Bond incident degree indices of stepwise irregular graphs

The bond incident degree (BID) index of a graph $ G $ is defined as $ BID_{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({\rm{mod}}\;4) $, and we give upper bounds on $ BID_{f} $ index in terms of the order $ n $ and the maximum degree $ \Delta $. Moreover, we completely characterize the extremal stepwise irregular graphs of order $ n $ with respect to $ BID_{f} $.