TY - JOUR
ID - 22426
TI - $mathcal{B}$-Partitions, determinant and permanent of graphs
JO - Transactions on Combinatorics
JA - TOC
LA - en
SN - 2251-8657
AU - Singh, Ranveer
AU - Bapat, Ravindra B.
AD - Department of Mathematics, Indian Institute of Technology Jodhpur, Jodhpur, India
AD - Stat-Math Unit, ISI Delhi
Y1 - 2018
PY - 2018
VL - 7
IS - 3
SP - 37
EP - 54
KW - $mathcal{B}$-partition
KW - signed graph
KW - mixed block graph
DO - 10.22108/toc.2017.105288.1508
N2 - 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.
UR - https://toc.ui.ac.ir/article_22426.html
L1 - https://toc.ui.ac.ir/article_22426_684cba1ff8383118f056e8041a6e743a.pdf
ER -