It is a well-known fact that a graph of diameter d has at least d+1 eigenvalues. A graph is d-extremal (resp. dα-extremal) if it has diameter d and exactly d+1 distinct eigenvalues (resp. α-eigenvalues), and a graph is split if its vertex set can be partitioned into a clique and a stable set. Such graphs have a diameter of at most three. If all vertex degrees in a split graph are either d˜ or d^, then we say it is (d˜,d^)-bidegreed. In this paper, we present a complete classification of the connected bidegreed 3α-extremal split graphs using the association of split graphs with combinatorial designs. This result is a natural generalization of Theorem 4.6 proved by Goldberg et al. and Proposition 3.8 proved by Song et al., respectively.
In 2010, Vukičević introduced an new graph invariant, the inverse sum indeg index of a graph, which has been studied due to its wide range of applications. Let Bnd be the class of bipartite graphs of order n and diameter d. In this paper, we mainly characterize the bipartite graphs in Bnd with the maximal inverse sum indeg index. Bipartite graphs with the largest, second-largest, and smallest inverse sum indeg indexes are also completely characterized.
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