Abstract
For a connected graph \( G \) with \( n \) vertices, the Lanzhou index of \( G \) is defined as \[ Lz(G)=\sum_{v\in V(G)}d(v)^2\,[n-1-d(v)], \] where \( d(v)\) is the degree of vertex \( v \) in \( G\). The extremal graphs with minimum (respectively,
maximum) Lanzhou index has been determined for trees, unicyclic graphs, bicyclic graphs and tricyclic graphs with \( n\) vertices, respectively. In this paper, by applying the majorization method, we determine the unique extremal graph
with minimum Lanzhou index for \( c\)-cyclic graph for \( n\ge 3c+4 \) vertices and \( c\ge 1\). Besides, we determine the unique extremal graph with maximum Lanzhou index in the class of \( c\)-cyclic graph with \( n\) vertices for \(
3\le c\le \frac{n}{13}\), and we also illustrate an example to show that the bound \( \frac{n}{13}\) is the best possible. This extends the corresponding results of [4,9–11,13].
