Hulls of codes from incidence matrices of connected regular graphs

The hulls of codes from the row span over F_p, for any prime p, of incidence matrices of connected k-regular graphs are examined, and the dimension of the hull is given in terms of the dimension of the row span of A+kI over \F_p, where A is an adjacency matrix for the graph. If p=2, for most classes of connected regular graphs with some further form of symmetry, it was shown that the hull is either {0} or has minimum weight at least 2k-2. Here we show that if the graph is strongly regular with parameter set (n,k,\lambda,\mu), then, unless k is even and $\mu$ is odd, the binary hull is non-trivial, of minimum weight generally greater than 2k-2, and we construct words of low weight in the hull; if k is even and $\mu$ is odd, we show that the binary hull is zero. Further, if a graph is the line graph of a $k$-regular graph, $k \ge 3$, that has an $\ell$-cycle for some $\ell \ge 3$, the binary hull is shown to be non-trivial with minimum weight at most $2\ell(k-2)$. Properties of the $p$-ary hulls are also established. }