Spectral radius and Hamiltonicity in generalized split graphs

A graph G ˆ is a split graph if its vertex set can be partitioned into a clique S and an independent set I , and the cross edges join a vertex in S to a vertex in I . A generalized split graph G is obtained from the split graph G ˆ by replacing each vertex v of I with a clique S v of size k ⩾ 2 , in which we add all edges between u and S v if u v ∈ E ( G ˆ ) . For convenience, denote the generalized split graph by G = ( S , ∪ v ∈ I S v ) . In this paper, we provide a sufficient spectral radius condition for the existence of a Hamiltonian cycle in a connected generalized split graph with | S | ⩾ max ⁡ { ( k + 3 ) | I | + 3 , 4 k 2 + 4 k − 2 } . Moreover, we characterize the extremal graph with the maximum spectral radius among all connected generalized split graphs G = ( S , ∪ v ∈ I S v ) with e ( S , ∪ v ∈ I S v ) = k ( | I | + q ) edges between S and ∪ v ∈ I S v , where q ∈ { 0 , 1 } ∪ [ k 2 + 7 k − 1 , | S | − 1 ] . These results extend the main results obtained by Zhu, Fan, and Lin (2025) [43] .