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The Idiosyncratic Polynomial of Digraphs

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Abstract

The idiosyncratic polynomial of a graph G with adjacency matrix A is the characteristic polynomial of the matrix \( A + y(J-A-I)\), where I is the identity matrix and J is the all-ones matrix. It follows from a theorem of Hagos (2000) combined with an earlier result of Johnson and Newman (1980) that the idiosyncratic polynomial of a graph is reconstructible from the multiset of the idiosyncratic polynomial of its vertex-deleted subgraphs. For a digraph D with adjacency matrix A, we define its idiosyncratic polynomial as the characteristic polynomial of the matrix \( A + y(J-A-I)+zA^{\top }\). By forbidding two fixed digraphs on three vertices as induced subdigraphs, we prove that the idiosyncratic polynomial of a digraph is reconstructible from the ordered multiset of the idiosyncratic polynomial of its induced subdigraphs on three vertices. As an immediate consequence, the idiosyncratic polynomial of a tournament is reconstructible from the collection of its 3-cycles. Another consequence is that all the transitive orientations of a comparability graph have the same idiosyncratic polynomial.

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Notes

  1. A flag-free digraph is a digraph which contains no flags as induced subdigraphs.

  2. Throughout this paper, the characteristic polynomial of a square matrix M is the determinant of \(xI-M\).

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Authors and Affiliations

  1. Laboratoire de Mathématiques Fondamentales et Appliquées, Faculté des Sciences Aïn Chock, Hassan II University of Casablanca, Casablanca, Maroc

    Edward Bankoussou-mabiala, Abderrahim Boussaïri, Abdelhak Chaïchaâ, Brahim Chergui & Soufiane Lakhlifi

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  1. Edward Bankoussou-mabiala
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  2. Abderrahim Boussaïri
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  3. Abdelhak Chaïchaâ
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  5. Soufiane Lakhlifi
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Correspondence to Abderrahim Boussaïri.

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Communicated by Ararat Harutyunyan.

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A The 3-Vertex Digraphs (up to Hemimorphy) and Their Idiosyncratic Polynomials

A The 3-Vertex Digraphs (up to Hemimorphy) and Their Idiosyncratic Polynomials

\( x^{3} - 3 \, y^{2} x - 2 \, y^{3} \)

\( x^{3} + \left( -2 \, y^{2} - y - z\right) x -y^{2}{\left( y + z + 1\right) } \)

\( x^{3} + \left( -y^{2} - 2 \, y - 2 \, z\right) x -y{\left( y^{2} + 2 \, y z + z^{2} + 1\right) } \)

\( x^{3} + \left( -y^{2} - 2 \, y - 2 \, z\right) x -2 \, y{\left( y + z\right) } \)

\( x^{3} + \left( -2 \, y^{2} - z^{2} - 2 \, z - 1\right) x -2 \, y^{2} {\left( z + 1\right) } \)

\( x^{3} + \left( -3 \, y - 3 \, z\right) x - y^{2} -{\left( y + z\right) } {\left( y + z + 1\right) } \)

\( x^{3} + \left( -3 \, y - 3 \, z\right) x -{\left( y + z + 1\right) }{\left( y^{2} + 2 \, y z + z^{2} - y - z + 1\right) } \)

\( x^{3} + \left( -y^{2} - z^{2} - y - 3 \, z - 1\right) x -y {\left( z + 1\right) }{\left( y + z + 1\right) } \)

\( x^{3} + \left( -z^{2} - 2 \, y - 4 \, z - 1\right) x -2 \, {\left( y + z\right) } {\left( z + 1\right) } \)

\( x^{3} + \left( -z^{2} - 2 \, y - 4 \, z - 1\right) x -{\left( z + 1\right) }{\left( y^{2} + 2 \, y z + z^{2} + 1\right) } \)

\( x^{3} + \left( -y^{2} - 2 \, z^{2} - 4 \, z - 2\right) x -2 \, y {\left( z + 1\right) }^{2} \)

\( x^{3} + \left( -2 \, z^{2} - y - 5 \, z - 2\right) x -{\left( y + z + 1\right) } {\left( z + 1\right) }^{2} \)

\( x^{3} + \left( -3 \, z^{2} - 6 \, z - 3\right) x -2 \, {\left( z + 1\right) }^{3} \)

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Bankoussou-mabiala, E., Boussaïri, A., Chaïchaâ, A. et al. The Idiosyncratic Polynomial of Digraphs. Ann. Comb. 26, 329–344 (2022). https://doi.org/10.1007/s00026-022-00572-9

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