A Complete Characterization of Bidegreed Split Graphs with Four Distinct <i>α</i>-Eigenvalues

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

A Complete Characterization of Bidegreed Split Graphs with Four Distinct &alpha;-Eigenvalues

It is a well-known fact that a graph of diameter d has at least d+1 eigenvalues. A graph is d-extremal (resp. d&alpha;-extremal) if it has diameter d and exactly d+1 distinct eigenvalues (resp. &alpha;-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&tilde; or d^, then we say it is (d&tilde;,d^)-bidegreed. In this paper, we present a complete classification of the connected bidegreed 3&alpha;-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

A Complete Characterization of Bipartite Graphs with Given Diameter in Terms of the Inverse Sum Indeg Index

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