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, 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, for G-designs which are degree-balanced but not orbit-balanced.

Balance, partial balance and balanced-type spectra in graph-designs / Bonisoli, Arrigo; Ruini, Beatrice. - In: JCMCC. JOURNAL OF COMBINATORIAL MATHEMATICS AND COMBINATORIAL COMPUTING. - ISSN 0835-3026. - 2015:93(2015), pp. 3-22.

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