Eigenvalues of a graph are the eigenvalues of its adjacency matrix. The multiset of eigenvalues is called its spectrum. There are many properties which can be explained using the spectrum like energy, connectedness, vertex connectivity, chromatic number, perfect matching etc. Laplacian spectrum is the multiset of eigenvalues of Laplacian matrix. The… (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.
A metric basis is a set W of vertices of a graph G(V, E) such that for every pair of vertices u, v of G, there exists a vertex w ∈ W with the condition that the length of a shortest path from u to w is different from the length of a shortest path from v to w. The minimum cardinality of a metric basis for G is called the metric dimension. A pair of vertices… (More)
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)