Non-existence of antipodal cages of even girth

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 $n-M(k,g).$ A $(k,g)$-cage is a $(k,g)$-graph with the fewest possible number of vertices, among all $(k,g)$-graphs. A graph of diameter $d$ is said to be antipodal if, for any vertices $u, v, w$ such that $d(u,v)=d$ and $d(u, w)=d$, it follows that $d(v, w)=d$ or $v=w.$ In [4] Biggs and Ito proved that any $(k,g)$-cage of even girth $g=2d\geq6$ and excess $e\leq k-2$ is a bipartite graph of diameter $d+1.$ In this paper we treat the $(k,g)$-cages of even girth and excess $e\leq k-2.$ Based on a spectral analysis we give a relation between the eigenvalues of the adjacency matrix $A$ and the distance matrix $A_{d+1}$ of such cages. Moreover, following the methodology used in [4] and [13], we prove the non-existence of the antipodal $(k,g)$-cages of excess $e$, where $k\geq e+2\geq4$ and $g=2d\geq14.$