Diameters of random circulant graphs

Abstract

Abstract. The diameter of a graph measures the maximal distance between any pair of vertices. The diameters of many small-world networks, as well as a variety of other random graph models, grow logarithmically in the number of nodes. In contrast, the worst connected networks are cycles whose diameters increase linearly in the number of nodes. In the present study we consider an intermediate class of examples: Cayley graphs of cyclic groups, also known as circulant graphs or multi-loop networks. We show that the diameter of a random circulant 2k-regular graph with n vertices scales as n 1/k , and establish a limit theorem for the distribution of their diameters. We obtain analogous results for the distribution of the average distance and higher moments.

On the lattice programming gap of the group problems

“…We show that the obtained lower bound is optimal and that the upper bound has the optimal order. The proofs are based on recent results of Marklof and Strombergson [20] on the diameters of circulant graphs and on the estimates of Fukshansky and Robins [10] for the Frobenius numbers.…”

“…Thus all bounds derived in this paper can be applied to the group relaxation induced by a selected row of (1.4). Note that in this special case the lattice programming gap gap(Λ, l) can be associated with the diameter of a directed circulant graph (see [20] for details). Furthermore, the results of [20] show that the lower bound (1.9) is a good predictor for the value of gap(Λ, l) for a 'typical' Λ.…”

“…Following notation from [20], let LG + k = (Z k , E) be the standard directed lattice graph with vertex set Z k . The edge set E consists of all directed edges (x, x + e j ), where x ∈ Z k and e 1 , .…”

Fast methods for designing circulant network topology with high connectivity and survivability

“…A circulant graph outperforms other topologies owing to its low message delay, high connectivity, and strong survivability [4][5][6][7][8]. Therefore, it has been widely employed in many technical fields such as telecommunication networks, computer networks, parallel processing systems, and social networks [9][10][11][12].…”

“…The connectivity and survivability of circulant graph networks can be evaluated by several metrics, such as average distance and connectivity ratio [2,[12][13][14][15]. The average distance is preferred as the prime optimization objective for it effectively represents network connectivity.…”