  In this paper we study the existence of homomorphisms $G\to H$ using
semidefinite programming. Specifically, we use the vector chromatic number of a
graph, defined as the smallest real number $t \ge 2$ for which there exists an
assignment of unit vectors $i\mapsto p_i$ to its vertices such that $\langle
p_i, p_j\rangle\le -1/(t-1),$ when $i\sim j$. Our approach allows to reprove,
without using the Erd\H{o}s-Ko-Rado Theorem, that for $n>2r$ the Kneser graph
$K_{n:r}$ and the $q$-Kneser graph $qK_{n:r}$ are cores, and furthermore, that
for $n/r = n'/r'$ there exists a homomorphism $K_{n:r}\to K_{n':r'}$ if and
only if $n$ divides $n'$. In terms of new applications, we show that the
even-weight component of the distance $k$-graph of the $n$-cube $H_{n,k}$ is a
core and also, that non-bipartite Taylor graphs are cores. Additionally, we
give a necessary and sufficient condition for the existence of homomorphisms
$H_{n,k}\to H_{n',k'}$ when $n/k = n'/k'$. Lastly, we show that if a
2-walk-regular graph (which is non-bipartite and not complete multipartite) has
a unique optimal vector coloring, it is a core. Based on this sufficient
condition we conducted a computational study on Ted Spence's list of strongly
regular graphs and found that at least 84% are cores.
