Translate 
Abstract
Let G be a simple connected graph with edge set E(G) and vertex set V(G). The weighted Mostar index of a graph G is defined as w+Mo(G)=∑e=uv∈E(G)(dG(u)+dG(v))|nu(e)−nv(e)|, where nu(e) denotes the number of vertices closer to u than to v for an edge uv in G. In this paper, we obtain the upper bound and lower bound of the weighted Mostar index among all bicyclic graphs and characterize the corresponding extremal graphs.
Highlights
The weighted Mostar index [14] of a graph G is defined as w+ Mo(G) = ∑ (dG(u) + dG(v))|nu(e) − nv(e)|
We extend this method to the weighted Mostar index and improve this method
B1 is the bicyclic graph with the maximum weighted Mostar index in Bn1, and it is clear that w+ Mo(B1) = 4(n + 1)(n − 3) + n(n − 2)(n − 5) = n3 − 3n2 + 2n − 12
Summary
All graphs considered in this paper are finite, undirected, connected and simple. We refer the readers to [1] for the terminology and notations. A connected graph is called a unicyclic graph if the number of edges equals the number of vertices. A connected graph is called a bicyclic graph if the number of edges is 1 more than the number of vertices. Asmat [16] obtained the upper bound of the weighted Mostar index for trees with a given diameter and the corresponding extremal graph. We obtain the upper bound and lower bound of the weighted Mostar index among all bicyclic graphs and characterize the corresponding extremal graphs. The proof mainly refers to the proof of the lower bound of Mostar index for bicyclic graphs in [12]. The method of proving the upper bound of Mostar index for bicyclic graphs by defining the deficit of edges in [12] is not applicable here. We use a method different from that in [12] to find the upper bound of weighted Mostar index among all bicyclic graphs
Sign-in/Register to view highlights & summary 
Published Version (
Free)
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
