Graphs of fixed order and given vertex connectivity with maximum bond incident degree indices
Year: 2026
Type: article
Abstract: Let $G$ be a graph and denote its edge set by $E(G)$. For a vertex $v_i\in V(G)$, let $d_i$ denote its degree. A broad class of numerical parameters for graphs is given by $\mathcal{BID}_\vartheta(G)=\sum_{v_i v_j\in E(G)} \vartheta(d_i,d_j),$where the function $\vartheta$ assigns a real value to each pair of degrees of endvertices of edges of $G$. Such graph parameters are known as bond incident degree (BID) indices. The family $\mathcal{BID}_\vartheta$ specializes to the generalized atom--bond connectivity index $\mathcal{ABC}_\alpha$ when $\vartheta(d_i,d_j)=\bigl((d_i+d_j-2)^{-1} d_i d_j \bigr)^{\alpha},$ for any real parameter $\alpha$; in particular, the choice $\alpha=\tfrac12$ yields the classical atom-bond connectivity index. In their work [Discrete Applied Mathematics 243 (2018) 115--124], Chen and Hao posed the problem of identifying the fixed-order connected graphs with prescribed edge or vertex connectivity that attain the maximum value of $\mathcal{ABC}_\alpha$ for $0<\alpha\le \tfrac12$. The present article resolves the aforementioned problem by providing a general result for $\mathcal{BID}_\vartheta$ under some suitable conditions imposed on $\vartheta$. These conditions are fulfilled not only by $\mathcal{ABC}_\alpha$ for $0<\alpha\le \tfrac12$, 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 Randi\'c index, and the elliptic Sombor index.
Source: Discrete Applied Mathematics
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,090
Funder: University of Hail
OpenAlex ID: W7166587587