3-manifold spine cyclic presentations with seldom seen Whitehead graphs

We consider a family of cyclic presentations and showthat, subject to certain conditions on the defining parameters, they are spines of closed 3-manifolds. These are new examples where the reduced Whitehead graphs are of the same type as those of the Fractional Fibonacci presentations; here the corresponding manifolds are often (but not always) hyperbolic.We also express a lens space construction in terms of a class of positive cyclic presentations that are spines of closed 3-manifolds. These presentations then furnish exampleswhere theWhitehead graphs are of the same type as those of the positive cyclic presentations of type Z, as considered byMcDermott.