The exponential distance matrix of bi-block graphs

TL;DR

This work extends the study of exponential distance matrices from trees and block graphs to bi-block graphs, deriving explicit closed forms for the determinant, inverse, and cofactor sums of the exponential distance matrix. It introduces a block-structure framework with matrices $\mathbf{A}$, $\mathbf{B}$ and a vertex-weight vector $\mu_G$, and leverages Schur complements and block decompositions to obtain universal formulas that depend only on block sizes. The authors also connect these results to a $q$-Laplacian generalization, showing that $\mathscr{F}^{-1}=\frac{1}{1-q^2}\mathscr{L}$ under invertibility, thus unifying exponential distance matrix theory with a natural spectral operator on bi-block graphs. Overall, the paper generalizes several known results and provides explicit, computable expressions for a broad class of graphs built from complete bipartite blocks.

Abstract

Let $G$ be a connected graph with vertex set $\{v_1, v_2, \ldots, v_\mathbf{n}\}$. As a variant of the classical distance matrix, the \emph{exponential distance matrix} was introduced independently by Yan and Yeh, and by Bapat et al. For a nonzero indeterminate $q$, the exponential distance matrix $\mathscr{F} = (\mathscr{F}_{ij})_{\mathbf{n} \times \mathbf{n}}$ of $G$ is defined by $\mathscr{F}_{ij} = q^{d_{ij}},$ where $d_{ij}$ denotes the distance between vertices $v_i$ and $v_j$ in $G$. A connected graph is said to be a \emph{bi-block graph} if each of its blocks is a complete bipartite graph, possibly of varying bipartition sizes. In this paper, we obtain explicit expressions for the determinant, inverse, and cofactor sum of the exponential distance matrix of bi-block graphs. As a consequence, some known results concerning the exponential distance matrix and the $q$-Laplacian matrix are generalized.

Theorems & Definitions (21)

• Proposition 2.1
• Proposition 2.2
• Lemma 2.3
• proof
• Proposition 3.1
• Lemma 3.2
• Lemma 3.3
• proof
• Theorem 3.4
• proof