On the Connected Subgraphs Arrangement of the Wheel Graph W5: Intersection Lattice, Supersolvability, and Freeness

Abstract

This paper presents a comprehensive analysis of the connected subgraphs arrangement  associated with the wheel graph . Following the definition by [1], this arrangement consists of hyperplanes  where  induces a connected subgraph of  and either contains the central vertex 0 or forms a connected interval on the cycle . We explicitly compute the full intersection lattice of this arrangement, contains all 1570 non-empty intersections. Using this lattice , we provide two distinct proofs of fundamental algebraic properties. First, we demonstrate that   is not supersolvable by exhibiting a specific element that fails to be modular. Second, using a localization argument centered on a non-chordal cyclic subarrangement, we apply Stanley’s theorem to prove that   is not free. Finally, we extend our discussion to the general family   for , conjecturing that these connected subgraphs arrangements are neither supersolvable nor free for all .