On split graphs with four distinct eigenvalues
Abstract
It is a well-known fact that a graph of diameter d has at least d+1 eigenvalues. Let us call a graph d-extremal if it has diameter d and exactly d+1 eigenvalues. Such graphs have been intensively studied by various authors. %Much attention has been devoted to the study of graphs that are extremal with respect to this relation: i.e have diameter d and exactly d+1 distinct eigenvalues. A graph is split if its vertex set can be partitioned into a clique and a stable set. Such a graph has diameter at most 3. We obtain a complete classification of the connected bidegreed 3-extremal split graphs. We also show how to construct certain families of non-bidegreed 3-extremal split graphs.
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