Arithmetical Structures On Fan Graphs
Dilli Ram Chhetri, Namita Behera, Raj Bhawan Yadav
TL;DR
This work develops a comprehensive description of arithmetical structures on Fan graphs $F_n$ by deriving explicit divisibility conditions for $r$-vectors, mapping to corresponding $d$-structures, and identifying the Laplacian arithmetical structure $(2n,2,\ldots,2)$ with $r=(1,\ldots,1)$. It then analyzes how clique-star transformations yield arrow-star graphs and related constructions, establishing recurrence methods to build $F_n$-structures from arm data and showing growth in the number of structures under these transforms. A key contribution is the decomposition of the fan's critical group as a product over its arms, with each arm's critical group drawn from a small set { $\mathbb{Z}_3$, $\mathbb{Z}_2$, $\{e\}$ }, and the extension of these ideas to stars via Egyptian-fraction correspondences. The work also connects the structures on $F_n$ to those on derived graphs like $AS_n$, $C_4F_n$, and corona graphs, offering a framework for enumerating arithmetical structures and applying Smith normal form techniques to understand the associated critical groups.
Abstract
In this paper, we study the arithmetical structures on Fan Graphs Fn. Let G be a finite and connected graph. An arithmetical structure on G is a pair (d, r) of positive integer vectors such that r is primitive (the greatest common divisor of its coefficients is 1) and (diag(d)-A)r = 0, where A represents the adjacency matrix of G. This work explores the combinatorial properties of the arithmetical structures associated with Fn. Further, we discuss the arrow-star graph, a structure derived from the fan graph, along with its properties. Additionally, we investigate the critical group linked to each such structure on Fn.