The structure of <f>(t,r)</f>-regular graphs of large order

A graph is <f>(t,r)</f>-regular iff it has at least one independent <f>t</f>-set of vertices and the open neighborhood of any such set contains exactly <f>r</f> vertices. Our goal is to show that when <f>t&ges;3</f> and the order is sufficiently large, then the structure of <f>(t,r)</f>-regular graphs is similar to, but not exactly the same as the structure of <f>(2,r)</f>-regular graphs as derived by Faudree and Knisley. That is, there is an “almost” complete kernel of order at most <f>r</f> surrounded by satellite cliques, all of the same order, which are “mostly” joined to the kernel. [Copyright &y& Elsevier]