2025/07/22 by Rosário Fernandes, Fernandes, Rosário
Computer Science · Engineering · Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #Graph theory and applications #Matrix Theory and Algorithms #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.2507.16593
openalex publication_date 2025/07/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let A be a reciprocal matrix of order n and w be its Perron eigenvector. To infer the efficiency of w for A, based on the principle of Pareto optimal decisions, we study the strong connectivity of a certain digraph associated with A and w. A reciprocal matrix B of order n+1 is an extension of A if the matrix A is obtained from B by removing its last row and column. We prove that there is no extension of a reciprocal matrix whose digraph associated with the extension and its Perron eigenvector has a source, as conjectured by Furtado and Johnson in ``Efficiency analysis for the Perron vector of a reciprocal matrix". As an application, considering n≥ 5 and A a matrix obtained from a consistent one by perturbing four entries above the main diagonal, x,y,z,a, and the corresponding reciprocal entries, in a way that there is a submatrix of size 2 containing the four perturbed entries and not containing a diagonal entry, we describe the relations among x,y,z,a with which A always has efficient Perron eigenvector.