Extremal graphs with maximum complementary second Zagreb index
Hui Gao
TL;DR
The paper addresses the problem of identifying graphs that maximize the complementary second Zagreb index $cM_{2}(G)$. It introduces a degree-oriented mixed-graph framework with edge orientation rules, partitions the vertex set into $X$ and $Y$, and defines two transformative operations that do not decrease $cM_{2}$, enabling a structural crackdown on extremal graphs. The main result proves that the extremal graphs are exactly the join $K_{m} \vee \overline{K_{n-m}}$ for some $1 \le m \le n-1$, thereby confirming the FO-25 conjecture. This advances the theory of complementary degree-based topological indices and provides a precise extremal classification with potential implications for chemical graph theory and related applications.
Abstract
Recently, a couple of degree-based topological indices, defined using a geometrical point of view of a graph edge, have attracted significant attention and being extensively investigated. Furtula and Oz [Complementary Topological Indices, \textit{MATCH Commun. Math. Comput. Chem.\/} \textbf{93} (2025) 247--263] introduced a novel approach for devising ``geometrical'' topological indices and focused special attention on the complementary second Zagreb index as a representation of the introduced approach. In the same paper, they also conjectured the maximal graphs of order $n$ with the maximum complementary second Zagreb index. In this paper, we confirm their conjecture.