Bond incident degree indices of stepwise irregular graphs

The bond incident degree (BID) index of a graph $G$ is defined as $BI{D}_f(G)=\sum _{uv\in E(G)}f(d(u),d(v))$, where $d(u)$ is the degree of a vertex $u$ and $f$ is a non-negative real valued symmetric function of two variables. A graph is stepwise irregular if the degrees of any two of its adjacent vertices differ by exactly one. In this paper, we give a sharp upper bound on the maximum degree of stepwise irregular graphs of order $n$ when $n\equiv 2(\mathrm{m}\mathrm{o}\mathrm{d}4)$, and we give upper bounds on $BI{D}_f$ index in terms of the order $n$ and the maximum degree $\mathrm{\Delta }$. Moreover, we completely characterize the extremal stepwise irregular graphs of order $n$ with respect to $BI{D}_f$.