\section{Introduction}

Throughout $G$ is a graph of order $n$ with adjacency matrix $A$.
Let $I$, $J$ and $D$ be the identity matrix, the all-ones matrix and the diagonal matrix with the degrees of $G$ on the diagonal,
respectively.
Any matrix of the form $U=\a A+\b I + \c J + \d D$ with $\a,\b,\c,\d\in\Re$, $\a\neq 0$ (as usuall, $\Re$ is the set of real numbers)
is called a {\em universal adjacency matrix} of $G$.
Several well-studied types of matrices associated to $G$ are a special case of a universal matrix.
For example, if $(\alpha,\beta,\c,\delta)=(1,0,0,0)$, $(-1,0,0,1)$, $(1,0,0,1)$, $(-2,-1,1,0)$ we obtain the adjacency,
the Laplacian, the signless Laplacian and the Seidel matrix, respectively,
and for $(\a,\b,\c,\d)=(-1,-1,1,0)$, $(1,n,-1,-1)$, $(-1,n-2,1,-1)$, $(2,1,-1,0)$
we get the matrices of the complement $\overline{G}$ of $G$ of the mentioned types.

If $G$ is regular of degree $d$, then there is an easy relation between the spectra of the various universal adjacency matrices.
Indeed, if $d=\x_1,\ \x_2,\ \ldots,\ \x_n$ are the eigenvalues of $A$, then
$\a d +\b+\c n+\d d,\ \a \x_2+\b+\d d,\ \ldots,\ \a \x_n+\b+\d d$ are the eigenvalues of $U$.
Let us denote the characteristic polynomial of a matrix $M$, $\det(\x I-M)$, by $\f(M,\x)$.
Then the relation between the spectrum of $A$ and the spectrum of $U$ can be formulated as:
\begin{equation}\label{UA}
\frac{\f(U,\a \x+\b+\d d)}{\a\x-\a d - \c n}
= \a^{n-1} \frac{\f(A,\x)}{\x-d}.
\end{equation}
In this note we consider the case that