Balance, Partial Balance and Balanced-type Spectra in Graph-Designs

For a given graph \( G \), the set of positive integers \( v \) for which a \( G \)-design exists is usually called the spectrum for \( G \) and the determination of the spectrum is sometimes called the spectrum problem. We consider the spectrum problem for \( G \)-designs satisfying additional conditions of balance, in the case where \( G \) is a member of one of the following infinite families of trees: caterpillars, stars, comets, lobsters, and trees of diameter at most \( 5 \). We determine the existence spectrum for balanced \( G \)-designs, degree-balanced and partially degree-balanced \( G \)-designs, and orbit-balanced \( G \)-designs. We also address the existence question for non-balanced \( G \)-designs, for \( G \)-designs which are either balanced or partially degree-balanced but not degree-balanced, and for \( G \)-designs which are degree-balanced but not orbit-balanced.