In this paper, necessary and sufficient conditions are given for the existence of the group inverse of the block matrix A A B 0 ¡ over any skew field, where A, B are both square and rank(B) ≥ rank(A). The representation of this group inverse and some relative additive results are also given.
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 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.
Let G be a connected graph of order n. The resistance matrix of G is defined as R G = (r ij (G)) n×n , where r ij (G) is the resistance distance between two vertices i and j in G. Eigenvalues of R G are called R-eigenvalues of G. If all row sums of R G are equal, then G is called resistance-regular. For any connected graph G, we show that R G determines the… (More)