Characteristic polynomial of a graph

Characteristic polynomial of a graph --- характеристический полином графа.

Given a graph [math]\displaystyle{ G }[/math], the polynomial [math]\displaystyle{ p_{G}(\lambda) = \det(\lambda I - A_{G}) }[/math], where [math]\displaystyle{ A_{G} }[/math] is the adjacency matrix of [math]\displaystyle{ G }[/math]. This clearly does not depend on the labeling of vertices. The roots of the characteristic polynomial, i.e. the eigenvalues of [math]\displaystyle{ A_{G} }[/math], are called the eigenvalues of the graph [math]\displaystyle{ G }[/math]. The spectrum of [math]\displaystyle{ G }[/math] is the set of solutions to [math]\displaystyle{ p(\lambda) = 0 }[/math] denoted by [math]\displaystyle{ S_{p} = \{\lambda_{1}, \lambda_{2}, \ldots, \lambda_{n}\} }[/math] with [math]\displaystyle{ \lambda_{1} \geq \lambda_{2} \geq \ldots, \lambda_{n} }[/math]. The first eigenvalue [math]\displaystyle{ \lambda_{1} }[/math] is called the index or spectral radius.