  The Moore bound constitutes both an upper bound on the order of a graph of
maximum degree $d$ and diameter $D=k$ and a lower bound on the order of a graph
of minimum degree $d$ and odd girth $g=2k+1$. Graphs missing or exceeding the
Moore bound by $\epsilon$ are called {\it graphs with defect or excess
$\epsilon$}, respectively.
  While {\it Moore graphs} (graphs with $\epsilon=0$) and graphs with defect or
excess 1 have been characterized almost completely, graphs with defect or
excess 2 represent a wide unexplored area.
  Graphs with defect (excess) 2 satisfy the equation $G_{d,k}(A) = J_n + B$
($G_{d,k}(A) = J_n-B$), where $A$ denotes the adjacency matrix of the graph in
question, $n$ its order, $J_n$ the $n\times n$ matrix whose entries are all
1's, $B$ the adjacency matrix of a union of vertex-disjoint cycles, and
$G_{d,k}(x)$ a polynomial with integer coefficients such that the matrix
$G_{d,k}(A)$ gives the number of paths of length at most $k$ joining each pair
of vertices in the graph.
  In particular, if $B$ is the adjacency matrix of a cycle of order $n$ we call
the corresponding graphs \emph{graphs with cyclic defect or excess}; these
graphs are the subject of our attention in this paper.
  We prove the non-existence of infinitely many such graphs. As the highlight
of the paper we provide the asymptotic upper bound of $O(\frac{64}3d^{3/2})$
for the number of graphs of odd degree $d\ge3$ and cyclic defect or excess.
This bound is in fact quite generous, and as a way of illustration, we show the
non-existence of some families of graphs of odd degree $d\ge3$ and cyclic
defect or excess.
  Actually, we conjecture that, apart from the M\"obius ladder on 8 vertices,
no non-trivial graph of any degree $\ge 3$ and cyclic defect or excess exists.
