Labeled graph rearrangements on matched and star products - Amir Barghi; Daryl DeFord (GTA Research Group, Univ. Newcastle, Indonesian Combinatorics Society and ITB, 2026)
In this paper we present enumerative results for Stirling numbers of the first kind for two graph products, the matched product and the m-star, using the combinatorial model of rearrangements. The kth Stirling number of the first kind for a simple graph G counts the number of ways to decompose G into exactly k vertex-disjoint cycles, including single vertices as 1-cycles, single edges as 2-cycles, and counting orientations for cycles of order three or higher. This naturally leads to the definition of the graphical factorial of G as the total number of such decompositions without any restrictions on the number of cycles involved. The matched product, motivated by a popular construction for modeling multiplex data, requires specifying a labeling of each component graph and naturally defines several families of graphs whose Stirling numbers of the first kind can be enumerated in terms of their component graphs. The m-star product of a graph G is defined as the join of G with the empty graph with m vertices. We compute the Stirling numbers of the first kind and factorials for the m-star of complete graphs, forests, and cycles, and we provide bounds for other families. The combinatorial proofs in the case of m-star products also motivate a generalization in terms of disjoint path decompositions of graphs, which provides another promising avenue of study.