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Let G = (V, E) be a connected graph and let d(u, v) denote the distance between vertices u, v ∈ V. A metric basis for G is a set B ⊆ V of minimum cardinality such that no two vertices of G have the same distances to all points of B. The cardinality of a metric basis of G is called the metric dimension of G, denoted by dim(G). In this paper we determine the… (More)

Let G = (V, A) be a directed graph without parallel arcs, and let S ⊆ V be a set of vertices. Let the sequence S = S0 ⊆ S1 ⊆ S2 ⊆ · · · be defined as follows: S1 is obtained from S0 by adding all out-neighbors of vertices in S0. For k 2, S k is obtained from S k−1 by adding all vertices w such that for some vertex v ∈ S k−1 , w is the unique out-neighbor of… (More)

The minimum rank of a simple graph G is the smallest possible rank over all symmetric real matrices A whose nonzero off-diagonal entries correspond to the edges of G. Using the zero forcing number, we prove that the minimum rank of the butterfly network is 1 9 (3r + 1)2 r+1 − 2(−1) r and that this is equal to the rank of its adjacency matrix.

Average distance is an important parameter for measuring the communication cost of computer networks. A popular approach for its computation is to first partition the edge set of a network into convex components using the transitive closure of the Djokovi´c-Winkler's relation and then to compute the average distance from the respective invariants of the… (More)