The cocktail party graph of order , also called the hyperoctahedral graph (Biggs 1993, p. 17) or Roberts graph, is the graph consisting of two rows of paired nodes in which all nodes but the paired ones are connected with a graph edge. It is the graph complement of the ladder rung graph , and the dual graph of the hypercube graph . It is the skeleton of the -cross polytope.This graph arises in the handshake problem. It is a complete n-partite graph that is denoted by Brouwer et al. (1989, pp. 222-223), and is distance-transitive, and hence also distance-regular.The cocktail party graph of order is isomorphic to the circulant graph . The -cocktail party graph is also the -Turán graph.Special cases are summarized in the following table.-cocktail party graph1empty graph 2square graph 3octahedral graph416-cell graphThe -cocktail party graph has independence polynomialwith corresponding recurrence equation..