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Let G be a graph with adjacency matrix A, let H(t) = exp(itA). G is called a periodic graph if there exists a time τ such that H(τ) is diagonal. If u and v are distinct vertices in G, we say that perfect state transfer occurs from u to v if there exists a time τ such that |H(τ) u,v | = 1. A necessary and sufficient condition for G is periodic is given. We… (More)
Let G be a weighted graph with Laplacian matrix L and signless Laplacian matrix Q. In this note, block representations for the group inverse of L and Q are given. The resistance distance in a graph can be obtained from the block representation of the group inverse of L.
Let K be a skew field and K n×n be the set of all n × n matrices over K. The purpose of this paper is to give some necessary and sufficient conditions for the existence and the representations of the group inverse of the block matrix A C B D under some conditions.