On the general sum-connectivity index of hypergraphs

Given a non-zero real number $\alpha$, the general sum-connectivity index ${\chi }_{\alpha }$ for graph $G$ is given by the sum ${\mathrm{\Sigma }}_{xy\in E(G)}(d_x+d_y{)}^{\alpha }$. Here, $d_x$ denotes the degree of a vertex $x$ in graph $G$, and $E(G)$ is the edge set of $G$. A hypertree is a connected hypergraph without cycles. In a $k$-uniform hypergraph, every hyperedge contains exactly $k$ vertices, and a hypergraph is termed linear if any two distinct hyperedges share at most one vertex. This study addresses analytical challenges inherent in hypergraphs by concentrating on key subclasses and introducing innovative perturbation methods to solve fundamental extremal problems within these frameworks. Through this approach, we aim to broaden the scope and utility of graph-theoretic techniques. Specifically, within the class of uniform linear hypergraphs, we characterize the extremal hypergraphs with respect to the ${\chi }_{\alpha }$ operation. Furthermore, we investigate extremal problems related to ${\chi }_{\alpha }$ in both general hypergraphs and bipartite hypergraphs, yielding new insights into their previously unexplored structural properties.