## 7.1 Introduction

In this chapter we are concerned with the eigenvalues of graphs and some of their chemical applications. Let *G* be a (simple) graph, with vertex set *V*(*G*) and edge set *E*(*G*). The number of vertices of *G* is *n*, and its vertices are labeled by *v*_{1}, *v*_{2},..., *v*_{n}. The adjacency matrix *A*(*G*) of the graph *G* is a square matrix of order *n*, whose (*i*, *j*)-entry is equal to 1 if the vertices *v*_{i} and *v*_{j} are adjacent and equal to zero otherwise.

The eigenvalues λ_{1}, λ_{2},..., λ_{n} of the adjacency matrix *A*(*G*) are said to be the *eigenvalues of the graph G* and to form its *spectrum*. Details of the spectral theory of graphs can be found in the seminal monograph [1].

The characteristic polynomial of the adjacency matrix, i.e., det(λ *I*_{n}– *A*(*G*)), where *I*_{n} is the unit matrix of order *n*, is said to be the *characteristic polynomial of the graph G* and will be denoted by *ϕ*(*G*, λ). From linear algebra it is known that the graph eigenvalues are just the solutions of the equation *ϕ*(*G*, λ) = 0.

One of the most remarkable chemical applications of graph theory is based on the close correspondence between the graph eigenvalues and the molecular orbital energy levels of π-electrons in conjugated hydrocarbons. For details, see [2–4]. If *G* is a molecular graph of a conjugated hydrocarbon with *n* vertices and λ_{1}, λ_{2},..., λ_{n} are its eigenvalues, then in the so-called Hüchkel molecular orbital (HMO) approximation [3,5], the energy of the *i*th molecular orbital is given by

where α and β are pertinent constants. In order ...