  The Moore bound $M(k,g)$ is a lower bound on the order of $k$-regular graphs
of girth $g$ (denoted $(k,g)$-graphs). The excess $e$ of a $(k,g)$-graph of
order $n$ is the difference $ n-M(k,g) $. In this paper we consider the
existence of $(k,g)$-bipartite graphs of excess $4$ via studying spectral
properties of their adjacency matrices. We prove that the $(k,g)$-bipartite
graphs of excess $4$ satisfy the equation $kJ=(A+kI)(H_{d-1}(A)+E)$, where $A$
denotes the adjacency matrix of the graph in question, $J$ the $n \times n$
all-ones matrix, $E$ the adjacency matrix of a union of vertex-disjoint cycles,
and $H_{d-1}(x)$ is the Dickson polynomial of the second kind with parameter
$k-1$ and of degree $d-1$. We observe that the eigenvalues other than $\pm k$
of these graphs are roots of the polynomials $H_{d-1}(x)+\lambda$, where
$\lambda$ is an eigenvalue of $E$. Based on the irreducibility of
$H_{d-1}(x)\pm2$ we give necessary conditions for the existence of these
graphs.
  If $E$ is the adjacency matrix of a cycle of order $n$ we call the
corresponding graphs \emph{graphs with cyclic excess}; if $E$ is the adjacency
matrix of a disjoint union of two cycles we call the corresponding graphs
\emph{graphs with bicyclic excess}. In this paper we prove the non-existence of
$(k,g)$-graphs with cyclic excess $4$ if $k\geq6$ and $k \equiv1 \!\! \pmod
{3}$, $g=8, 12, 16$ or $k \equiv2 \!\! \pmod {3}$, $g=8,$ and the non-existence
of $(k,g)$-graphs with bicyclic excess $4$ if $k\geq7$ is odd number and $g=2d$
such that $d\geq4$ is even.
