The well-studied eccentric connectivity index directly consider the contribution of all edges in a graph. By considering the total eccentricity sum of all non-adjacent vertex, Hua et al. proposed a new topological index, namely, eccentric connectivity coindex of a connected graph. The eccentric connectivity coindex of a connected graph $G$ is defined as

$${\overline{\xi }}^c(G)=\sum _{uv\notin E(G)}({\epsilon }_G(u)+{\epsilon }_G(v)).$$

Where ${\epsilon }_G(u)$ (resp. ${\epsilon }_G(v)$) is the eccentricity of the vertex $u$ (resp. $v$). In this paper, some extremal problems on the ${\overline{\xi }}^c$ of graphs with given parameters are considered. We present the sharp lower bounds on ${\overline{\xi }}^c$ for general connecteds graphs. We determine the smallest eccentric connectivity coindex of cacti of given order and cycles. Also, we characterize the graph with minimum and maximum eccentric connectivity coindex among all the trees with given order and diameter. Additionally, we determine the smallest eccentric connectivity coindex of unicyclic graphs with given order and diameter and the corresponding extremal graph is characterized as well.

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.