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