Forward Difference Properties of the (n, k)-Star Graph and Some Other Interconnection Networks
Eddie Cheng, Ethan Gibbons, Ke Qiang Qiu, Zhizhang Shen · 2023
An important invariant of an interconnection network is its surface area, the number of vertices at distance i from a node. Although much work has been done to obtain formulas for the surface areas for many interconnection networks, most of the formulas are not in the so-called closed form except for a very few trivial graphs. It is known that for an interconnection network, if its surface area satisfies the so-called forward difference property, then for any specific distance i, its surface area of radius i in closed form (a polynomial of degree i) can be obtained, provided that we have i + 1 initial values of the surface area of radius i. This property is known to hold for the hypercube and the star graph. We show in this paper that the property also holds for the (n, k)-star graph, 1 ≤ k ≤ n − 1, a family of interconnection networks that also include the star graph when k = n − 1. We then show that the technique we use for the result is general that can also be used to prove the property for some other networks.