𝕮-inverse of graphs and mixed graphs
This article introduces a generalization of the concept of inverse graphs applicable to both graphs and mixed graphs. Given a graph GG with adjacency matrix A(G)A\left(G), the inverse graph G−1{G}^{-1} is defined such that its adjacency matrix is similar to the inverse of A(G)A\left(G) through a dia...
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| Format: | Article |
| Language: | English |
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De Gruyter
2025-02-01
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| Series: | Open Mathematics |
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| Online Access: | https://doi.org/10.1515/math-2024-0104 |
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| _version_ | 1823860531104055296 |
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| author | Alomari Omar Abudayah Mohammad Ghanem Manal |
| author_facet | Alomari Omar Abudayah Mohammad Ghanem Manal |
| author_sort | Alomari Omar |
| collection | DOAJ |
| description | This article introduces a generalization of the concept of inverse graphs applicable to both graphs and mixed graphs. Given a graph GG with adjacency matrix A(G)A\left(G), the inverse graph G−1{G}^{-1} is defined such that its adjacency matrix is similar to the inverse of A(G)A\left(G) through a diagonal matrix with entries of ±1\pm 1. While this diagonal matrix may or may not exist for graphs with nonsingular adjacency matrices, our study extends the concept to include mixed graphs as well. It has been proven that for certain unicyclic graphs, such a diagonal matrix does not exist. Motivated by this, we generalized the definition of inverse graphs to include mixed graphs, allowing us to find inverse mixed graphs for a class previously shown to lack one. |
| format | Article |
| id | doaj-art-cf15bba5341446bfadf59219db53ab0a |
| institution | Kabale University |
| issn | 2391-5455 |
| language | English |
| publishDate | 2025-02-01 |
| publisher | De Gruyter |
| record_format | Article |
| series | Open Mathematics |
| spelling | doaj-art-cf15bba5341446bfadf59219db53ab0a2025-02-10T13:24:35ZengDe GruyterOpen Mathematics2391-54552025-02-012313165317610.1515/math-2024-0104𝕮-inverse of graphs and mixed graphsAlomari Omar0Abudayah Mohammad1Ghanem Manal2School of Electrical Engineering and Information Technology, German Jordanian University, Amman, JordanSchool of Electrical Engineering and Information Technology, German Jordanian University, Amman, JordanDepartment of Mathematics, The University of Jordan, Amman, JordanThis article introduces a generalization of the concept of inverse graphs applicable to both graphs and mixed graphs. Given a graph GG with adjacency matrix A(G)A\left(G), the inverse graph G−1{G}^{-1} is defined such that its adjacency matrix is similar to the inverse of A(G)A\left(G) through a diagonal matrix with entries of ±1\pm 1. While this diagonal matrix may or may not exist for graphs with nonsingular adjacency matrices, our study extends the concept to include mixed graphs as well. It has been proven that for certain unicyclic graphs, such a diagonal matrix does not exist. Motivated by this, we generalized the definition of inverse graphs to include mixed graphs, allowing us to find inverse mixed graphs for a class previously shown to lack one.https://doi.org/10.1515/math-2024-0104mixed graphsα-hermitian adjacency matrixinverse matrixmixed graphsbipartite mixed graphsunicyclic bipartite mixed graphsperfect matching05c2005c50 |
| spellingShingle | Alomari Omar Abudayah Mohammad Ghanem Manal 𝕮-inverse of graphs and mixed graphs Open Mathematics mixed graphs α-hermitian adjacency matrix inverse matrix mixed graphs bipartite mixed graphs unicyclic bipartite mixed graphs perfect matching 05c20 05c50 |
| title | 𝕮-inverse of graphs and mixed graphs |
| title_full | 𝕮-inverse of graphs and mixed graphs |
| title_fullStr | 𝕮-inverse of graphs and mixed graphs |
| title_full_unstemmed | 𝕮-inverse of graphs and mixed graphs |
| title_short | 𝕮-inverse of graphs and mixed graphs |
| title_sort | 𝕮 inverse of graphs and mixed graphs |
| topic | mixed graphs α-hermitian adjacency matrix inverse matrix mixed graphs bipartite mixed graphs unicyclic bipartite mixed graphs perfect matching 05c20 05c50 |
| url | https://doi.org/10.1515/math-2024-0104 |
| work_keys_str_mv | AT alomariomar cinverseofgraphsandmixedgraphs AT abudayahmohammad cinverseofgraphsandmixedgraphs AT ghanemmanal cinverseofgraphsandmixedgraphs |