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.

The σ -irregularity, a variant of the well-established Albertson irregularity, is a topological invariant defined for a graph G = ( V , E ) as σ ( G ) = ∑ u v ∈ E ( d ( u ) − d ( v ) ) 2 , where d ( u ) and d ( v ) denote the degrees of vertices u and v , respectively. Recent research has successfully characterized chemical trees with the maximum σ -irregularity. In this paper, we expand upon this research by establishing several structural properties of maximal trees with prescribed maximum degree Δ . Application of these properties enables us to characterize maximal trees with Δ = 5 . We establish that extremal trees contain only vertices of degrees 1 , 2 and Δ . Moreover, the number of edges with both end-vertices having the degree 2 or Δ is very small, so almost all edges have the (second) maximum possible contribution to σ -irregularity. We believe this property or similar should extend to maximal trees for any value of Δ , so this is an interesting direction for further research.

Principal component analysis (PCA) is commonly used to compute a bounding box of a point set in Rd. The popularity of this heuristic lies in its speed, easy implementation and in the fact that usually, PCA bounding boxes quite well approximate the minimum-volume bounding boxes.Since there are examples of discrete points sets in the plane, showing that the worst case ratio of the volume ofthe PCA bounding box and the volume of the minimum-volume bounding box tends to infinity,we consider PCA bounding boxes for continuous sets, especially for the convex hull of a point set. Here, we contributenew upper bounds on the approximation factor of PCA bounding boxesof convex sets in R2 and R3.