On the limit of the sequence [formula omitted] for a multipartite tournament [formula omitted

Abstract

For an integer k≥2, let A be a Boolean block matrix with blocks Aij for 1≤i,j≤k such that Aii is a zero matrix and Aij+AjiT is a matrix with all elements 1 but not both corresponding elements of Aij and AjiT equal to 1 for i≠j.Jung et al. (2023) studied the matrix sequence {Am(AT)m}m=1∞. This paper, an extension of the one above, was initiated by the observation that {Am(AT)m}m=1∞ converges if A has no zero rows. We compute the limit of the matrix sequence {Am(AT)m}m=1∞ if A has no zero rows. To this end, we take a graph theoretical approach: noting that A is the adjacency matrix of a multipartite tournament D, we compute the limit of the graph sequence Cm(D)m=1∞ when D has no vertices of outdegree 0.