A Criterion for the t-Vertex Condition of Graphs

Combining two concepts of regularity of graphs, namely k-isoregularity and the t-vertex condition, a generalization of a classical result by Hestenes and Higman is presented. As an application it is shown that two infinite series of graphs constructed by Brouwer, Ivanov, and Klin which are not rank 3 satisfy the 5-vertex condition.