# $\mathcal{B}$-partitions, application to determinant and permanent of graphs

## TL;DR

This paper introduces the concept of $\\mathcal{B}$-partitions in graphs to compute determinants and permanents, providing formulas and calculations for various classes of graphs including block graphs and signed graphs.

## Contribution

It defines $\\mathcal{B}$-partitions and derives formulas for determinants and permanents of graphs using these partitions, extending previous methods to new graph classes.

## Key findings

- Derived formulas for determinants and permanents using $\\mathcal{B}$-partitions.
- Calculated determinants and permanents for specific graph classes.
- Extended methods to signed and mixed graphs.

## Abstract

Let $G$ be a graph(directed or undirected) having $k$ number of blocks. A $\mathcal{B}$-partition of $G$ is a partition into $k$ vertex-disjoint subgraph $(\hat{B_1},\hat{B_1},\hdots,\hat{B_k})$ such that $\hat{B}_i$ is induced subgraph of $B_i$ for $i=1,2,\hdots,k.$ The terms $\prod_{i=1}^{k}\det(\hat{B}_i),\ \prod_{i=1}^{k}\text{per}(\hat{B}_i)$ are det-summands and per-summands, respectively, corresponding to the $\mathcal{B}$-partition. The determinant and permanent of a graph having no loops on its cut-vertices is equal to summation of det-summands and per-summands, respectively, corresponding to all possible $\mathcal{B}$-partitions. Thus, in this paper we calculate determinant and permanent of some graphs, which include block graph with negatives cliques, signed unicyclic graph, mix complete graph, negative mix complete graph, and star mix block graphs.