On a conjecture concerning the extensions of a reciprocal matrix
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916856436097024 |
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| author | Fernandes, Rosário |
| author_facet | Fernandes, Rosário |
| contents | 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 $n\geq 5$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_16593 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On a conjecture concerning the extensions of a reciprocal matrix Fernandes, Rosário Combinatorics 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 $n\geq 5$ 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. |
| title | On a conjecture concerning the extensions of a reciprocal matrix |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2507.16593 |