The chromatic polynomial is a concept of algebraic graph theory studied for the first time by George David Birkhoff in 1912. Depending on the number of colors, it evaluates all the possible distinct colorations of the vertices of a graphic data. Using the edge elimination and contraction method, the chromatic polynomial can be expressed as the sum of simpler subgraphs. It can therefore be evaluated recursively. Many relevant properties can also be proved inductively. George originally studied the chromatic polynomial in an attempt to prove the four-color problem, which states that any map can be colored using at most 4 colors such that no two adjacent regions are the same color. In the context of graph theory, any map can be viewed as a planar graph. George wanted to prove that 4 is not a chromatic (root) number for planar graphs in general. Currently, the roots and coefficients of the chromatic polynomial remain important topics of study in modern graph theory. Chapter 1 will formally introduce the problem of vertex coloring. Some of the most important results related to vertex coloring will be discussed. Chapter 2 is a rigorous study of the chromatic polynomial in the context of algebraic graph theory. Chapters 3 and 4 are based on personal research. Some interpretations of the coefficients and conditions on the roots of the chromatic polynomial will be discussed. Chapter 1 Vertex coloring problem The structure of a graph Γ is defined by its vertices and their pairwise relations specified by the edges. A vertex coloring is an assignment of colors to the vertices of Γ and thus defines a partition of its vertex set. Definition 1.1. Let Γ = (V, E) be a finite graph, where V is the set of vertices and E the set of edges. A vert...... at the center of the sheet......(c) = M4(e) = M5(b) = M5(d).M4 : V → C M5 : V → CFigure 1.4: the disconnected, the bipartite graph Γ3 cannot be uniquely colored. If Γ is uniquely colorable, then there exists one and only one feasible partition of V into χ(Γ) = r independent sets. Thus, if M is an r-coloring of Γ, then every vertex in any color class Vi of M must have at least one edge connected to any other color class. Thus,│E│ ≥ ((r-1)│V│)/2 .In fact, a better limit can be found, which leads to the next proposition.Proposition 1.7. Let Γ = (V, E) be a finite graph and χ(Γ) = r. If Γ is uniquely colorable, then│E│ ≥ (r-1)│V│ - (r(r-1))/2 .Proof. Let P = {V1, V2, …, Vr} be the unique partition of V into r independent subsets. Note that every subgraph Γ'(Vi ) induced by any two color classes Vi and Vj must be