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A $(k,g)$-cage is a $k$-regular simple graph of girth $g$ with minimumpossible number of vertices. In this paper, $(k,g)$-cages which are Mooregraphs are referred as minimal $(k,g)$-cages. A simple connected graph iscalled distance regular(DR) if all its vertices have the same intersectionarray. A bipartite graph is called distance biregular(DBR) if all the verticesof the same partite set admit the same intersection array. It is known thatminimal $(k,g)$-cages are DR graphs and their subdivisions are DBR graphs. Inthis paper, for minimal $(k,g)$-cages we give a formula for distance spectralradius in terms of $k$ and $g$, and also determine polynomials of degree$[\frac{g}{2}]$, which is the diameter of the graph. This polynomial gives alldistance eigenvalues when the variable is substituted by adjacency eigenvalues.We show that a minimal $(k,g)$-cage of diameter $d$ has $d+1$ distinct distanceeigenvalues, and this partially answers a problem posed in [5]. We prove thatevery DBR graph is a $2$-partitioned transmission regular graph and then give aformula for its distance spectral radius. By this formula we obtain thedistance spectral radius of subdivision of minimal $(k,g)$-cages. Finally wedetermine the full distance spectrum of subdivision of some minimal$(k,g)$-cages.

On the distance spectrum of minimal cages and associated distance biregular graphs