  We investigate two conjectured spectral graph theoretic strengthenings of
Tur\'an's theorem. Let $\mu_1 \ge \ldots \ge \mu_n$ denote the eigenvalues of a
graph $G$ with $n$ vertices, $m$ edges and clique number $\omega(G)$. The
concise version of Tur\'an's theorem is that $n/(n - d)$ is a lower bound for
the clique number $\omega(G)$, where $d$ is the average degree. Our first
conjecture is that $d$ can be replaced in this bound with $\sqrt{s^+}$, where
$s^+$ is the sum of the squares of the positive eigenvalues. We prove this
conjecture for triangle-free, weakly perfect and Kneser graphs and for almost
all graphs. We have also used various software tools to search for a
counter-example. Nikiforov proved a spectral version of Tur\'an's theorem that
\[ \mu_1^2 \le \frac{2m(\omega(G) - 1)}{\omega(G)}, \] and Bollob\'as and
Nikiforov conjectured that for $G \not = K_n$ \[ \mu_1^2 + \mu_2^2 \le
\frac{2m(\omega(G) - 1)}{\omega(G)}. \] For our second conjecture, we propose
that for all graphs $(\mu_1^2 + \mu_2^2)$ in this inequality can be replaced by
the sum of the squares of the $\omega(G)$ largest eigenvalues, provided they
are positive. We prove the conjecture for weakly perfect, Kneser, and classes
of strongly regular graphs. We also provide experimental evidence and describe
how the bound can be applied. Liu and Ning published a wide-ranging paper
entitled ``Unsolved Problems in spectral graph theory'', and these two
conjectures were placed second and fourth in their list of such problems.
