Czechoslovak Mathematical Journal, first online, pp. 1-18
Extremal values with respect to vertex-degree-based topological indices
in unicyclic and bicyclic graphs
Ke ZHAO, Xuan ZHAO
Received February 27, 2026. Published online October 6, 2026.
Abstract: Let graph $G$ be an ordered pair $(V(G),E(G))$ consisting of a set $V(G)$ of vertices and a set $E(G)$ of edges. The formal definition of a vertex-degree-based topological index (VDB topological index) of $G$ is $TI_f(G)=\sum_{v_iv_j\in E(G)}f(d_{v_i},d_{v_j})$, where $f(x,y)> 0$ is a symmetric real function with $x \ge1$ and $y\ge1$, and $d_{v_i}$ denotes the degree of vertex $v_i$ in $G$. We present several conditions on the function $f(x,y)$ and prove that if a VDB topological index satisfies these conditions, then its extremal graphs are almost regular. Based on this conclusion, we derive the maximum VDB indices of unicyclic and bicyclic graphs, and characterize the corresponding extremal graphs. As an application, we verify that there are eight VDB topological indices that satisfy these conditions.
Keywords: VDB index; unicyclic graph; bicyclic graph; extremal value
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Affiliations: Ke Zhao, Xuan Zhao (corresponding author), School of Mathematics and Statistics, Ningxia University, 217 Wencui North Street, Yinchuan, P. R. China 750021, e-mail: Zk13561185151@163.com, xuanzhao@nxu.edu.cn
