Extremal Mostar Index of Graphs with Given Number of Cut Edges

The Mostar index of a connected graph $G$ is defined as $$Mo(G)=\sum _{uv\in E(G)}\mid n_u(uv)-n_v(uv)\mid ,$$ where for an edge $e=uv$, $n_u(e)$ denotes the number of vertices of $G$ that are closer to $u$ than to $v$. In this paper, we determine the maximum possible Mostar index among all connected graphs of order $n$ with exactly $k$ cut edges, where $1\le k\le n-1$. We prove that the maximum value is given by $k(n-2)+(n-k-1)k$, and the unique extremal graph is $K_{n-k}^k$ (a complete graph on $n-k$ vertices with $k$ pendant edges attached to a single vertex). We also establish a sharp lower bound and characterise the extremal graphs for the minimum value. Furthermore, we extend the results to graphs with a given cyclomatic number and a given number of cut edges. Our findings complete the extremal characterisation of the Mostar index for this fundamental graph class.