A $q$-analogue of distance matrix of bi-block graphs
Joyentanuj Das
TL;DR
We study the $q$-distance matrix $\mathscr{D}(G)$ of a bi-block graph $G$, where the off-diagonal entry encodes distance via the $q$-analogue $[\alpha]=1+q+\cdots+q^{\alpha-1}$. Using Schur complements, block decompositions, and Li-type cofactor formulas, we derive explicit determinant and inverse formulas, starting from the base case of the complete bipartite block $K_{s,t}$ and extending to general bi-block graphs by induction on the number of blocks. The main contributions are closed-form expressions for the determinant and cofactors, and an explicit inverse $\mathscr{D}^{-1} = -\mathscr{L} + \frac{1}{\lambda_G}\mathbf{x}\mathbf{x}^T$, with $\lambda_G$ and $\mathscr{L}$ defined in terms of block data and the vectors $\mathbf{x}$, $\mathbf{y}$; these results hold under suitable nondegeneracy conditions on $q$. These findings extend prior results for block graphs and bi-block graphs to the $q$-analogue setting, providing exact, computable formulas for analyzing $q$-distance structures on bi-block graphs. The work highlights structural decompositions that connect block-level determinants to the global matrix via induced leaf-block recurrences.
Abstract
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.