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

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.

💡 Research Summary

This paper introduces a novel combinatorial framework called “B-partition” for computing the determinant and permanent of the adjacency matrix of a graph. The method leverages the block structure of a graph, providing a way to break down the calculation for complex graphs into simpler, manageable parts.

The core idea is to partition a graph into vertex-disjoint induced subgraphs, each derived from one of the graph’s blocks (maximal biconnected subgraphs). Such a partition is called a B-partition. The authors prove a fundamental lemma (Lemma 3.2) stating that for a weighted (signed or directed) graph with no loops on its cut-vertices, the determinant (and permanent) is equal to the sum, over all possible B-partitions, of the product of the determinants (or permanents) of the subgraphs in each partition. These product terms are named “det-summands” and “per-summands.”

A key theoretical contribution is establishing a one-to-one correspondence (Lemma 3.3) between these B-partitions and the k-tuples of non-negative integers used in a known determinant formula for simple block graphs (Theorem 3.1). This bridges a known combinatorial formula with the new, more general, and interpretable B-partition concept. The framework is then used to derive the permanent formula for balanced signed block graphs (Theorem 3.4).

The power of the B-partition method is demonstrated through its application to several non-trivial graph classes:

1. Signed Block Graphs with Negative Cliques: The determinant is computed for graphs where each block is a complete signed graph containing vertex-disjoint negative cliques of equal size, with positive edges incident to cut-vertices.
2. Signed Unicyclic Graphs: These are connected graphs with number of edges equal to number of vertices, essentially a cycle with trees attached. Formulas for determinant and permanent are derived, distinguishing between cases where the central cycle is balanced or unbalanced.
3. Mixed Complete Graphs (mK_n): Defined as complete graphs where every possible arc (directed edge) with any sign (±1) is present between non-adjacent vertices of a base directed cycle. The eigenvalues of these graphs and their all-negative counterparts are analyzed first, leading to determinant expressions via B-partitions.
4. Mixed Star Block Graphs: Graphs where several mixed complete graphs share a single common cut-vertex. A recursive application of the B-partition principle yields a formula for their determinant.

In summary, this paper presents B-partition as a unifying and powerful tool that reduces the problem of computing graph determinants/permanents to a combinatorial counting problem over the graph’s decomposable structure. It provides explicit formulas for previously untreated graph families, offering new insights at the intersection of graph theory, linear algebra, and combinatorics. The results have potential implications in spectral graph theory, chemical graph theory (e.g., Hückel theory), and the study of graph polynomials.