter |_n/2_|. The chromatic number is 2 for n even and 4 for n odd (Godsil 2004). Godsil observes that the independence number of the folded n-cube graph F_n is given by

 alpha(F_n)=2^(n-2)-1/4(1-(-1)^n)(n-1; (n-1)/2),

a result which follows from Cvetkovic's eigenvalue bound to establish an upper bound and a direct construction of the independent set by looking at vertices at an odd (resp., even) distance from a fixed vertex when n is odd (resp., even) (S. Wagon, pers. comm.).

Folded cube graphs are distance-regular and distance-transitive.

The following table summarizes special cases.

The bipartite double graph of the folded n-cube graph is the hypercube graph Q_n.

See also

Clebsch Graph, Halved Cube Graph, Hypercube Graph, Kummer Graph

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Brouwer, A. E. "Folded 6-Cube and Graphs with the Same Parameters.", A. E.; Cohen, A. M.; and Neumaier, A. "Halved and Folded Cubes." §9.2D in Distance-Regular Graphs. New York: Springer-Verlag, pp. 264-265, 1989.Choudam, S. A. and Nandini, R. U. "Complete Binary Trees in Folded and Enhanced Cubes." Networks 43, 266-272, "Folded Cubes.", A. and Latifi, S. "Properties and Performance of Folded Hypercubes." IEEE Trans. Parallel Distrib. Syst. 2, 31-42, 1991.Godsil, C. "Folded Cubes" and "Eigenvalues and Folded Cubes." §7.6 and 7.7 in Interesting Graphs and Their Colourings. Unpublished manuscript, pp. 70-73, 2006.van Bon, J. "Finite Primitive Distance-Transitive Graphs." Europ. J. Combin. 28, 517-532, 2007.van Dam, E. and Haemers, W. H. "An Odd Characterization of the Generalized Odd Graphs." CentER Discussion Paper Series, No. 2010-47, SSRN 1596575. 2010.Varvarigos, E. "Efficient Routing Algorithms for Folded-Cube Networks." Proc. 14th Int. Phoenix Conf. on Computers and Communications. IEEE, pp. 143-151, 1995.

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Folded Cube Graph

Cite this as:

Weisstein, Eric W. "Folded Cube Graph." From MathWorld--A Wolfram Web Resource.

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