Abstract:

Let $G$ be a connected graph. Gao et al. first introduced the invariant of $G$: $\Delta M(G)=M_{o}(G)-irr(G)$, and raised a problem: how to determine the extremal graphs among all connected graphs of order $n$ with respect to $\Delta M(G)$, where $M_{o}(G)$ and $irr(G)$ stand for the Mostar index and irregularity of $G$, respectively. In this paper, we characterize the upper bounds of $\Delta M(G)$ over all unicyclic graphs and bicyclic graphs with diameter 3, and determine the extremal graphs.

Key words: Mostar index, Irregular, Unicyclic graph, Bicyclic graph, Diameter