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All Journal Electronic Journal of Graph Theory and Applications (EJGTA)
Barghi, Amir
St. Michael's College
Electrical & Electronics Engineering

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Labeled graph rearrangements on matched and star products Barghi, Amir; DeFord, Daryl
Electronic Journal of Graph Theory and Applications (EJGTA) Vol 14, No 1 (2026): Electronic Journal of Graph Theory and Applications
Publisher : GTA Research Group, Univ. Newcastle, Indonesian Combinatorics Society and ITB

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.5614/ejgta.2026.14.1.7

Abstract

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.
Co-Authors DeFord, Daryl