Graphs of fixed order and given vertex connectivity with maximum bond incident degree indices

Let G be a graph and denote its vertex set and edge set by V ( G ) and E ( G ) , respectively. For a vertex v i ∈ V ( G ) , let d i denote its degree. A broad class of numerical parameters for graphs is given by BID ϑ ( G ) = ∑ v i v j ∈ E ( G ) ϑ ( d i , d j ) , where the function ϑ is symmetric and it assigns a real value to each pair of degrees of adjacent vertices of G . Such graph parameters are known as bond incident degree (BID) indices. The family BID ϑ specializes to the general atom–bond connectivity index ABC α when ϑ ( d i , d j ) = ( ( d i + d j − 2 ) d i − 1 d j − 1 ) α , for any real parameter α ; in particular, the choice α = 1 2 yields the classical atom–bond connectivity index. In their work (Chen and Hao, 2018), Chen and Hao posed the problem of identifying those graphs from the class of all connected graphs of fixed order with prescribed edge or vertex connectivity that attain the maximum value of ABC α for any α with 0 < α ≤ 1 2 . The present article resolves the aforementioned problem by providing a general result for BID ϑ under some suitable conditions imposed on ϑ . These conditions are fulfilled not only by ABC α for 0 < α ≤ 1 2 , but also by many other existing particular BID indices, such as the reformulated first Zagreb index, the Sombor index and its reduced form, the Euler–Sombor index, the inverse sum indeg index, the Zagreb–Sombor index, the reciprocal sum-connectivity index, the reciprocal Randić index, and the elliptic Sombor index.