Let G be a graph of order n ≥1 with chromatic polynomial pG(k). (a) Prove that the constant term

step 1 So for this problem, we need to require G is a simple graph. Otherwise, the result is not true.

Let G be a graph of order n ≥1 with chromatic polynomial...

Let $G$ be a graph of order $n \geq 1$ with chromatic polynomial $p_{G}(k)$. (a) Prove that the constant term of $p_{G}(k)$ equals $0 .$ (b) Prove that the coefficient of $k$ in $p_{G}(k)$ is nonzero if and only if $G$ is connected. (c) Prove that the coefficient of $k^{n-1}$ in $p_{G}(k)$ equals $-m$, where $m$ is the number of edges of $G$.

Let $G$ be a graph of order $n \geq 1$ with chromatic polynomial $p_{G}(k)$.
(a) Prove that the constant term of $p_{G}(k)$ equals $0 .$
(b) Prove that the coefficient of $k$ in $p_{G}(k)$ is nonzero if and only if $G$ is connected.
(c) Prove that the coefficient of $k^{n-1}$ in $p_{G}(k)$ equals $-m$, where $m$ is the number of edges of $G$.

Key Concepts

Chromatic Polynomial

The chromatic polynomial of a graph is a function that counts the number of proper colorings using k colors. It is a polynomial of degree equal to the number of vertices. Fundamental properties include that the constant term is zero (since no proper coloring exists with zero colors) and that its coefficients encode structural information about the graph. The polynomial is derived using combinatorial arguments such as deletion–contraction or the Inclusion–Exclusion Principle, making it a central tool in the study of graph colorings.

Graph Connectivity

Connectivity in a graph refers to whether there is a path between every pair of vertices. This property significantly impacts the chromatic polynomial; for instance, the coefficient of k in the polynomial is nonzero if and only if the graph is connected. This relationship illustrates how the global structure of a graph influences the colorability and the algebraic properties of its chromatic polynomial.

Graph Structure and Edge Count

The structure of a graph, particularly the number of edges, plays a critical role in determining specific coefficients in the chromatic polynomial. For example, the coefficient of the term k^(n-1) is equal to minus the number of edges. This reflects how each edge imposes a restriction on how vertices can be colored, directly linking the graph’s combinatorial properties to the algebraic expression of its chromatic polynomial.

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