On the 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 e=n−M(k,g). A (k,g)-cage is a (k,g)-graph with the fewest possible number of vertices. 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. Biggs and Ito proved that any (k,g)-cage of even girth g=2d≥6 and excess e≤k−2 is a bipartite graph of diameter d+1. In this paper we treat (k,g)-cages of even girth and excess e≤k−2. Based on spectral analysis we give a relation between the eigenvalues of the adjacency matrix A and the distance matrix Ad+1 of such cages. Applying matrix theory, we prove the non-existence of antipodal (k,g)-cages of excess e, for k≥e+2≥4 and g=2d≥14.