| Summary: | In the study of eigenvalues, multiplicities, and graphs, the minimum number of multiplicities equal to 1 in a real symmetric matrix with graph G, U(G), is an important constraint on the possible multiplicity lists among matrices in 𝒮(G). Of course, the structure of G must determine U(G), but, even for trees, this linkage has proven elusive. If T is a tree, U(T) is at least 2, but may be much greater. For linear trees, recent work has improved our understanding. Here, we consider nonlinear trees, segregated by diameter. This leads to a new combinatorial construct called a core, for which we are able to calculate U(T). We suspect this bounds U(T) for all nonlinear trees with the given core. In the process, we develop considerable combinatorial information about cores.
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