A $q$-analogue of distance matrix of bi-block graphs
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866914067834208256 |
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| author | Das, Joyentanuj |
| author_facet | Das, Joyentanuj |
| contents | A $q$-analogue of the distance matrix, referred to as the \emph{$q$-distance matrix}, is obtained from the distance matrix by replacing each nonzero entry $α$ with the sum $1+q+\cdots+q^{α-1}$. This notion was introduced independently by Bapat, Lal, and Pati~\cite{Ba-Lal-Pati}, and by Yan and Yeh~\cite{Yan}. A connected graph is called a \emph{bi-block graph} if each of its blocks is a complete bipartite graph. In this paper, we derive explicit formulas for the determinant and the inverse of the $q$-distance matrix of bi-block graphs. These results both generalize the corresponding formulas for the distance matrix of bi-block graphs obtained in~\cite{Hou3} and extend the results for block graphs in~\cite{Xing} to the class of bi-block graphs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_26060 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A $q$-analogue of distance matrix of bi-block graphs Das, Joyentanuj Combinatorics 05C12, 05C50 A $q$-analogue of the distance matrix, referred to as the \emph{$q$-distance matrix}, is obtained from the distance matrix by replacing each nonzero entry $α$ with the sum $1+q+\cdots+q^{α-1}$. This notion was introduced independently by Bapat, Lal, and Pati~\cite{Ba-Lal-Pati}, and by Yan and Yeh~\cite{Yan}. A connected graph is called a \emph{bi-block graph} if each of its blocks is a complete bipartite graph. In this paper, we derive explicit formulas for the determinant and the inverse of the $q$-distance matrix of bi-block graphs. These results both generalize the corresponding formulas for the distance matrix of bi-block graphs obtained in~\cite{Hou3} and extend the results for block graphs in~\cite{Xing} to the class of bi-block graphs. |
| title | A $q$-analogue of distance matrix of bi-block graphs |
| topic | Combinatorics 05C12, 05C50 |
| url | https://arxiv.org/abs/2509.26060 |