Some topological properties of star graphs: The surface area and volume

Distance formula and shortest paths for the (n,k)-star graphs

“…By copy, we mean copy the exchange of the symbols. See Table 1 for an example when the chosen associated permutation of [4,2,11,1,5,12,6,7,14,3] is [4,2,11,1,5,12,6,7,14,3,8,9,13,10,15]. Clearly q is an associated permutation of ½1; 2; .…”

“…; k and we are done. (Table 2 provides another example when we pick [4,2,11,1,5,12,6,7,14,3,15,10,9,13,8] to be the associated permutation.) h It follows immediately from the above proposition that finding distances in S n;k reduces to finding distances in S n .…”

“…It is easy to see that such an extended permutation is indeed a permutation and that each of the external symbols is in a distinct cycle in the cycle decomposition of p E . In the above example, note that we purposely write each of 11,12,13,14,15 in the last position of the cycle containing it. Indeed for the cycle containing each external symbol, we will write the external symbol last in the cycle.…”

“…We first note that given permutations p and q, the distance between them is the same as the distance between a permutation a and the identity, as S n is vertex-transitive, where a is the permutation of the symbols in p with respect to q. [4,2,11,1,5,12,6,7,14,3,8,10,9,13,15] and q to be [1,2,3,4,5,6,7,8,9,10,11,12,13,15,14]. Then a, the corresponding permutation of the symbols in p with respect to q, is precisely 1 2 3 4 5 6 7 8 9 10 11 12 13 15 14 4 2 11 1 5 12 6 7 14 3 8 10 9 13 15 :…”

“…The second problem is to count how many distinct shortest paths there are between two specified vertices. The first problem has attracted interest from various researchers for a long time [3,6,12,[19][20][21] and the problem has recently been solved for both the star graphs and the ðn; kÞ-star graphs. In this section, we solve the second problem for the ðn; kÞ-star graphs.…”