I'm having some trouble with the an problem out of Bondy and Murty's Graph Theory (2008):
1.1.21 b) Show that rational eigenvalues of a graph are integral.
I understand that this is a statement about the adjacency matrix of the graph, however I'm currently learning linear algebra so I don't know what it means for an eigenvalue to be 'integral'. After several searches on the internet and various texts on the subject, as well as the B&M book itself, I cannot find any description whatsoever as to what it means for an eigenvalue to be integral. All I have been able to turn up was this PDF : https://core.ac.uk/download/files/153/6714827.pdf, the second page of which proves that question I've been posed.
However, the PDF does not actually clarify what being integral means, and additionally I don't understand how the lemmas used to imply the solution . Does anyone know what integral means in this situation, or where I can read about this?
Integral eigenvalues: all eigenvalues are integers.
Suppose $$P(x)=x^n+a_{n-1}x^{n-1}+...+a_1x+a_0$$ is the characteristic polynomial. Let $p/q$ is a rational root of $P(x)$ where $p$ and $q$ co-prime integers. Then we have $$-a_0q^{n-1}=\frac{p}{q}\left(p^{n-1}+a_{n-1}p^{n-2}q+...+a_1q^{n-1}\right).$$ Here LHS expression is an integer while the second factor in the RHS is also an integer. Therefore $p/q$ has to be an integer. That is - 'a rational root of a monic polynomial with integer coefficients is always an integer'.