Degree-based graphical indices of $k$-cyclic graphs

Let $G$ be a graph with edge set $E(G)$. Let $d_x$ denote the degree of a vertex $x$ in $G$. For a nonnegative integer $k$, a connected graph of order $n$ and size $n+k-1$ is called a $k$-cyclic graph. This paper is concerned with $k$-cyclic graphs and their graphical indices of the form $BI{D}_f(G)=\sum _{uv\in E(G)}f(d_u,d_v)$, where $f$ is a symmetric function whose outputs are real numbers. Particularly, the graphs minimizing or maximizing $BI{D}_f$ among all $k$-cyclic graphs with a given order are studied under certain constraints on $f$. Various existing indices meet these constraints, and hence the obtained results hold for those indices; more precisely, one of the obtained results covers the recently developed elliptic Sombor and Zagreb-Sombor indices, while another result covers the recently introduced Euler-Sombor index.