On bipartite cages of excess 4

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.