Designing a protocol to exchange a secret key is one of the most fundamental subjects in cryptography. Using a random deal of cards, pairs of card players (agents) can share secret keys that are information-theoretically secure against an eavesdropper. A key set protocol, which uses a random deal of cards, can perform an Eulerian secret key exchange, in which the pairs of players sharing secret...

Computing the average shortest-path length of a large scale-free network needs much memory space and computation time. Hence, parallel computing must be applied. In order to solve the load-balancing problem for coarse-grained parallelization, the relationship between the computing time of a single-source shortest-path length of node and the features of node is studied. We present a dynamic prog...

Define the average path length in a connected graph as the sum of the length of the shortest path between all pairs of nodes, divided by the total number of pairs of nodes. Letting SN denote the sum of the shortest path lengths between all pairs of nodes in a complete m-ary tree of depth N , we derive a first-order linear but non-homogeneous recurrence relation for SN , from which a closed-form...

This paper presents a theorem that asserts that average edge length of the minimum spanning tree of a complete graph on n+ 1 vertices is less than or equal to the average edge length of all the n+ 1 minimum spanning trees of the induced graph on n vertices. The result is also in compliance with results given by Frieze and Steele.  1999 Elsevier Science B.V. All rights reserved.

Let X1,X2, . . . be a sequence of random variables which are stationary and ergodic (see Breiman [1968, p. 109]) with values in a set S. Let μk be the stationary distribution for a sequence of length k so that for any fixed collection A1, A2, . . . , Ak of subsets of s, μk(A1, A2, . . . , Ak) = P[Xn ∈ A1,Xn+1 ∈ A2, . . . ,Xn+k−1 ∈ Ak] is independent of n, n = 1, 2, . . . . Now let B be a subset...