Graphs of Reduced Words and Some Connections

The family of graphs of reduced words of a certain subcollection of permutations in the union ${\cup }_{n\ge 4}{\mathfrak{S}}_n$ of symmetic groups is investigated. The subcollection is characterised by the hook cycle type $(n-2,1,1)$ with consecutive fixed points. A closed formula for counting the vertices of each member of the family is given and the vertex-degree polynomials for the graphs with their generating series is realised. Lastly, some isomorphisms of these graphs with various combinatorial objects are established.