On the k-subgraphs of the Generalized n-cubes

Graphs are used in modeling interconnections networks and measuring their properties. Knowing and understanding the graph theoretical/combinatorial properties of the underlying networks are necessary in developing more efficient parallel algorithms as well as fault-tolerant communication/routing algorithms [1] The hypercube is one of the most versatile and efficient networks yet discovered for parallel computation. One generalization of the hypercube is the n-cube Q(n,m) which is a graph whose vertices are all the binary ntuples, such that two vertices are adjacent whenever they differ in exactly m coordinates. The k-subgraph of the Generalized n-cube Qk(n,m) is the induced subgraph of the n-cube Q(n,m) where q=2, such that a vertex v ∈ V(Qk(n,m)) if and only if v ∈ V(Q(n,m)) and v is of parity k. This paper presents some degree properties of Qk(n,m) as well as some isomorphisms it has with other graphs, namely: 1) ) 2 , ( 1 n Qn− is isomorphic to Kn 2) ) 2 , ( i n Qk is isomorphic to Gi(n,k) 3) QS(n,2i) is isomorphic to SGi(n) . Key-Words: n-cubes, graphs, Johnson graphs, hypercube