$\mathcal{B}$-Partitions, determinant and permanent of graphs

Abstract

Let $G$ be a graph (directed or undirected) having $k$ number of blocks $B_1, B_2,\hdots,B_k$. A $\mathcal{B}$-partition of $G$ is a partition consists of $k$ vertex-disjoint subgraph $(\hat{B_1},\hat{B_1},\hdots,\hat{B_k})$ such that $\hat{B}_i$ is an induced subgraph of $B_i$ for $i=1,2,\hdots,k.$ The terms $\prod_{i=1}^{k}\det(\hat{B}_i),\ \prod_{i=1}^{k}\text{per}(\hat{B}_i)$ represent the det-summands and the per-summands, respectively, corresponding to the $\mathcal{B}$-partition $(\hat{B_1},\hat{B_1},\hdots,\hat{B_k})$. The determinant (permanent) of a graph having no loops on its cut-vertices is equal to the summation of the det-summands (per-summands), corresponding to all possible $\mathcal{B}$-partitions. In this paper, we calculate the determinant and the permanent of classes of graphs such as block graph, block graph with negatives cliques, signed unicyclic graph, mixed complete graph, negative mixed complete graph, and star mixed block graphs.

Keywords

• $mathcal{b}$-partition
• signed graph
• mixed block graph