On the singularity and the inverse of 3-colored digraphs
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866913112190353408 |
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| author | Zaffer, Md Isheteyak |
| author_facet | Zaffer, Md Isheteyak |
| contents | This article considers the class of connected 3-colored digraphs. Let $G$ be a 3-colored digraph and $A(G)$ be its adjacency matrix. $G$ is said to be non-singular (resp. singular) if $A(G)$ is a non-singular (resp. singular) matrix. A connected digraph is k-cyclic if it has $n$ vertices and $n+k-1$ edges. The main objective of this article is to provide a characterization of non-singular 3-colored unicyclic and bicyclic digraphs. If $A(G)$ is non-singular and $A(G)^{-1}$ has a $zero$ diagonal, then $A(G)^{-1}$ can be realized as the adjacency matrix of a digraph with complex weights. Therefore, we also identify all 3-colored bicyclic digraphs such that the diagonal of $A(G)^{-1}$ is zero. Furthermore, we study the invertibility of these digraphs and identify all those bicyclic 3-colored digraphs whose inverse is also a 3-colored digraph. We conduct the same study for the class of unicyclic 3-colored digraphs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_24274 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the singularity and the inverse of 3-colored digraphs Zaffer, Md Isheteyak Combinatorics 05C50 (Primary) 15A09, 05C22, 15C75, 05C20 (Secondary) This article considers the class of connected 3-colored digraphs. Let $G$ be a 3-colored digraph and $A(G)$ be its adjacency matrix. $G$ is said to be non-singular (resp. singular) if $A(G)$ is a non-singular (resp. singular) matrix. A connected digraph is k-cyclic if it has $n$ vertices and $n+k-1$ edges. The main objective of this article is to provide a characterization of non-singular 3-colored unicyclic and bicyclic digraphs. If $A(G)$ is non-singular and $A(G)^{-1}$ has a $zero$ diagonal, then $A(G)^{-1}$ can be realized as the adjacency matrix of a digraph with complex weights. Therefore, we also identify all 3-colored bicyclic digraphs such that the diagonal of $A(G)^{-1}$ is zero. Furthermore, we study the invertibility of these digraphs and identify all those bicyclic 3-colored digraphs whose inverse is also a 3-colored digraph. We conduct the same study for the class of unicyclic 3-colored digraphs. |
| title | On the singularity and the inverse of 3-colored digraphs |
| topic | Combinatorics 05C50 (Primary) 15A09, 05C22, 15C75, 05C20 (Secondary) |
| url | https://arxiv.org/abs/2602.24274 |