On the Graphical $r$-Stirling Numbers of the First Kind for Specific Graph Families

This paper investigates the $\mathbf{\text{graphical}}r\mathbf{\text{-Stirling}}\mathbf{\text{numbers}}\mathbf{\text{of}}\mathbf{\text{the}}\mathbf{\text{first}}\mathbf{\text{kind}}$, denoted by $\text{str}Gk$, which enumerate partitions of a vertex set $V(G)$ into $k$ disjoint cycles such that $r$ specified vertices occupy distinct blocks. We establish closed-form expressions and recursive identities for fundamental graph families, including $\mathbf{\text{Path}}$ ($P_n$), $\mathbf{\text{Cycle}}$ ($C_n$), $\mathbf{\text{Star}}$ ($S_n$), $\mathbf{\text{Wheel}}$ ($W_n$), and $\mathbf{\text{Fan}}$ ($F_n$) graphs. A primary focus of this study is the $\mathbf{\text{statistical}}\mathbf{\text{characterization}}$ of the cycle distribution. We derive explicit formulas for the $\mathbf{\text{mean}}$ and $\mathbf{\text{variance}}$ of these numbers, extracted from the structural properties of the $r$-cycle polynomials. These results provide a rigorous measure of the average cycle density and variability across different graph topologies, bridging the gap between algebraic combinatorics and the structural analysis of restricted permutations.