On the limit of the sequence {Cm(D)}m=1∞ for a multipartite tournament D

• Published in: Discrete Applied Mathematics Vol. 340; pp. 1 - 13
• Main Authors: Jung, Ji-Hwan, Kim, Suh-Ryung, Yoon, Hyesun
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 15-12-2023
• Subjects: Index of imprimitivity; Limit of a Boolean matrix sequence; Limit of a graph sequence; m-step competition graph; Multipartite tournament; Sets of imprimitivity; Sets of imprimitivity; m-step competition graph; Index of imprimitivity; Limit of a Boolean matrix sequence; Limit of a graph sequence; Multipartite tournament
• ISSN: 0166-218X; 1872-6771
• DOI: 10.1016/j.dam.2023.06.044

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.