On regular graphs with a tree of diameter 3 as a star complement

We investigate the regular graphs with a star complement H which is a tree of diameter 3. Thus H is a double star $D(m,n)$, i.e. a tree with two vertices of degree m and n greater than 1, and all other vertices of degree 1. We determine all the regular graphs G that arise when either (a) $\mu \in \{0,1\}$ or (b) $m=n$ and μ is an integer less than −1. It is also proved that for $m=n$ and $\mu \ge 2$, the degree of G must be n; moreover,$n\ge \frac{\mu \left(\mu \right(2\mu (\mu +1)-3)+3)-1}{2\mu -1},$ when μ is an integer.