Extremal Mostar Index of Graphs with Given Number of Cut Edges

TL;DR

This paper determines the maximum and minimum Mostar index values for connected graphs with a fixed number of cut edges, identifying extremal graphs and extending to graphs with specified cyclomatic numbers.

Contribution

It provides exact extremal values and characterizations for the Mostar index in graphs with given cut edges and extends results to graphs with fixed cyclomatic number.

Findings

Maximum Mostar index is k(n-2)+(n-k-1)k for the extremal graph K_{n-k}^k.

Unique extremal graph for maximum is K_{n-k}^k.

Sharp lower bounds and characterizations for minimum Mostar index are established.

Abstract

The Mostar index of a connected graph \(G\) is defined as \[ Mo(G)=\sum_{uv\in E(G)}\bigl|n_u(uv)-n_v(uv)\bigr|, \] 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.