On degree-based graph invariants of fixed-order unicyclic graphs with prescribed maximum degree

Consider a graph $G$ having edge set $E$, and denote by $d_x$ the degree of a vertex $x$ in $G$. A unicyclic graph is defined as a connected graph containing exactly one cycle. This work focuses on unicyclic graphs of a fixed order and examines the graph invariants of such graphs of the form $BI{D}_{\varphi }(G)=\sum _{yz\in E}\varphi (d_y,d_z)$, where $\varphi$ is a symmetric and real-valued function. Such graph invariants are known as BID (bond incident degree) indices. The main objective is to determine the graphs that either minimize or maximize the quantity $BI{D}_{\varphi }$ among fixed-order unicyclic graphs with prescribed maximum degree, under specific assumptions on the function $\varphi$. These assumptions are satisfied by several classical and modern degree-based graph invariants. Generally, the results obtained are applicable to a wide range of such invariants. In particular, one of the obtained results covers the harmonic and sum-connectivity indices, while another applies to the recently proposed Sombor and Euler–Sombor indices as well as their reduced versions.