Vertex Neighbor Sum Rule

Let g be a smaller graph whose vertices have labels a, b, c... If vertices of a larger graph G satisfy the neighbor sums of a smaller graph g, or linear combinations of the same sum rules, then necessarily the eigenvalues of G include the eigenvalues of g. 

What Randic, Guo, and Kleiner noticed is that if two graphs g and h are bipartite and isospectral, so will be graphs G and H obtained by the ring enlarganent procedure. From this follows  

Conjecture If graph g yields non-isomorphic bipartite enlargements Gj, Gj, etc., the derived 

As was noticed, at that time, some isospectral graphs g and h generate isospectral pairs G, H via the ring enlargement procedure, and some do not. Why The answer is not known, so we have an open problem  

Under what conditions will a pair of isospectral graphs g, h upon enlargement produce isospectral pairs G and H  

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