In general, graph polynomials are a way to encode information about a graph in a polynomial. One of the most famous examples is the chromatic polynomial of a graph which outputs the number of ways to color a graph with $k$ colors when evaluated at $k$. This was first introduced by Birkhoff in 1912 as a way to prove the then 4-Color Conjecture. His approach was to combine the combinatorial problem of graph coloring with the analytic properties of finding roots of polynomials to show that if $G$ was a planar graph, then its chromatic polynomial never had a root at $4$. Although Birkhoff was unsuccessful in proving the 4-Color Conjecture using the chromatic polynomial, his work paved the way for using analytic techniques of a graph polynomial to uncover information about the underlying graph.
Many of the polynomials that have been studied in the last century since Birkhoff have been generating polynomials, i.e. the coefficients of these polynomials count subsets with certain properties of different sizes. One such polynomial has been the main object of my academic interest: the independence polynomial. The independence polynomial was introduce as a generalization of the matching polynomial was by Gutman and Harary in 1983. It is defined as follows: For a graph $G$, an independent set is a subset of $V(G)$ that are all pair-wise nonadjacent, the independence number is the cardinality of the largest independent set in $G$. Let $i_k$ denote the number of independent sets with $k$ vertices in $G$. Then the independence polynomial of $G$, denoted $i(G,x)$, is defined by $$ i(G,x)=\sum_{k=0}^{\alpha(G)}i_kx^k. $$ Questions then arise on the shape of the coefficient sequence as this contains information about the graph. A polynomial $p(x)=\sum\limits_{k=0}^na_kx^k$ is unimodal if it is first nondecreasing and then nondecreasing, more precisely, there exists a $j$ such that $a_0\le a_1\le\cdots\le a_{j-1}\le a_j\ge a_{j+1}\ge\cdots \ge a_{n-1}\ge a_n$. It is said to be log-concave if $a_k^2\ge a_{k-1}a_{k+1}$ for all $k=1,2,\ldots ,n-1$. It is not too difficult to see that any log-concave polynomial with all positive coefficients is also unimodal (the ratios of consecutive coefficients are always nondecreasing, so while they are bigger than $1$, the coefficients are nondecreasing and after they drop below $1$, the coefficients are nonincreasing). A more surprising fact is that a polynomial with all positive coefficients that has all real roots is necessarily log-concave and therefore unimodal. This suggests that looking at the roots, in particular the nature of these roots, can provide insight into the underlying graph. There are other more general results (see one that was used in my paper 2018 paper on unimodality) that do not restrict the roots to being real and therefore justify looking at all roots of the independence polynomial in the complex plane. We call the roots of $i(G,x)$ the independence roots of $G$. Problems I have been interested in have been using to roots to prove certain families of graphs have log-concave independence polynomials, determining families of graphs with independence roots in the right half of the complex plane, and independence polynomials with independence roots on the imaginary axis.
There are many other generating graph polynomials, for example, the matching (matching-generating), domination, edge cover, and zero forcing polynomials are generating polynomials for the numbers of matchings, dominating sets, edge covers, and zero forcing sets of the graph, respectively. Open questions related to the roots and coefficient sequences of various generating graph polynomials are listed below in no particular order:
