Edward J Farrell is a celebrated mathematician of the African Diaspora. In 1978, he introduced a general class of graph polynomials, called Farrell-polynomials. Let $F$ be a family of connected graphs (meaning there exists a path between each pair of nodes – a path can be made up of edges and other nodes). With each element $\alpha$ belonging to $F$, we can associate an indeterminate or weight $w_{\alpha}$. An $F$-cover of a graph $G$ is a spanning subgraph, all of whose components belong to $F$. A spanning subgraph is a subgraph which contains all nodes of the original graph. With each $F$-cover $C$ of $G$, we associate the weight $w(C)=\prod w_{\alpha}$, where the product is taken over all the components $\alpha$ in $C$. The $F$-polynomial of $G$ is $F(G;\underline{w})=\sum w(C)$, where $\underline{w}$ is a vector of indeterminates defined by $w_{\alpha}$ and the summation is taken over all the $F$-covers in $G$.

A tree is a graph in which any pair of nodes is connected by exactly one path. An $m$-star $S_m$ is a tree with $m+1$ nodes, containing a node of valency (degree) $m$ called the centre of the star, meaning it is joined to $m$ other nodes. These $m$ nodes of valency 1 are called tips. A proper star is a star that contains at least one edge. A 0-star is a component node and a 1-star is a component edge as shown below.  Continue reading →