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.
The set of eigenvalues of a graph together with their multiplicities is called the spectrum of. The knowledge of spectrum can be used to obtain various topological properties of graphs like connectedness, toughness and many more. In this paper we use MATLAB to completely describe the spectrum of Sierpiński graphs and Sierpiński triangles, thus adding to the… (More)