The maximum Balaban index (sum-Balaban index) of unicyclic graphs. (English) Zbl 1313.05190
Summary: The Balaban index of a connected graph \(G\) is defined as
\[
J(G)=\frac{|E(G)|}{\mu +1}\sum_{e=uv\in E(G)}\frac{1}{{^{\sqrt{D_{G}(u)D_{G}(v)}}}},
\]
and the sum-Balaban index is defined as
\[
SJ(G)=\frac{|E(G)|}{\mu+1}\sum_{e=uv\in E(G)}\frac{1}{{^{\sqrt{D_{G}(u) D_{G}(v)}}}},
\]
where \(D_{G}(u)=\sum_{w\in V(G)}d_{G}(u, w), \) and \(\mu\) is the cyclomatic number of \(G\). In this paper, the unicyclic graphs with the maximum Balaban index and the maximum sum-Balaban index among all unicyclic graphs on \(n\) vertices are characterized respectively.
MSC:
05C35 | Extremal problems in graph theory |
05C50 | Graphs and linear algebra (matrices, eigenvalues, etc.) |
05C12 | Distance in graphs |
05C40 | Connectivity |
05C38 | Paths and cycles |